Perturbation theory for parabolic differential equations

Habilitationsschrift

vorgelegt

der Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden

von

Dr. rer. nat. Hendrik Vogt

geboren am 8. 6. 1971 in Oldenburg (Oldb)

Eingereicht am 9. 6. 2010

Tag der Probevorlesung: 10. 2. 2011

Die Habilitationsschrift wurde in der Zeit von Dezember 2008 bis Juni 2010 am Institut für Analysis angefertigt.

Contents

Introduction 3

1 Perturbation theory for propagators 9 1.1 Strongly measurable propagators . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Miyadera perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Closed Miyadera perturbations . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4 The Trotter product formula . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Absorption propagators 29 2.1 Abstract theory of absorption propagators . . . . . . . . . . . . . . . . . . 30 2.2 Miyadera class for potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3 The backward Miyadera condition . . . . . . . . . . . . . . . . . . . . . . . 42 2.4 Logarithmically convex functions . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 Interpolation inequalities for absorption propagators . . . . . . . . . . . . . 49 2.6 Consistent propagators on weighted L_{p}-spaces . . . . . . . . . . . . . . . . 56

3 Kernel bounds for propagators 58 3.1 Kernel bounds for ultracontractive propagators . . . . . . . . . . . . . . . 58 3.2 Stability of the boundary behaviour . . . . . . . . . . . . . . . . . . . . . . 64 3.3 Kernel estimates on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 74

Bibliography 83

Introduction

The validity of Gaussian bounds for fundamental solutions of second-order parabolic equations in divergence form with non-smooth coefficients goes back to Aronson. In [Aro67], Aronson studied the parabolic equation

∂_{t}u = ∇ · (a(t, x)∇u)  (0.1)

with a uniformly parabolic measurable coefficient a: [0, ∞)×ℝ^{n} → ℝ^{n×n}, i.e., a is bounded and there exists ε > 0 such that ∑_{j,k=1}^{n} a_{jk}(t, x)ξ_{j}ξ_{k} ⩾ ε|ξ|^{2} for all (t, x) ∈ [0, ∞) × ℝ^{n} and all ξ ∈ ℝ^{n}. He proved that there are constants C_{1}, C_{2}, c_{1}, c_{2} > 0 such that the fundamental solution g of (0.1) satisfies the two-sided Gaussian bounds

C_{1}(t − s)^{−n/2}e^{−c_{1} \frac{|x−y|^{2}}{t−s}} ⩽ g(t, x; s, y) ⩽ C_{2}(t − s)^{−n/2}e^{−c_{2} \frac{|x−y|^{2}}{t−s}}  (0.2)

for all (t, x), (s, y) ∈ [0, ∞) × ℝ^{n} with t > s; see [Aro67; Rem. 5] for an explanation why these estimates are global in time, i.e., why no restriction t − s ⩽ T is needed.

In [Aro68], Aronson studied a more general parabolic equation that includes lower order terms with measurable coefficients, in particular, a multiplication term (potential) was allowed that was supposed to satisfy a certain integrability property. Starting from the publication of [Aro68] the topic has a rich history, and numerous extensions and generalisations have been obtained. For results in the Euclidean space setting see, e.g., [EiPo84,FaSt86,Dav87a,Dav89,Str92,Sem99,LiSe00,Dan00]; further generalisations will be discussed below.

For the case of Schrödinger operators −Δ + V, in [AiSi82] the Kato class of potentials was shown to be the appropriate class for the L_{1}-perturbation theory of the corresponding C_{0}-semigroups, and in [Sim82] it was shown that the fundamental solution of the perturbed heat equation still satisfies upper and lower Gaussian estimates. It was much later, however, that parabolic equations with more general time-dependent potentials were studied and the non-autonomous Kato class as the proper extension of the Kato class was introduced; cf. [Zha96,Zha97,ScVo99,Gul02,Gul04].

Essentially following [Zha96] we say that a potential V ∈ L_{1,loc}([0, ∞) × ℝ^{n}) is in the non-autonomous Kato class NK if

N^{+}(V) := lim_{α→0} N_{α}^{+}(V) = 0 and N^{−}(V) := lim_{α→0} N_{α}^{−}(V) = 0,

where

N_{α}^{±}(V) = sup_{x,s} ∫_{0}^{α} ∫_{ℝ^{n}} k_{t}(x − y)|V (s ± t, y)| dy dt (0 < α ⩽ ∞),

with the free heat kernel k_{t}(x) = (4πt)^{−n/2}e^{− \frac{|x|^{2}}{4t}}, and V (τ, ·) := 0 for τ < 0. We say that V is in the enlarged non-autonomous Kato class N̂K if N^{+}(V) < ∞ and N^{−}(V) < ∞. (Observe that this differs from [ScVo99], where only N^{+}(V) < ∞ was required.) Note that N^{+}(V) = N^{−}(V) in the case of time-independent V. In fact, NK just reduces to the Kato class in this case, and N̂K to the extended Kato class (cf. [ScVo99]).

In [Zha97], Zhang studied the fundamental solution of the parabolic equation

∂_{t}u = ∇ · (a(t, x)∇u) − V (t, x)u.  (0.3)

He proved that if a is uniformly parabolic and V_{λ} := V (·/λ, ·) ∈ NK for a suitable λ > 0, then the fundamental solution satisfies the two-sided Gaussian bounds (0.2) for s < t ⩽ s + 1. (The framework of [Zha97] is actually the more general one of parabolic equations with uniformly subelliptic principal part, see below.) Of course the above bounds can be extended to all t > s by means of the reproducing kernel property. However, this leads to an additional factor e^{ωt} in the upper bound, and e^{−ωt} in the lower bound; in general the bounds are not global in time.

As is observed in [ScVo99], for the well-posedness of the Cauchy problem for (0.3) in L_{1}(ℝ^{n}) only the condition N^{+}(V_{λ}) < λ/M is needed, where λ, M are such that the fundamental solution g_{0} corresponding to V = 0 satisfies g_{0}(t, x; s, y) ⩽ Mk_{λ(t−s)}(x − y). This is derived by means of the non-autonomous Miyadera perturbation theorem [RRSV00; Thm. 3.4]; in fact, V satisfies the required Miyadera smallness condition under the above assumption. However, a condition controlling N^{+}(V_{λ}) (a forward Kato condition) is not sufficient for obtaining Gaussian bounds: Zhang’s result also requires a backward Kato condition, i.e., a condition on N^{−}(V_{λ}). The latter condition is responsible for the wellposedness of the adjoint Cauchy problem

−∂_{t}u = ∇ · (a^{⊤}(t, x)∇u) − V (t, x)u, u(T, ·) = f

in L_{1}(ℝ^{n}), where T > 0 (cf. Section 2.3).

It was noted in [Zha97] that the Gaussian bounds (0.2) for the fundamental solution of (0.3) remain valid for s < t ⩽ s + 1 if V_{λ} ∈ N̂K and the quantities N^{±}(V_{λ}) are sufficiently small, for a suitable λ > 0. The results of this thesis imply that the upper bound holds if the negative part V^{−} of V satisfies N^{±}(V_{λ}^{−}) < λ/M (with λ, M as in the previous paragraph); cf. Theorem 3.10. It follows from the general approach of absorption propagators that we use that no restrictions on the positive part V^{+} of V are needed. The lower bound in (0.2) holds under the assumption N^{±}(V_{λ}^{+}) < ∞ and the condition that the Cauchy problem for (0.3) is well-posed in L_{1}(ℝ^{n}); cf. Theorem 3.12. In order to obtain global bounds (0.2), we require the same conditions as above but with N_{∞}^{±} in place of N^{±}. For the upper bound this essentially is proved in [Zha97]. The global lower bound under these conditions is due to [LVV06; Rem. 3.13].

The aim of this thesis is to develop a perturbation method that enables one to derive the above results in the general framework of positive propagators on L_{p}-spaces, thus allowing for a much wider range of applications. A (linear) propagator, or evolution family, describes the time evolution of a system with time-dependent driving force. The perturbation by a time-dependent potential that we are going to study can be thought of as an operator addition to the local infinitesimal generators of the propagator. Since the latter do not exist, in general, the perturbed propagators are constructed by requiring the validity of a Duhamel formula. We refer to the introduction of [RRSV00] for a more extensive discussion.

This approach using the Duhamel formula and the resulting Dyson-Phillips series works rather directly in the case of bounded perturbations (not necessarily by potentials); see [LVV06; Sec. 1.2]. (In [RäSc99; Sec. 2], bounded perturbations are studied by means of evolution semigroups.) In Chapter 1 we show that the approach can also be implemented in the more general case of Miyadera perturbations; thus we entirely avoid the use of evolution semigroups.

For the special case of positive propagators on L_{p}-spaces and perturbation by potentials, one can still go further by approximating a general potential by bounded potentials and using monotonicity. This approach, known as the theory of absorption semigroups and absorption propagators, is well-established for C_{0}-semigroups (cf. [Voi86], [Voi88], [ArBa93], [Man01]). In the non-autonomous case, perturbations by positive potentials have been studied in [RäSc99], and only in [LVV06] the study of sign-changing potentials was initiated.

In Chapter 2 we further develop the theory of absorption propagators. One of the new aspects is that we enlarge the class of admissible perturbations, thus allowing perturbation by highly oscillating potentials. Moreover, we study strongly measurable propagators (as opposed to strongly continuous propagators). This makes the development of the theory more natural since the class of strongly measurable propagators is closed under strong convergence.

The main ingredient in our proof of stability of Gaussian bounds under perturbation by potentials is an interpolation inequality for absorption propagators (see Section 2.5) that is well-known in the case of Schrödinger semigroups, in which probabilistic tools can be used for the proof. We provide a purely analytical argument that is based on a Trotter product formula for strongly continuous propagators, provided in Section 1.4. Then in Section 3.1 we show how the interpolation inequality can be used to prove both upper and lower Gaussian type bounds, in the general context of positive ultracontractive propagators.

Above we have only discussed parabolic equations on the whole space ℝ^{n}. Suppose now that Ω is a bounded connected open subset of ℝ^{n}, let T ∈ (0, ∞], and consider the parabolic equation with homogeneous Dirichlet boundary condition

∂_{t}u = ∇ · (a(t, x)∇u),  u = 0 on (0, T) × ∂Ω,  (0.4)

where a: [0, T)×Ω → ℝ^{n×n} is measurable and uniformly parabolic. It follows from general domination principles that the fundamental solution of (0.4) satisfies the upper bound in (0.2). The corresponding lower bound, however, does not hold, and the upper bound does not reflect the boundary behaviour. Moreover, the long-time behaviour is different since Ω is bounded. The latter is the reason why in the following we assume that T < ∞ unless stated otherwise.

The autonomous case was studied in detail by Davies, based on intrinsic ultracontractivity estimates from [DaSi84]. Under the assumption that n ⩾ 3 and that a and ∂Ω are sufficiently smooth, it follows from [Dav87b; Thm. 3 and Thm. 9] that there exist C, c > 0 such that the fundamental solution g of (0.4) satisfies the Gaussian type upper bound

g(t, x; s, y) ⩽ C(t − s)^{−n/2}(\frac{d(x)d(y)}{t − s} ∧ 1) · exp(−c\frac{|x − y|^{2}}{t − s} )  (0.5)

for all (t, x), (s, y) ∈ [0, T) × Ω with t > s, where d(x) denotes the distance from x to the boundary of Ω. An analogous lower bound remained an open question; only in the simpler situation of the long-time behaviour, precise upper and lower bounds were provided.

The missing lower bound was first proved in [Zha02] for the case of the heat equation and C^{1,1}-boundary, again under the assumption n ⩾ 3. The latter assumption was removed in [Son04a]. It was noted in [Zha02; Rem. 1.3] that the lower bound also holds for space-dependent Hölder continuous coefficients. In [Son04b], global upper and lower bounds of the type (0.5) are proved for the case of the heat equation, assuming that n ⩾ 3 and that Ω is the subgraph of a bounded C^{1,1}-function Γ: ℝ^{n−1} → ℝ.

The case of time-dependent coefficients was first investigated by Riahi in [Ria01] for the half space and later in [Ria05] for bounded Ω with C^{1,1}-boundary. The most general result we know of is due to Cho. In [Cho06] he proves (0.5) and a matching lower bound under the assumption that Ω has C^{1,α}-boundary, for some α ∈ (0, 1), and that the coefficients are Dini continuous with respect to the parabolic distance |(t, x) − (s, y)| = |t − s|^{1/2} + |x − y|, i.e., |a_{jk}(X) − a_{jk}(Y)| ⩽ ω(|X − Y |) for all X, Y ∈ [0, T) × Ω and j, k = 1, …, n, for some increasing function ω: [0, ∞) → [0, ∞) satisfying ∫_{0}^{1} ω(t)/t dt < ∞. (Without some regularity of a and ∂Ω, the kernel estimate (0.5) does not hold in general.)

We point out that already in [Dav87b], the parabolic equation contained a (timeindependent) potential V satisfying a suitable condition on infinitesimal form smallness. In [Ria03] it is proved that (0.5) and also the corresponding lower bound are stable under perturbation by potentials from NK. This does not cover, however, the boundary singularities of the potentials considered in [Dav87b]. Only in [Ria07] a proper generalisation was found: It follows from [Ria07; Cor. 5.7] that the kernel bound (0.5) and a matching lower bound hold if the parabolic equation (0.4) is perturbed by a potential that is locally in NK and that satisfies, for some α > 0 and some small enough c, ε > 0, the estimate

sup_{x,s} ∫_{0}^{α} ∫_{ℝ^{n}} \frac{d(y)}{d(x)}γ_{t}^{c}(x, y)|V (s ± t, y)| dy dt ⩽ ε,

where γ_{t}^{c}(x, y) = t^{−n/2}(\frac{d(x)d(y)}{t} ∧ 1) exp(−c\frac{|x−y|^{2}}{t−s} ). We point out that Riahi’s result deals with the more general case of perturbation by measures, and that an additional drift term is included whose coefficient is a vector-valued measure.

Owing to the method of [Ria07], the constants c and ε in the above condition on V are not sharp. In Section 3.2 we prove stability of kernel estimates of the type (0.5) in a more general setting, under a condition on V that is sharp up to a factor of 2.

In Section 3.3, the concluding section of the thesis, we show that our methods can also be applied to stability of kernel estimates on complete Riemannian manifolds with Ricci curvature bounded below. The main breakthrough in this context was accomplished by Li and Yau, who in [LiYa86] proved global upper and lower bounds for the heat kernel in the case of non-negative Ricci curvature. It is shown in [Sal92] that these estimates remain valid if one replaces the heat equation with a uniformly parabolic equation with timedependent measurable coefficients. For further developments see [DaPa89, Dav93, Stu95, Gri99, Gri06]. In particular, in [Gri06; Sec. 10], Grigor’yan has obtained sharp estimates for the long time behaviour in the time-independent symmetric case.

Estimates analogous to the Li-Yau estimate also hold for uniformly subelliptic operators of Hörmander type; see [Sán84], [JeSá86] for compact manifolds and [KuSt87],

[KuSt88] for the setting of ℝ^{n}. In fact, it is the latter setting in which the stability results of [Zha97] are proved, not only the uniformly parabolic setting described in the beginning. In Section 3.3 we prove stability of kernel estimates in a more general framework that includes the above applications, under a miminal assumption on the perturbation.

The outline of the thesis is as follows. In Chapter 1 we investigate the general perturbation theory for strongly measurable propagators. The notion of strongly measurable propagators is introduced in Section 1.1, and some basic properties are proved. In Section 1.2 we prove the Miyadera perturbation theorem for strongly measurable propagators and related results; the special case of closed Miyadera perturbations is studied in Section 1.3 in more detail. In Section 1.4 we present a convergence result for bounded perturbations and derive the Trotter product formula for strongly continuous propagators.

Chapter 2 is devoted to the theory of absorption propagators, i.e., to the particular case of perturbation of positive propagators on L_{p}-spaces by real-valued potentials. The general theory is developed in Section 2.1, and in Section 2.2 the relation to the Miyadera perturbation theorem is established. In Section 2.3 we introduce a backward Miyadera condition that is responsible for local L_{∞}-boundedness of perturbed propagators. The interpolation inequalities for absorption propagators that are mentioned above are studied in Sections 2.5 and 2.6; the proofs are based on the notion of logarithmically convex operator-valued functions that is presented in Section 2.4.

In Chapter 3 we apply the abstract theory of Chapter 2 to our main subject, the stability of kernel estimates for strongly continuous propagators under perturbation by potentials. Section 3.1 deals with general kernel estimates in the setting of ultracontractive propagators. In Section 3.2 we prove refined results for more special Gaussian type estimates involving boundary terms. Finally, in Section 3.3 we show that our method can also be adapted to cover the case of heat kernel estimates on Riemannian manifolds.

Acknowledgements. I would like to thank Jürgen Voigt for his encouragement and for many stimulating discussions. Morover, I want to thank Vitali Liskevich, Markus Haase, Michiel van den Berg and Alexander Grigor’yan for valuable discussions at different occasions.

8

Chapter 1

Perturbation theory for strongly measurable propagators

Strongly continuous propagators are a standard device in the study of non-autonomous evolution equations. Here, more generally, we investigate strongly measurable propagators since they arise naturally as absorption propagators; cf. Section 2.1. Only at the end of the chapter, strong continuity will play a crucial role. This will have the consequence that in the kernel estimates in Chapter 3 we will have to start from an unperturbed propagator that is strongly continuous.

In Section 1.1 we give a short introduction to strongly measurable and strongly continuous propagators. In Sections 1.2 and 1.3 we prove the Miyadera perturbation theorem for strongly measurable propagators and related results concerning Duhamel formulas and Dyson-Phillips series. Section 1.4 deals with the special case of bounded perturbations; there we prove a Trotter product formula for strongly continuous propagators.

1.1 Strongly measurable propagators

Let J ⊆ ℝ be an interval, and let D_{J} := {(t, s) ∈ J × J; t ⩾ s}. For t ∈ ℝ we use the notation J_{⩾t} := {s ∈ J; s ⩾ t}, and analogously J_{>t}, J_{⩽t}, J_{<t}.

Let X be a (real or complex) Banach space. A propagator on X (with parameter interval J) is a function U: D_{J} → L(X) satisfying

U(t, t) = I, U(t, r) = U(t, s) U(s, r) (t ⩾ s ⩾ r in J).

With a slight abuse of language, we say that U is a strongly measurable propagator if additionally U is separately strongly measurable, i.e., U(·, s) is strongly measurable on J_{⩾s}, for all s ∈ J, and U(t, ·) is strongly measurable on J_{⩽t}, for all t ∈ J. This definition is explained by the fact that separate strong measurability of propagators implies joint strong measurability, but not vice versa if X is not separable, as shown in the following lemma and in Example 1.3 below.

1.1 Lemma. (a) Let U be a strongly measurable propagator on X. Then U is jointly strongly measurable.

(b) Let U be a propagator on X, and assume that U is jointly strongly measurable. Then U(t, ·) is strongly measurable on J_{⩽t}, for all t ∈ J. If, in addition, X is separable then U(·, s) is strongly measurable on J_{⩾s}, for all s ∈ J, so U is a strongly measurable propagator.

Proof. (a) Let τ ∈ J and R_{τ} := J_{⩾τ} × J_{⩽τ}. Then (t, s) ↦ U(t, s) = U(t, τ)U(τ, s) is strongly measurable on R_{τ}, by Lemma 1.2 below. The assertion follows since ⋃_{τ∈J∩ℚ} R_{τ} has full measure in D_{J}.

(b) Let t ∈ J and x ∈ X; we prove that U(t, ·)x is measurable on J_{⩽t}. In the case t = inf J there is nothing to show. If t > inf J then U(τ, ·)x is measurable on J_{⩽τ} for a.e. τ ∈ J_{<t}, so U(t, ·)x = U(t, τ)U(τ, ·)x is measurable on J_{⩽τ} for a.e. τ ∈ J_{<t} and hence measurable on J_{⩽t}.

Assume now that X is separable, and let X_{0} be a countable dense subset of X. Then there exists a null set N ⊆ J such that U(·, s)x is measurable on J_{⩾s} for all s ∈ J \ N and all x ∈ X_{0}. It follows that U(·, s) is strongly measurable on J_{⩾s} for all s ∈ J \ N, and with an argument analogous to the above we obtain the same for all s ∈ J, which completes the proof.

1.2 Lemma. Let f: J → X be measurable, and let B: J → L(X) be strongly measurable. Then t ↦ B(t)f(t) is strongly measurable. If, in addition, A: J → L(X) is strongly measurable then t ↦ B(t)A(t) is strongly measurable.

Proof. Let f_{n}: J → X be step functions such that f_{n} → f a.e. as n → ∞. Then t ↦ B(t)f_{n}(t) is measurable for each n ∈ ℕ and B(·)f_{n}(·) → B(·)f(·) a.e. as n → ∞, which implies the first assertion. The second assertion is immediate from the first one.

The following example shows that one cannot dispense with the separability assumption in Lemma 1.1(b).

1.3 Example. Let J := Ω := [0, 1], and let μ be the counting measure on Ω. We define U: D_{J} → L(L_{1}(μ)) by U(t, t) := I for all t ∈ J and

U(t, s)f := f(s)δ_{t} (0 ⩽ s < t ⩽ 1, f ∈ L_{1}(μ)),

where δ_{t} = 𝟙_{{t}} is the indicator function of {t}. One easily checks that U is a propagator on L_{1}(μ), and U is jointly strongly measurable since U(t, s)f = 0 for a.e. s ∈ J, for every f ∈ L_{1}(μ). However, the function t ↦ U(t, 0)δ_{0} = δ_{t} is not measurable, so U is not a strongly measurable propagator.

Unlike strong measurability, strong continuity of propagators is defined in the obvious way. A propagator U on X with parameter interval J is called a strongly continuous propagator if U: D_{J} → L(X) is strongly continuous. Clearly, a strongly continuous propagator is separately strongly continuous, but it follows from Example 1.5 below that the converse is not true. The next lemma shows that the converse is true for locally bounded propagators. (The remainder of the section is taken from [LVV06; Sec. 1].)

1.4 Lemma. ([Gul04; Thm. 2.2]) Let U be a propagator on X. Suppose that (i) U is locally bounded; (ii) for any s ∈ J the mapping J_{⩾s} ∋ t ↦ U(t, s) is strongly continuous; (iii) for any t ∈ J the mapping J_{⩽t} ∋ s ↦ U(t, s) is strongly continuous at s = t. Then U is strongly continuous.

Proof. Fix (t_{0}, s_{0}) ∈ D_{J}, and let J_{0} := J_{<s_{0}} (J_{0} := {s_{0}} in case s_{0} = inf J). Due to (iii) the set ⋃_{r∈J_{0}} U(s_{0}, r)X is dense in X. Thus, due to (i), it suffices to show that the mapping D_{J} ∋ (t, s) ↦ U(t, s)x_{0} is continuous at (t_{0}, s_{0}), for x_{0} := U(s_{0}, r)x with r ∈ J_{0} and x ∈ X. Let t_{n} → t_{0}, s_{n} → s_{0}, without loss of generality s_{n} ⩾ r for all n ∈ ℕ. Denote x_{n} := U(s_{n}, r)x. Condition (ii) implies that x_{n} → x_{0}. Using (i) and (ii) again we conclude that

U(t_{n}, s_{n})x_{0} = U(t_{n}, r)x + U(t_{n}, s_{n})(x_{0} − x_{n}) → U(t_{0}, r)x = U(t_{0}, s_{0})x_{0}.

The uniform boundedness theorem implies that strongly continuous propagators are locally bounded. The following example shows that separate strong continuity of a propagator is not sufficient to obtain local boundedness (as was claimed in [CuPr78; Def. 2.32]). In particular it shows that condition (i) of Lemma 1.4 is needed for the conclusion.

1.5 Example. We indicate an example on the Hilbert space ℓ_{2}.

Let ((a_{n}, b_{n}))_{n∈ℕ} be a sequence of pairwise disjoint subintervals of [0, 1], and for each n ∈ ℕ let a_{n} < s_{n} < t_{n} < b_{n}. For n ∈ ℕ let φ_{n}: [0, 1] → [\frac{1}{n}, 1] be a continuous function, φ_{n} = 1 on [0, a_{n}], φ_{n}(s_{n}) = \frac{1}{n}, φ_{n}(t_{n}) = 1 and φ_{n} = \frac{1}{n} on [b_{n}, 1]. We define U: D_{[0,1]} → L(ℓ_{2}) by

U(t, s)((x_{n})_{n}) := (\frac{φ_{n}(t)}{φ_{n}(s)}x_{n})_{n} ,

for 0 ⩽ s ⩽ t ⩽ 1 and (x_{n})_{n} ∈ ℓ_{2}.

Then U is a propagator on ℓ_{2} with parameter interval [0, 1]. For (t, s) ∈ D_{[0,1]} and n ∈ ℕ observe that \frac{φ_{n}(t)}{φ_{n}(s)} ⩽ 1 except when a_{n} < s < t < b_{n}. From the pairwise disjointness of the intervals (a_{n}, b_{n}) it thus follows that [0, t] ∋ s ↦ U(t, s) is strongly continuous for all t ∈ [0, 1], and that [s, 1] ∋ t ↦ U(t, s) is strongly continuous for all s ∈ [0, 1]. However, ||U(t_{n}, s_{n})|| = n for all n ∈ ℕ, so U is not bounded.

1.6 Remark. Note that, in the previous example, the unboundedness of U occurs close to the diagonal. In fact, as in the proof of Lemma 1.1(a) one obtains that a separately strongly continuous propagator U with parameter interval J is strongly continuous on J_{⩾τ} × J_{⩽τ}, for all τ ∈ J, and hence strongly continuous on {(t, s) ∈ D_{J}; t − s ⩾ δ}, for all δ > 0.

1.2 Miyadera perturbations of strongly measurable propagators

In this and the next section we investigate Miyadera perturbations of strongly measurable propagators. In [RRSV00] (see also [LVV06; Sec. 1.3]), the method of evolution semigroups is used to prove the Miyadera perturbation theorem for strongly continuous propagators. We do not assume strong continuity of the propagator, so the corresponding evolution semigroup need not be strongly continuous either. That is why in the following it is not suitable to make use of evolution semigroups.

Let X be a Banach space, and let U be a locally bounded strongly measurable propagator on X with parameter interval J ⊆ ℝ. A Miyadera perturbation of U (with constants (α, γ)) is a family (B(t))_{t∈J} of linear operators in X satisfying the following condition.

(M) For all s ∈ J, the set

X_{s} := {x ∈ X; B(·)U(·, s)x is a.e. defined and measurable on J_{⩾s}}

is a dense subspace of X, and there exist α ∈ (0, ∞] and γ ⩾ 0 such that

∫_{s}^{t} ||B(τ)U(τ, s)x|| dτ ⩽ γ||x|| ((t, s) ∈ D_{J}, t − s ⩽ α, x ∈ X_{s}).

We say that (B(t))_{t∈J} is a small Miyadera perturbation of U if α, γ can be chosen such that γ < 1. The case α = ∞ is included since it is important for the global behaviour of perturbed propagators; see, e.g., Remark 3.11(c) and Remark 3.24(a).

1.7 Remarks. (a) We say that U is exponentially bounded if there exist M ⩾ 1 and ω ∈ ℝ such that

||U(t, s)|| ⩽ Me^{ω(t−s)} ((t, s) ∈ D_{J}),

or equivalently, if sup{||U(t, s)||; (t, s) ∈ D_{J}, t−s ⩽ 1} < ∞. If U is exponentially bounded and B: J → L(X) is bounded and strongly measurable, then by Lemma 1.2 one easily sees that (B(t))_{t∈J} is an infinitesimally small Miyadera perturbation of U, i.e., the infimum of all possible γ in condition (M) is zero.

(b) Let (B(t))_{t∈J} be a Miyadera perturbation of U. Since U is a propagator, we obtain from the definition of the spaces X_{s} that U(t, s)X_{s} ⊆ X_{t} for all (t, s) ∈ D_{J}. Further we show that, for all (t, s) ∈ D_{J}, there exists γ_{t,s} ⩾ 0 such that

∫_{s}^{t} ||B(τ)U(τ, s)x|| dτ ⩽ γ_{t,s}||x|| (x ∈ X_{s}).  (1.1)

This is trivial in the case α = ∞, so assume that α < ∞. Choose n ∈ ℕ such that (n − 1)α ⩽ t − s < nα, and let t_{j} := s + jα for j = 0, …, n − 1 and t_{n} := t. Then

∫_{s}^{t} ||B(τ)U(τ, s)x|| dτ = ∑_{j=0}^{n−1} ∫_{t_{j}}^{t_{j+1}} ||B(τ)U(τ, t_{j})U(t_{j}, s)x|| dτ ⩽ ∑_{j=0}^{n−1} γ||U(t_{j}, s)x||

for all x ∈ X_{s}, so (1.1) holds with γ_{t,s} = γ ∑_{j=0}^{n−1} ||U(t_{j}, s)||.

(c) Miyadera perturbations of propagators have first been studied in [RRSV00] (see also [RRS96] for non-autonomous Miyadera perturbations of C_{0}-semigroups). We point out that in [RRSV00], strong continuity of U is assumed, and the authors work with a slightly different condition that at the first sight seems more general than condition (M) above. Namely, they assume the estimate in (M) only for x ∈ Y_{s}, where (Y_{s})_{s∈J} is a given family of dense subspaces of X satisfying Y_{s} ⊆ X_{s} and U(t, s)Y_{s} ⊆ Y_{t} for all (t, s) ∈ D_{J} ([RRSV00; p. 350], case p = 1). Obviously, this is an assumption on the parts B̃ B(t) := B(t)|_{D(B(t))∩Y_{t}} of B(t) only.

We now show that in this situation (B̃ B(t))_{t∈J} satisfies condition (M) with the same constants α, γ as in the above condition on (B(t))_{t∈J}. First observe that for the spaces X̃ X_{s} corresponding to (B̃ B(t))_{t∈J} we have Y_{s} ⊆ X̃ X_{s}, so that X̃ X_{s} is dense in X. Now let (t, s) ∈ D_{J} with t − s ⩽ α, and let x ∈ X̃ X_{s}. Then U(σ, s)x ∈ Y_{σ} for a.e. σ ∈ [s, t], by the definition of X̃ X_{s}. For every such σ we have

∫_{σ}^{t} ||B̃ B(τ)U(τ, σ)U(σ, s)x|| dτ ⩽ γ||U(σ, s)x||.

It follows that ∫_{s}^{t} ||B̃ B(τ)U(τ, s)x|| dτ ⩽ γ lim sup_{σ→s} ||U(σ, s)|| · ||x||, so

A_{t,s}x := B̃ B(·)U(·, s)x ∈ L_{1}(s, t; X)

defines a bounded operator from X̃ X_{s} to L_{1}(s, t; X). Since Y_{s} is dense in X, we conclude that the estimate ||A_{t,s}x||_{1} = ∫_{s}^{t} ||B̃ B(τ)U(τ, s)x|| dτ ⩽ γ||x||, which was assumed for all x ∈ Y_{s}, extends to all x ∈ X̃ X_{s}. (Note that the last part of the argument is not needed if U is strongly continuous, as assumed in [RRSV00].)

The following Miyadera perturbation theorem is the main result of this section. It generalises [LVV06; Thm. 1.16(a)], where the result is proved in the case of strongly continuous propagators. Also in that case, a slightly weaker version has already been shown in [RRSV00; Thm. 3.4(a), Cor. 3.5]. In these papers, the approach of evolution semigroups is used to derive the result from the Miyadera perturbation theorem for C_{0}- semigroups [Voi77; Thm. 1]; see also [Miy66; Thm. 2] for the orginal version of the theorem.

1.8 Theorem. Let (B(t))_{t∈J} be a small Miyadera perturbation of U. Then there exists a unique locally bounded strongly measurable propagator U_{B} on X satisfying the Duhamel formula

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U_{B}(t, τ)B(τ)U(τ, s)x dτ  ((t, s) ∈ D_{J}, x ∈ X_{s}), (1.2)

with X_{s} as defined in condition (M).

If U is strongly continuous then U_{B} is strongly continuous, too.

For the proof we will use the following lemma, which brings a new perspective on condition (M). For the convenience of a simplified notation, we extend U to a strongly measurable function from J × J to L(X) by setting U(t, s) := 0 for t < s.

1.9 Lemma. Let (B(t))_{t∈J} be a Miyadera perturbation of U, and assume that J is compact. Then

A(s)x := B(·)U(·, s)x  (x ∈ X_{s})  (1.3)

extends to a bounded operator A(s) from X to L_{1}(J; X), for each s ∈ J, and the function A: J → L(X; L_{1}(J; X)) thus defined is bounded and strongly measurable.

Proof. Note that U is bounded since J is compact. We thus obtain from Remark 1.7(b) that there exists c ⩾ 0 such that

||A(s)x||_{1} = ∫_{s}^{sup J} ||B(τ)U(τ, s)x|| dτ ⩽ c||x||

for all s ∈ J and all x ∈ X_{s}. Therefore, (1.3) determines a bounded operator A(s): X → L_{1}(J; X), for each s ∈ J, and the resulting function A is bounded.

Let now (t, s) ∈ D_{J}. For x ∈ X_{s} and a.e. τ ∈ J_{⩾t} we have

A(s)x(τ) = B(τ)U(τ, t)U(t, s)x = A(t)U(t, s)x(τ).

This implies that 𝟙_{J_{⩾t}}A(s)x = A(t)U(t, s)x for all x ∈ X_{s} and hence for all x ∈ X. Since U(t, ·)x: J_{⩽t} → X is measurable, it follows that

J_{⩽t} ∋ s ⟼ 𝟙_{J_{⩾t}}A(s)x ∈ L_{1}(J; X)

is measurable, for all x ∈ X and all t ∈ J. From this we deduce the strong measurability of A, using that 𝟙_{J_{⩾t}}A(s)x → A(s)x in L_{1}(J; X) as t ↓ s.

The following result will be used to establish strong continuity of perturbed propagators that satisfy both Duhamel formulas. For the proof of the strong continuity assertion of Theorem 1.8 we will only need Lemma 1.11(a) below. In these two results, (B(t))_{t∈J} is a family of linear operators in X, and Y_{s} is a dense subspace of X, for each s ∈ J.

1.10 Proposition. ([LVV06; Prop. 1.18]) Assume that U is strongly continuous, and let W be a locally bounded propagator on X with parameter interval J. Assume that, for all (t, s) ∈ D_{J} and all x ∈ Y_{s}, the functions W (t, ·)B(·)U(·, s)x and B(·)W (·, s)x are integrable on [s, t], and

W (t, s)x = U(t, s)x + ∫_{s}^{t} W (t, τ)B(τ)U(τ, s)x dτ,  (1.4)
W (t, s)x = U(t, s)x + ∫_{s}^{t} U(t, τ)B(τ)W (τ, s)x dτ.  (1.5)

Then W is strongly continuous.

Proof. This is an immediate consequence of Lemma 1.4 and the following lemma.

1.11 Lemma. ([LVV06; Lemma 1.19]) Assume that U is strongly continuous, and let W be a locally bounded propagator on X with parameter interval J.

(a) Assume that, for all (t, s) ∈ D_{J}, x ∈ Y_{s}, the function W (t, ·)B(·)U(·, s)x is integrable on [s, t], and (1.4) holds. Then W (t, ·) is strongly continuous on J_{⩽t} for all t ∈ J.

(b) Assume that, for all (t, s) ∈ D_{J}, x ∈ Y_{s}, the function B(·)W (·, s)x is integrable on [s, t], and (1.5) holds. Then W (·, s) is strongly continuous on J_{⩾s} for all s ∈ J.

Proof. (a) Fix t ∈ J and s_{0} ∈ J_{⩽t}. Let r ∈ J_{<s_{0}} (r = s_{0} in case s_{0} = inf J), x ∈ Y_{r} and x_{s} := U(s, r)x for s ∈ [r, t]. By (1.4) we have

x_{s} = W (s, r)x − ∫_{r}^{s} W (s, τ)B(τ)U(τ, r)x dτ.

Applying W (t, s) to this equation and using (1.4) again, we obtain that

W (t, s)x_{s} = W (t, r)x − ∫_{r}^{s} W (t, τ)B(τ)U(τ, r)x dτ
= U(t, r)x + ∫_{s}^{t} W (t, τ)B(τ)U(τ, r)x dτ.

(Note that we cannot obtain this by setting x = x_{s} in (1.4) if U(s, r)Y_{r} ⊈ Y_{s}.) Thus, by the dominated convergence theorem,

W (t, s)x_{s} → U(t, r)x + ∫_{s_{0}}^{t} W (t, τ)B(τ)U(τ, r)x dτ = W (t, s_{0})x_{s_{0}}

as [r, t] ∋ s → s_{0}. Since x_{s} → U(s_{0}, r)x = x_{s_{0}}, we conclude that

W (t, s)x_{s_{0}} = W (t, s)x_{s} + W (t, s)(x_{s_{0}} − x_{s}) → W (t, s_{0})x_{s_{0}}

as [r, t] ∋ s → s_{0}. The set of elements x_{s_{0}} under consideration is dense in X, so the assertion follows.

(b) Let s ∈ J and x ∈ Y_{s}. Since

W (t, s)x = U(t, s)x + ∫_{J} 𝟙_{[s,t]}(τ)U(t, τ)B(τ)W (τ, s)x dτ (t ∈ J_{⩾s})

and U is strongly continuous (and hence locally bounded), W (·, s)x is continuous on J_{⩾s} by the dominated convergence theorem. This shows the assertion since Y_{s} is dense in X and W is locally bounded.

Proof of Theorem 1.8. We will first show: If U_{B} is an element of the space

M := {V : D_{J} → L(X); V is locally bounded,

V is separately and jointly strongly measurable}

that satisfies the Duhamel formula (1.2), then U_{B} is a propagator. Then we will show that there exists a unique U_{B} ∈ M satisfying (1.2). The assertion about strong continuity is proved in the last step.

Let α ∈ (0, ∞), γ ∈ [0, 1) be as in condition (M), and let U_{B} ∈ M satisfy (1.2). (Note that the integral in (1.2) exists since B(·)U(·, s)x is integrable on (s, t) by (1.1) and U_{B}(t, ·) is strongly measurable and bounded on (s, t); see also Lemma 1.2.) Clearly, U_{B}(t, t) = I for all t ∈ J. Given (t, s) ∈ D_{J} we show that U_{B}(t, s)U_{B}(s, r) = U_{B}(t, r) for all r ∈ [s − α, s] ∩ J; then it follows by induction that U_{B} is a propagator.

Let r ∈ [s − α, s] ∩ J and x ∈ X_{r}. Then U(s, r)x ∈ X_{s}, so by (1.2) we obtain that

U_{B}(t, s)U(s, r)x = U(t, r)x + ∫_{s}^{t} U_{B}(t, τ)B(τ)U(τ, r)x dτ.  (1.6)

Moreover,

U_{B}(s, r)x = U(s, r)x + ∫_{r}^{s} U_{B}(s, τ)B(τ)U(τ, r)x dτ.

Applying U_{B}(t, s) to both sides of the latter formula and using (1.6), we infer that

U_{B}(t, s)U_{B}(s, r)x = U(t, r)x + ∫_{s}^{t} U_{B}(t, τ)B(τ)U(τ, r)x dτ
+ ∫_{r}^{s} U_{B}(t, s)U_{B}(s, τ)B(τ)U(τ, r)x dτ.

We now subtract (1.2), used with r in place of s, from the preceding equation to obtain that

U_{B}(t, s)U_{B}(s, r)x − U_{B}(t, r)x = ∫_{r}^{s}(U_{B}(t, s)U_{B}(s, τ) − U_{B}(t, τ))B(τ)U(τ, r)x dτ.

For the operators D_{τ} := U_{B}(t, s)U_{B}(s, τ) − U_{B}(t, τ) ∈ L(X) we thus have

||D_{r}x|| ⩽ sup_{τ∈[r,s]} ||D_{τ}|| ∫_{r}^{s} ||B(τ)U(τ, r)x|| dτ (r ∈ [s, s − α] ∩ J, x ∈ X_{r}).

By the Miyadera condition we conclude that ||D_{r}|| ⩽ γ sup_{τ∈[r,s]} ||D_{τ}|| for all r ∈ [s, s−α]∩J. Therefore, D_{r} = 0 for all r ∈ [s, s − α] ∩ J, and we have shown that U_{B} is a propagator.

For the proof of existence and uniqueness we assume without loss of generality that J is compact; then U is bounded. Below we will show that

ΦV (t, s)x := ∫_{s}^{t} V (t, τ)A(s)x(τ) dτ (V ∈ M, (t, s) ∈ D_{J}, x ∈ X)

(with A as in Lemma 1.9) defines an operator Φ: M → M, and that Φ is a strict contraction on M if M is endowed with the Morgenstern norm

||V ||_{λ} := sup{e^{−λ(t−s)}||V (t, s)||; (t, s) ∈ D_{J}},

for λ > 0 large enough. Since (M, ||·||_{λ}) is a Banach space and U ∈ M, it then follows that there exists a unique U_{B} ∈ M such that U_{B} = U + ΦU_{B}, and this identity is equivalent to the validity of (1.2) since

ΦU_{B}(t, s)x = ∫_{s}^{t} U_{B}(t, τ)B(τ)U(τ, s)x dτ ((t, s) ∈ D_{J}, x ∈ X_{s})

by the definition of A(s).

Let V ∈ M and t ∈ J. Since V (t, ·) is strongly measurable and bounded on J_{⩽t}, we can define a bounded operator C_{V} (t): L_{1}(J; X) → X by

C_{V} (t)f := ∫_{inf J}^{t} V (t, τ)f(τ) dτ.

Then

ΦV (t, s)x = ∫_{inf J}^{t} V (t, τ)A(s)x(τ) dτ = C_{V} (t)A(s)x

for all s ∈ J_{⩽t} and all x ∈ X, where we have used that A(s)x(τ) = 0 for τ ∈ J_{<s}. From the (joint) strong measurability of V it follows that C_{V} : J → L(L_{1}(J; X), X) is strongly measurable. Since A: J → L(X; L_{1}(J; X)) is strongly measurable by Lemma 1.9, we conclude that ΦV is separately and jointly strongly measurable (see Lemma 1.2).

In order to show that Φ is a strict contraction on (M, || · ||_{λ}), we endow L_{1}(J; X) with the norm given by ||f||_{1,λ} := ||t ↦ e^{−λt}f(t)||_{1}. Let M := sup{||U(t, s)||; (t, s) ∈ D_{J}}. Then for s ∈ J and x ∈ X_{s} we obtain as in Remark 1.7(b), with t_{j} = s + αj, that

\begin{aligned}||A(s)x||_{1,λ} & = ∫_{s}^{sup J} e^{−λτ}||B(τ)U(τ, s)x|| dτ ⩽ ∑_{j=0}^{n−1} e^{−λt_{j}}γ||U(t_{j}, s)x|| \\ & ⩽ γ(1 + M ∑_{j=1}^{n−1} e^{−λαj})e^{−λs}||x|| ⩽ γ̃_{λ}e^{−λs}||x||, \end{aligned}  (1.7)

where γ̃_{λ} := γ(1 + \frac{M}{e^{λα}−1}) → γ < 1 as λ → ∞, and n ∈ ℕ is the smallest number such that t_{n} ∉ J. Moreover, for V ∈ M, t ∈ J and f ∈ L_{1}(J; X) we can estimate

||C_{V} (t)f|| ⩽ ∫_{inf J}^{t} ||V (t, τ)f(τ)|| dτ ⩽ ∫_{inf J}^{t} ||V ||_{λ}e^{λ(t−τ)}||f(τ)|| dτ ⩽ ||V ||_{λ}e^{λt}||f||_{1,λ}.

It follows that ||ΦV (t, s)|| = ||C_{V} (t)A(s)|| ⩽ γ̃_{λ}e^{λ(t−s)}||V ||_{λ} for all (t, s) ∈ D_{J}, so that ΦV ∈ M and ||ΦV ||_{λ} ⩽ γ̃_{λ}||V ||_{λ}. We conclude that Φ is a strict contraction on (M, || · ||_{λ}) for λ large enough, and the proof of the first assertion is complete.

Assume now that U is strongly continuous. Then U_{B}(t, ·) is strongly continuous on J_{⩽t} for all t ∈ J, by Lemma 1.11(a). Moreover, it follows from the dominated convergence theorem that Φ maps the closed subspace

M_{1} := {V ∈ M; V (·, s) is strongly continuous on J_{⩾s} (s ∈ J)}

of M to itself, so U_{B} ∈ M_{1}. By Lemma 1.4 we conclude that U_{B} is strongly continuous.

1.12 Remarks. (a) Let (B(t))_{t∈J} be a small Miyadera perturbation of U, and let Φ: M → M be as in the above proof. Observe that for the definition of Φ, the compactness of J is actually not needed since for V ∈ M one only has to define ΦV on D_{K} for compact subintervals K of J. Since (I − Φ)U_{B} = U, we obtain that the perturbed propagator U_{B} is given by the Dyson-Phillips series ∑_{k=0}^{∞} U_{k}, where U_{k} := U_{k,B} := Φ^{k}U: D_{J} → L(X) is recursively given by U_{0} = U and

U_{k}(t, s)x = ∫_{s}^{t} U_{k−1}(t, τ)B(τ)U(τ, s)x dτ ((t, s) ∈ D_{J}, x ∈ X_{s})

for k ∈ ℕ. It follows from the proof of Theorem 1.8 that U_{B}(t, s) = ∑_{k=0}^{∞} U_{k}(t, s) converges absolutely, uniformly for (t, s) in compact subsets of D_{J}.

(b) The idea of the estimation in (1.7) is not new. In [Voi77; p. 168], a similar estimate is used in the proof of the Miyadera perturbation theorem for C_{0}-semigroups, but the purpose there is to determine the generator of the perturbed semigroup. Here the use of the Morgenstern norm allows us to show immediately that the perturbed propagator satisfies the Duhamel formula for all (t, s) ∈ D_{J}, and to obtain absolute convergence of the Dyson-Phillips series on all of D_{J}.

In the literature, the Duhamel formula is usually first shown for t − s ⩽ α only; then for larger t−s an induction argument is used that is based on a variant of Proposition 1.13 below. The same applies to absolute convergence of the Dyson-Phillips series; see [Rha92; Prop. 2.3] for a result in the context of C_{0}-semigroups. Here we reverse arguments: We use the absolute convergence of the Dyson-Phillips series on D_{J} for an easy proof of Proposition 1.13.

(c) Clearly, Theorem 1.8 also holds if (B(t))_{t∈J} is just locally Miyadera small, i.e., if (B(t))_{t∈K} is a small Miyadera perturbation of U|_{D_{K}} for all compact subintervals K ⊆ J. This holds, e.g., if B: J → L(X) is locally bounded and strongly measurable. In the simple case that B(t) = cI for all t ∈ J, for some c ∈ 𝕂, one easily checks that the unique propagator U_{B} satisfying the Duhamel formula (1.2) is given by U_{B}(t, s) = e^{c(t−s)}U(t, s) for all (t, s) ∈ D_{J}.

Note that the constituents U_{k} of the Dyson-Phillips series are also defined if (B(t))_{t∈J} is a Miyadera perturbation of U that is not necessarily small.

1.13 Proposition. Let (B(t))_{t∈J} be a Miyadera perturbation of U, let U_{k} be the constituents of the corresponding Dyson-Phillips series, and let r, s, t ∈ J satisfy r ⩽ s ⩽ t. Then

U_{k}(t, r) = ∑_{j=0}^{k} U_{j}(t, s)U_{k−j}(s, r)

for all k ∈ ℕ_{0}.

Proof. There exists θ_{0} > 0 such that θB = (θB(t))_{t∈J} is a small Miyadera perturbation of U for all |θ| < θ_{0}. Observe that U_{k,θB} = θ^{k}U_{k,B} for all k ∈ ℕ_{0} and all θ ∈ ℝ. For |θ| < θ_{0} we thus obtain that

∑_{k=0}^{∞} θ^{k}U_{k,B}(t, r) = U_{θB}(t, r) = U_{θB}(t, s)U_{θB}(s, r)

= ∑_{k=0}^{∞} ∑_{j=0}^{k} θ^{j}U_{j,B}(t, s)θ^{k−j}U_{k−j,B}(s, r),

which implies the assertion.

In the next result we show an estimate for the norm of U_{B}(t, s), for (t, s) ∈ D_{J}. One can use formula (1.7) from the proof of Theorem 1.8 to derive this estimate, but we prefer to give a separate argument that relies solely on (1.2).

1.14 Proposition. Let (B(t))_{t∈J} be a small Miyadera perturbation of U with constants (α, γ), γ < 1, and assume that M := sup{||U(t, s)||; (t, s) ∈ D_{J}} < ∞. Then

||U_{B}(t, s)|| ⩽ \frac{M}{1−γ} e^{ω(t−s)} ((t, s) ∈ D_{J}),

where ω = \frac{1}{α} ln(1 + \frac{γ}{1−γ} M).

Proof. We are going to prove that

c_{j} := sup\{||U_{B}(t, s)||; (t, s) ∈ D_{J}, t − s ⩽ jα\} ⩽ \frac{M}{1−γ} (1 + \frac{γ}{1−γ} M)^{j−1}

for all j ∈ ℕ; then the assertion follows by choosing j such that (j − 1)α ⩽ t − s ⩽ jα. Assuming the above estimate for all j < n, for some n ∈ ℕ, we show that it also holds for j = n.

Let (t, s) ∈ D_{J} with t − s ⩽ nα, and let t_{j} := s + \frac{j}{n}(t − s) for j = 0, …, n. Similarly as in Remark 1.7(b) we obtain from

U_{B}(t, s)x = U(t, s)x + ∑_{j=1}^{n} ∫_{t_{n−j}}^{t_{n−j+1}} U_{B}(t, τ)B(τ)U(τ, t_{n−j})U(t_{n−j}, s)x dτ (x ∈ X_{s})

and the definition of the c_{j} that

||U_{B}(t, s)|| ⩽ M + ∑_{j=1}^{n−1} c_{j}γM + c_{n}γ.

By the definition of c_{n} we infer that c_{n} ⩽ M(1 + γ ∑_{j=1}^{n−1} c_{j}) + γc_{n}. Since

γ ∑_{j=1}^{n−1} c_{j} ⩽ \frac{γ}{1−γ} M ∑_{j=0}^{n−2}(1 + \frac{γ}{1−γ} M)^{j} = (1 + \frac{γ}{1−γ} M)^{n−1} − 1

by the induction hypothesis, we conclude that (1 − γ)c_{n} ⩽ M(1 + \frac{γ}{1−γ} M)^{n−1}.

1.15 Remark. For Miyadera perturbations of C_{0}-semigroups, an analogous estimate is proved in [Voi77; Thm. 1c)], but with ω = \frac{1}{α} ln \frac{M}{1−γ}. In the case M > 1, Proposition 1.14 yields a better estimate; in particular, the exponential growth factor ω given in the proposition tends to 0 as γ → 0.

1.3 Closed Miyadera perturbations

Throughout this section, U is again a locally bounded strongly measurable propagator on a Banach space X with parameter interval J ⊆ ℝ. Given a small Miyadera perturbation (B(t))_{t∈J} of U, we now want to show the second Duhamel formula

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U(t, τ)B(τ)U_{B}(τ, s)x dτ  ((t, s) ∈ D_{J}, x ∈ X) (1.8)

for the perturbed propagator U_{B}. In the autonomous case, if U comes from a C_{0}-semi-group and B is relatively bounded with respect to the generator, this easily follows from the fundamental theorem of calculus. Here we can only show (1.8) in the case that almost every operator B(t) is closed; see Theorem 1.20(a) below. We start with some preparations.

1.16 Lemma. Let (B(t))_{t∈J} be a family of operators in X. Assume that almost every B(t) is closed, that B(·)x is a.e. defined and measurable on J, for all x ∈ X, and that there exists c ⩾ 0 such that ||B(·)x||_{1} ⩽ c||x|| for all x ∈ X. Let (Ω, μ) be a σ-finite measure space and f ∈ L_{1}(μ; X). Then the function (t, s) ↦ B(t)f(s) is a.e. defined and measurable on J × Ω.

Proof. The assertion is clear if f is a simple function. For f ∈ L_{1}(μ; X) there exists a sequence (f_{n}) of simple functions in L_{1}(μ; X) such that f_{n} → f a.e. and in L_{1}. By the assumption ||B(·)x||_{1} ⩽ c||x|| we obtain that g_{n}(t, s) := B(t)f_{n}(s) defines a Cauchy sequence (g_{n}) in L_{1}(J × Ω; X), so there exist g ∈ L_{1}(J × Ω; X) and a subsequence (g_{n_{k}}) such that g_{n_{k}} → g a.e. Since a.e. B(t) is closed and f_{n_{k}} → f a.e., we conclude that B(t)f(s) = g(t, s) for a.e. (t, s) ∈ J × Ω, which implies the assertion.

The next lemma says in particular that the spaces X_{s} in condition (M) (that also occur in the Duhamel formula (1.2)) are equal to X if (B(t))_{t∈J} is a closed Miyadera perturbation of U, by which we mean that (B(t))_{t∈J} is a Miyadera perturbation of U and a.e. B(t) is closed.

1.17 Lemma. Let (B(t))_{t∈J} be a closed Miyadera perturbation of U. Then X_{s} = X for all s ∈ J. If J is compact then Af(t, s) := B(t)U(t, s)f(s) defines a bounded operator A: L_{1}(J; X) → L_{1}(D_{J}; X), and for all f ∈ L_{1}(J; X) we have

C_{U}(t)f = ∫_{inf J}^{t} U(t, s)f(s) ds ∈ D(B(t)), B(t)C_{U}(t)f = ∫_{inf J}^{t} B(t)U(t, s)f(s) ds

for a.e. t ∈ J.

Proof. Without loss of generality we assume the compactness of J also for the proof of the first assertion. Let s ∈ J, and let A(s) be the operator defined in Lemma 1.9. Then A(s)|_{X_{s}} is bounded and densely defined. Moreover, from the definition of X_{s} and the closedness of a.e. B(t) we easily infer that A(s)|_{X_{s}} is closed and hence everywhere defined, i.e., X_{s} = X.

Let now f ∈ L_{1}(J; X). Given τ ∈ J, we apply Lemma 1.16 with B(t)U(t, τ) in place of B(t) and U(τ, ·)f(·) in place of f to obtain that Af is a.e. defined and measurable on J_{⩾τ} × J_{⩽τ}. It follows that Af is a.e. defined and measurable on D_{J}. Moreover,

||Af||_{1} ⩽ ∫_{J} ∫_{s}^{sup J} ||B(t)U(t, s)f(s)|| dt ds ⩽ ∫_{J} ||A(s)||||f(s)|| ds ⩽ sup_{s∈J} ||A(s)|| · ||f||,

by Fubini’s theorem, so A is a bounded operator by Lemma 1.9.

Finally, the integrability of Af implies that B(t)U(t, ·)f(·) is integrable on J_{⩽t}, for a.e. t ∈ J, so the last assertion follows from the closedness of a.e. B(t), by Hille’s theorem.

The next result expresses, roughly speaking, that one can iterate Duhamel formulas. An important feature is that the Miyadera perturbations are not assumed to be small.

1.18 Proposition. Let V, W be locally bounded strongly measurable propagators on X with parameter interval J. Let B = (B(t))_{t∈J} be a Miyadera perturbation of U and B_{1} = (B_{1}(t))_{t∈J} a closed Miyadera perturbation of V, and assume that

V (t, s)x = U(t, s)x + ∫_{s}^{t} V (t, τ)B(τ)U(τ, s)x dτ (x ∈ X_{s}),
W (t, s)x = V (t, s)x + ∫_{s}^{t} W (t, τ)B_{1}(τ)V (τ, s)x dτ (x ∈ X)

for all (t, s) ∈ D_{J}, where the spaces X_{s} are from the Miyadera condition for U and B. Then B + B_{1} is a Miyadera perturbation of U, and

W (t, s)x = U(t, s)x + ∫_{s}^{t} W (t, τ)(B + B_{1})(τ)U(τ, s)x dτ  (1.9)

for all (t, s) ∈ D_{J} and all x ∈ X_{s}. (Here B + B_{1} = (B(t) + B_{1}(t))_{t∈J}, where B(t) + B_{1}(t) is the operator sum on D(B(t)) ∩ D(B_{1}(t)).)

In particular, if B is a closed Miyadera perturbation and B_{1} = −B, then W = U.

Proof. Let (t, s) ∈ D_{J} and x ∈ X_{s}; then f := B(·)U(·, s)x ∈ L_{1}(s, t; X). By Lemma 1.17 we obtain that

(V − U)(τ, s)x = ∫_{s}^{τ} V (τ, σ)f(σ) dσ ∈ D(B_{1}(τ))

and

B_{1}(τ)(V − U)(τ, s)x = ∫_{s}^{τ} B_{1}(τ)V (τ, σ)f(σ) dσ

for a.e. τ ∈ (s, t), and that (τ, σ) ↦ B_{1}(τ)V (τ, σ)f(σ) is integrable on D_{(s,t)}. If (α, γ), (α, γ_{1}) are Miyadera constants of B, B_{1}, respectively, and t − s ⩽ α, then by Fubini’s theorem it follows that

∫_{s}^{t} ||B_{1}(τ)(V − U)(τ, s)x|| dτ ⩽ ∫_{s}^{t} ∫_{σ}^{t} ||B_{1}(τ)V (τ, σ)f(σ)|| dτ dσ ⩽ ∫_{s}^{t} γ_{1}||f(σ)|| dσ

and hence ∫_{s}^{t} ||B_{1}(τ)U(τ, s)x|| dτ ⩽ γ_{1}γ||x|| + γ_{1}||x||, and we infer that B + B_{1} is a Miyadera perturbation of U. Moreover,

∫_{s}^{t} W (t, τ)B_{1}(τ) ∫_{s}^{τ} V (τ, σ)f(σ) dσ dτ = ∫_{s}^{t} ∫_{σ}^{t} W (t, τ)B_{1}(τ)V (τ, σ)f(σ) dτ dσ. (1.10)

Since

W (t, s)x = U(t, s)x + ∫_{s}^{t} V (t, τ)B(τ)U(τ, s)x dτ + ∫_{s}^{t} W (t, τ)B_{1}(τ)V (τ, s)x dτ,

we conclude from (1.10) that

− W (t, s)x + U(t, s)x + ∫_{s}^{t} W (t, τ)(B + B_{1})(τ)U(τ, s)x dτ

= ∫_{s}^{t}(W − V)(t, σ)B(σ)U(σ, s)x dσ − ∫_{s}^{t} W (t, τ)B_{1}(τ)(V − U)(τ, s)x dτ = 0,

and the proof is complete.

As an easy consequence of Proposition 1.18 we obtain the following uniqueness result for perturbed propagators satisfying both Duhamel formulas.

1.19 Corollary. Let (B(t))_{t∈J} be a closed Miyadera perturbation of U, and let V be a locally bounded strongly measurable propagator on X with parameter interval J satisfying

V (t, s)x = U(t, s)x + ∫_{s}^{t} V (t, τ)B(τ)U(τ, s)x dτ  ((t, s) ∈ D_{J}, x ∈ X), (1.11)
V (t, s)x = U(t, s)x + ∫_{s}^{t} U(t, τ)B(τ)V (τ, s)x dτ  ((t, s) ∈ D_{J}, x ∈ X). (1.12)

Then V is the only locally bounded strongly measurable propagator satisfying (1.11), and the only one satisfying (1.12).

Proof. Let Ṽ V be another locally bounded strongly measurable propagator satisfying (1.11), with V replaced by Ṽ V. Then from (1.12), in the form

U(t, s)x = V (t, s)x + ∫_{s}^{t} U(t, τ)(−B)(τ)V (τ, s)x dτ ((t, s) ∈ D_{J}, x ∈ X),

it follows by Proposition 1.18 that Ṽ V = V. Uniqueness in (1.12) is proved in the same way.

We now prove, for closed Miyadera perturbations, that the second Duhamel formula (1.8) holds and that one can iterate Miyadera perturbations. In the case of strongly continuous propagators, the validity of (1.8) is due to [RRSV00; Thm. 3.4(c)]; see also [LVV06; Thm. 1.16(b)]. In the autonomous case of C_{0}-semigroups, the first assertion of part (b) of the following theorem is due to [OSSV96; Lemma 1.1]; there the closedness assumption is not needed.

1.20 Theorem. Let B = (B(t))_{t∈J} be a closed Miyadera perturbation of U with constants (α, γ).

(a) Then the constituents U_{k} of the corresponding Dyson-Phillips series satisfy the recursion formula

U_{k}(t, s)x = ∫_{s}^{t} U(t, τ)B(τ)U_{k−1}(τ, s)x dτ  (k ∈ ℕ, (t, s) ∈ D_{J}, x ∈ X). (1.13)

If γ < 1 then the perturbed propagator U_{B} from Theorem 1.8 is the unique locally bounded strongly measurable propagator satisfying the second Duhamel formula (1.8).

(b) Assume that γ < 1, and let B_{1} = (B_{1}(t))_{t∈J} be a closed Miyadera perturbation of U with constants (α, γ_{1}). Then B_{1} is a Miyadera perturbation of U_{B} with constants (α, \frac{γ_{1}}{1−γ}). If γ + γ_{1} < 1 (so that \frac{γ_{1}}{1−γ} < 1) then B + B_{1} is a small Miyadera perturbation of U, and U_{B+B_{1}} = (U_{B})_{B_{1}}.

1.21 Remarks. (a) Part (a) of the above theorem says in particular that the integrals in (1.13) and (1.8) are defined for all (t, s) ∈ D_{J} and all x ∈ X. We will in fact show more strongly that B(·)U_{k}(·, s)x and B(·)U_{B}(·, s)x are integrable on (s, t). For the latter function this also follows from part (b), which implies in particular that B is a Miyadera perturbation of U_{B}.

(b) We observe that the second assertion of part (a) of Theorem 1.20 can be derived from part (b): By the latter, there exists n ∈ ℕ such that \frac{1}{n}B is a small Miyadera perturbation of U_{\frac{k}{n} B}, and (U_{\frac{k}{n} B})_{− \frac{1}{n} B} = U_{\frac{k−1}{n} B}, for k = 1, …, n. Applying Proposition 1.18 n−1 times, we thus obtain that

U(t, s)x = U_{B}(t, s)x + ∫_{s}^{t} U(t, τ)(−B(τ))U_{B}(τ, s)x dτ

for all (t, s) ∈ D_{J} and all x ∈ X, which proves (1.8). Below we will give an independent proof based on (1.13).

(c) In [Sch02; Thm. 4.4], in the context of strongly continuous propagators, a perturbation theorem is proved that comprises versions of both the Miyadera and Desch-Schappacher theorems. From this result it follows, by [Sch02; Rem. 4.6(a)], that the second Duhamel formula (1.8) also holds for non-closable B(t) if instead of the Miyadera condition (M) one assumes more restrictively that there exist p > 1 and α, γ > 0 such that

∫_{s}^{t} ||B(τ)U(τ, s)x||^{p} dτ ⩽ γ^{p}||x|| ((t, s) ∈ D_{J}, t − s ⩽ α, x ∈ X_{s}).

In order to make this possible, the operators B(t) are suitably extended such that the integral in (1.8) is defined.

Proof of Theorem 1.20. (a) Without loss of generality assume that J is compact; then U is bounded. Let M̃ M be the space of all functions W: D_{J} → L(X) for which x ↦ B(·)W (·, s)x defines a bounded operator from X to L_{1}(J_{⩾s}; X), for all s ∈ J. For W ∈ M̃ M and (t, s) ∈ D_{J},

ΨW (t, s)x := ∫_{s}^{t} U(t, σ)B(σ)W (σ, s)x dσ

defines an operator ΨW (t, s) ∈ L(X).

Let W ∈ M̃ M; we show that ΨW ∈ M̃ M. Let s ∈ J, x ∈ X and f := B(·)W (·, s)x. Then f ∈ L_{1}(J_{⩾s}; X), so ΨW (τ, s)x = ∫_{s}^{τ} U(τ, σ)f(σ) dσ ∈ D(B(τ)) and

B(τ)ΨW (τ, s)x = ∫_{s}^{τ} B(τ)U(τ, σ)f(σ) dσ

for a.e. τ ∈ J_{⩾s}, by Lemma 1.17. Moreover, (τ, σ) ↦ B(τ)U(τ, σ)f(σ) is integrable on D_{J_{⩾s}}. Thus, with the weighted norm || · ||_{1,λ} on L_{1}(J_{⩾s}; X) from the proof of Theorem 1.8 we obtain by Fubini’s theorem and (1.7) that

||B(·)ΨW (·, s)x||_{1,λ} ⩽ ∫_{J_{⩾s}} e^{−λτ} ∫_{s}^{τ} ||B(τ)U(τ, σ)f(σ)|| dσ dτ

= ∫_{J_{⩾s}} ∫_{J_{⩾σ}} e^{−λτ}||B(τ)U(τ, σ)f(σ)|| dτ dσ = ∫_{J_{⩾s}} ||A(σ)f(σ)||_{1,λ} dσ

⩽ ∫_{J_{⩾s}} γ̃_{λ}e^{−λσ}||f(σ)|| dσ = γ̃_{λ}||B(·)W (·, s)x||_{1,λ},  (1.14)

and it follows that ΨW ∈ M̃ M.

Let Φ: M → M be as in the proof of Theorem 1.8. Let (t, s) ∈ D_{J} and x ∈ X. We use Fubini’s theorem as above to obtain for V ∈ M that

\begin{aligned}∫_{s}^{t} V (t, τ)B(τ)ΨW (τ, s)x dτ & = ∫_{s}^{t} ∫_{σ}^{t} V (t, τ)B(τ)U(τ, σ)B(σ)W (σ, s)x dτ dσ \\ & = ∫_{s}^{t} ΦV (t, σ)B(σ)W (σ, s)x dσ. \end{aligned} (1.15)

Observe that U ∈ M̃ M since X_{σ} = X for all σ ∈ J. For k ∈ ℕ it thus follows that

\begin{aligned}U_{k}(t, s)x = Φ^{k}U(t, s)x & = ∫_{s}^{t} Φ^{k−1}U(t, σ)B(σ)U(σ, s)x dσ = … \\ & = ∫_{s}^{t} U(t, τ)B(τ)Ψ^{k−1}U(τ, s)x dτ = Ψ^{k}U(t, s)x, \end{aligned}  (1.16)

which proves the first assertion.

Assume now that γ < 1, and choose λ > 0 such that γ̃_{λ} < 1. From (1.14) and (1.16) we infer that

||B(·)U_{k}(·, s)x||_{1,λ} = ||B(·)Ψ^{k}U(·, s)x||_{1,λ} ⩽ γ̃_{λ}^{k}||B(·)U(·, s)x||_{1,λ}  (1.17)

for all k ∈ ℕ, so ∑_{k=0}^{∞} B(·)U_{k}(·, s)x converges absolutely in L_{1}(J_{⩾s}; X) and hence a.e. on J_{⩾s}. Since ∑_{k=0}^{∞} U_{k} = U_{B} and a.e. B(τ) is closed, we thus obtain that

U_{B}(τ, s)x ∈ D(B(τ)) and B(τ)U_{B}(τ, s)x = ∑_{k=0}^{∞} B(τ)U_{k}(τ, s)x

for a.e. τ ∈ J_{⩾s}. We conclude that ∑_{k=0}^{∞} U(t, ·)B(·)U_{k}(·, s)x = U(t, ·)B(·)U_{B}(·, s)x in L_{1}(s, t; X) and hence

U_{B}(t, s)x = ∑_{k=0}^{∞} Ψ^{k}U(t, s)x = U(t, s)x + ∑_{k=1}^{∞} ∫_{s}^{t} U(t, τ)B(τ)U_{k−1}(τ, s)x dτ

= U(t, s)x + ∫_{s}^{t} U(t, τ)B(τ)U_{B}(τ, s)x dτ.

The uniqueness assertion follows from Corollary 1.19, so the proof of part (a) is complete.

(b) Let (t, s) ∈ D_{J} with t − s ⩽ α, and let x ∈ X. By an argument analogous to the one that leads to (1.14), we obtain for W ∈ M̃ M that

∫_{s}^{t} ||B_{1}(τ)ΨW (τ, s)x|| dτ ⩽ γ_{1} ∫_{s}^{t} ||B(τ)W (τ, s)x|| dτ.  (1.18)

Applying (1.18) with B_{1} = B, we infer that

∫_{s}^{t} ||B(τ)Ψ^{k−1}U(τ, s)x|| dτ ⩽ γ^{k−1} ∫_{s}^{t} ||B(τ)U(τ, s)x|| dτ ⩽ γ^{k}||x||

for all k ∈ ℕ. With W = U_{k−1} = Ψ^{k−1}U, (1.18) thus yields

∫_{s}^{t} ||B_{1}(τ)U_{k}(τ, s)x|| dτ ⩽ γ_{1} ∫_{s}^{t} ||B(τ)U_{k−1}(τ, s)x|| dτ ⩽ γ_{1}γ^{k}||x||.

The latter inequality (without the middle term) is clear for k = 0, so as in (a) we conclude that B_{1}(·)U_{B}(·, s)x ∈ L_{1}(s, t; X) and

∫_{s}^{t} ||B_{1}(τ)U_{B}(τ, s)x|| dτ ⩽ γ_{1} ∑_{k=0}^{∞} γ^{k}||x|| = \frac{γ_{1}}{1 − γ} ||x||,

which proves the first assertion.

Since both B and B_{1} satisfy condition (M) with X_{s} = X, we obtain that B + B_{1} is a Miyadera perturbation of U with constants (α, γ + γ_{1}). Now assume that γ + γ_{1} < 1, so that (B_{1}(t))_{t∈J} is a small Miyadera perturbation of U_{B}. Then by Proposition 1.18 we obtain that

(U_{B})_{B_{1}}(t, s)x = U(t, s)x + ∫_{s}^{t}(U_{B})_{B_{1}}(t, τ)(B + B_{1})(τ)U(τ, s)x dτ

for all (t, s) ∈ D_{J} and all x ∈ X, so the last assertion U_{B+B_{1}} = (U_{B})_{B_{1}} follows from uniqueness in the first Duhamel formula.

1.4 The Trotter product formula

In this section let U be a bounded strongly measurable propagator on a Banach space X with parameter interval J ⊆ ℝ. We are going to study the special case of bounded perturbations in more detail. In particular, in Theorem 1.26 below, we present a Trotter product formula for perturbed propagators. Throughout the section we assume that B: J → L(X) is bounded and strongly measurable.

1.22 Remark. Let c := ||B||_{∞}, and choose M ⩾ 1 such that ||U(t, s)|| ⩽ M for all (t, s) ∈ D_{J}. Then from the recursion formula (1.13) we obtain by induction that

||U_{k}(t, s)|| ⩽ M(Mc)^{k} \frac{(t − s)^{k}}{k!} (k ∈ ℕ_{0})

and hence ||U_{B}(t, s)|| ⩽ Me^{Mc(t−s)} for all (t, s) ∈ D_{J}.

The following convergence result for perturbed propagators is a generalisation of [RäSc99; Prop. 2.3(b)] and [LVV06; Prop. 1.8]. For simplicity it is formulated for bounded propagators and bounded perturbations, the generalisation to locally bounded propagators with locally bounded perturbations is straightforward (cf. Remark 1.12(c)).

1.23 Proposition. For each n ∈ ℕ let U^{n} be a bounded strongly measurable propagator on X with parameter interval J, and let B_{n}: J → L(X) be bounded and strongly measurable. Assume that

M := sup{||U^{n}(t, s)||; n ∈ ℕ, (t, s) ∈ D_{J}} < ∞

and c:= sup_{n∈ℕ} ||B_{n}||_{∞} < ∞, that U^{n}(t, s) → U(t, s) strongly as n → ∞, for all (t, s) ∈ D_{J}, and that B_{n}(·)x → B(·)x in L_{1,loc}(J; X) as n → ∞, for all x ∈ X. Then

U_{k,B_{n}}^{n}(t, s) → U_{k,B}(t, s) and U_{B_{n}}^{n}(t, s) → U_{B}(t, s)  (1.19)

strongly for all k ∈ ℕ_{0} and all (t, s) ∈ D_{J}.

If, in addition, U is strongly continuous and the strong convergence U^{n}(t, s) → U(t, s) is uniform for (t, s) in compact subsets of D_{J}, then the strong convergences in (1.19) are also uniform for (t, s) in compact subsets of D_{J}.

Proof. Without loss of generality assume that J is compact. As an initial step we define operators B, B_{n} ∈ L(L_{1}(J; X)) by Bf(t) := B(t)f(t) and B_{n}f(t) := B_{n}(t)f(t), for all n ∈ ℕ. The assumption on convergence of the B_{n} implies that B_{n}f → Bf in L_{1}(J; X) for all simple functions f: J → X. Since (B_{n}) is a bounded sequence in L(L_{1}(J; X)) and the simple functions are dense in L_{1}(J; X), it follows that B_{n} → B strongly as n → ∞.

We now prove the first convergence assertion in (1.19) by induction on k. For k = 0 there is nothing to show. Assume that the assertion holds for some k ∈ ℕ_{0}. Let (t, s) ∈ D_{J} and x ∈ X. Using the initial observation, the induction hypothesis and the estimate

||U_{k,B_{n}}^{n}(t, τ)|| ⩽ M(Mc)^{k} \frac{(t − s)^{k}}{k!}  (n ∈ ℕ, τ ∈ [s, t])  (1.20)

from Remark 1.22, we obtain that

f_{k}^{n}(t, s) := U_{k,B_{n}}^{n}(t, ·)B_{n}(·)U^{n}(·, s)x → U_{k,B}(t, ·)B(·)U(·, s)x =: f_{k}(t, s)

in L_{1}(s, t; X) as n → ∞. Therefore,

U_{k+1,B_{n}}^{n}(t, s)x = ∫_{s}^{t} f_{k}^{n}(t, s)(τ) dτ → ∫_{s}^{t} f_{k}(t, s)(τ) dτ = U_{k+1,B}(t, s)x  (1.21)

as n → ∞, and the first part of (1.19) is proved. From this we infer, taking into account (1.20) with τ = s, that

U_{B_{n}}^{n}(t, s)x = ∑_{k=0}^{∞} U_{k,B_{n}}^{n}(t, s)x → ∑_{k=0}^{∞} U_{k,B}(t, s)x = U_{B}(t, s)x  (1.22)

as n → ∞.

For the second part of the proof assume that U is strongly continuous and that U^{n}(t, s) → U(t, s) strongly as n → ∞, uniformly for (t, s) ∈ D_{J}. We prove by induction on k that the strong convergence U_{k,B_{n}}^{n}(t, s) → U_{k,B}(t, s) is uniform for (t, s) ∈ D_{J}. Again, for k = 0 there is nothing to show. Assuming uniform strong convergence for some k ∈ ℕ_{0}, we proceed with the notation from the first part of the proof. We are going to show that the convergence f_{k}^{n}(t, s) → f_{k}(t, s) is uniform for (t, s) ∈ D_{J}; then it follows that the convergences in (1.21) and (1.22) are uniform for (t, s) ∈ D_{J}, too. Here and in the following, we tacitly extend functions defined on subintervals of J by zero to functions on J.

The uniform strong convergence U^{n} → U implies that

g_{n}(s) := U^{n}(·, s)x → U(·, s)x =: g(s)

in L_{1}(J; X) as n → ∞, uniformly for s ∈ J. Moreover, the function g: J → L_{1}(J; X) thus defined is continuous since U is strongly continuous. By the initial observation, the induction hypothesis and estimate (1.20) we obtain as in the first part of the proof, for all f ∈ L_{1}(J; X), that

T_{n}(t)f := U_{k,B_{n}}^{n}(t, ·)B_{n}(·)f → U_{k,B}(t, ·)B(·)f =: T (t)f

in L_{1}(J; X), uniformly for t ∈ J. By Lemma 1.24(b) below, applied with K = L = J and X = Y = L_{1}(J; X), we conclude that

f_{k}^{n}(t, s) = T_{n}(t)g_{n}(s) → T (t)g(s) = f_{k}(t, s)

as n → ∞, uniformly for (t, s) ∈ D_{J}, and the proof is complete.

1.24 Lemma. Let K be a compact topological space, and let X, Y be Banach spaces. Let (g_{n}) be a sequence in ℓ_{∞}(K; X) such that g_{n} → g ∈ C(K; X) uniformly.

(a) Let (T_{n}) be a sequence in L(X, Y) such that T_{n} → T ∈ L(X, Y) strongly as n → ∞. Then T_{n} ∘ g_{n} → T ∘ g uniformly as n → ∞.

(b) Let L be a non-empty set, let T, T_{n}: L → L(X, Y) be bounded functions, for all n ∈ ℕ, and assume that T_{n}(t) → T (t) strongly as n → ∞, uniformly for t ∈ L. Then T_{n}(t)g_{n}(s) → T (t)g(s) as n → ∞, uniformly for t ∈ L and s ∈ K.

Proof. (a) The sequence (T_{n}) is bounded by the uniform boundedness theorem. Since g(K) is compact, we obtain that the convergence T_{n}x → T x is uniform for x ∈ g(K). It follows that

T_{n}g_{n}(t) − T g(t) = T_{n}(g_{n}(t) − g(t)) + (T_{n} − T)g(t) → 0

as n → ∞, uniformly for t ∈ K.

(b) follows from (a), applied with Y := ℓ_{∞}(L; Y) and T , T_{n}: X → Y,

T_{n}x := [t ↦ T_{n}(t)x] → [t ↦ T (t)x] =: T x

in Y as n → ∞, for all x ∈ X.

For the remainder of the section we assume that U is strongly continuous.

1.25 Remarks. (a) An important application of the above convergence result is the approximation of strongly continuous propagators by “discrete” propagators: For n ∈ ℕ let P_{n} be a discrete subset of J with inf P_{n} = inf J and sup P_{n} = sup J, and define φ_{n}: J → J by φ_{n}(s) := min{t ∈ P_{n}; t ⩾ s}. Assume that φ_{n} → id_{J} uniformly as n → ∞. Then for n ∈ ℕ we define a propagator U^{n} on X with parameter interval J by

U^{n}(t, s) := U(φ_{n}(t), φ_{n}(s)).

It follows from the strong continuity of U that U^{n}(t, s) → U(t, s) strongly as n → ∞, uniformly for (t, s) in compact subsets of D_{J}.

For n ∈ ℕ we now define B_{n}: J → L(X) as follows. If K is one of the connected components of J \ P_{n} and t ∈ K ∪ {sup K}, then we set

B_{n}(t) := \frac{1}{|K|} ∫_{K} B(τ) dτ (strong integral),

where |K| denotes the length of K, and B_{n}(inf J) := 0 if inf J ∈ J. Then B_{n}(·)x → B(·)x in L_{1,loc}(J; X) as n → ∞, for all x ∈ X, so we can apply Proposition 1.23 to obtain (1.19) with uniform strong convergence for (t, s) in compact subsets of D_{J}.

(b) We keep the notation from part (a). Let s_{0} < t_{0} be two consecutive points in P_{n}. Then U^{n}(t, τ) = I and B_{n}(τ) = B_{n}(t_{0}) for all (t, τ) ∈ D_{(s_{0},t_{0}]}. For the constituents of the Dyson-Phillips series for U_{B_{n}}^{n} we thus obtain by the recursion formula (1.13) that

U_{k+1,B_{n}}^{n}(t, s_{0})x = ∫_{s_{0}}^{t} U^{n}(t, τ)B_{n}(τ)U_{k,B_{n}}^{n}(τ, s_{0})x dτ = B_{n}(t_{0}) ∫_{s_{0}}^{t} U_{k,B_{n}}^{n}(τ, s_{0})x dτ

for all k ∈ ℕ_{0}, t ∈ (s_{0}, t_{0}] and all x ∈ X. By induction we infer that

U_{k,B_{n}}^{n}(t, s_{0}) = \frac{(t − s_{0})^{k}}{k!} B_{n}(t_{0})^{k}U(t_{0}, s_{0}) = \frac{(t − s_{0})^{k}}{k!} B_{n}(t)^{k}U(t, s_{0}) (k ∈ ℕ_{0})

and hence U_{B_{n}}^{n}(t, s_{0}) = e^{(t−s_{0})B_{n}(t)}U(t, s_{0}) for all t ∈ (s_{0}, t_{0}]. Therefore, if t_{0} < t_{1} < … < t_{m} are m+1 consecutive points in P_{n}, then we obtain by the propagator property that

U_{B_{n}}^{n}(t_{m}, t_{0}) = e^{(t_{m}−t_{m−1})B_{n}(t_{m})}U(t_{m}, t_{m−1}) ⋯ e^{(t_{1}−t_{0})B_{n}(t_{1})}U(t_{1}, t_{0}).

From the above application of Proposition 1.23 we immediately obtain the following Trotter product formula for perturbed propagators.

1.26 Theorem. Assume that U is strongly continuous, and let s, t ∈ J with s < t. For n ∈ ℕ let t_{0}^{n}, …, t_{n}^{n} ∈ [s, t] satisfy s = t_{0}^{n} < t_{1}^{n} < ⋯ < t_{n}^{n} = t, and assume that sup{t_{k}^{n} − t_{k−1}^{n}; k = 1, …, n} → 0 as n → ∞. For n ∈ ℕ and k = 1, …, n let

B_{k}^{n} := ∫_{t_{k−1}^{n}}^{t_{k}^{n}} B(τ) dτ (strong integral).

Then

e^{B_{n}^{n}}U(t_{n}^{n}, t_{n−1}^{n})e^{B_{n−1}^{n}}U(t_{n−1}^{n}, t_{n−2}^{n}) ⋯ e^{B_{1}^{n}}U(t_{1}^{n}, t_{0}^{n}) → U_{B}(t, s)

strongly as n → ∞.

Absorption propagators

In this chapter we further develop the theory of absorption propagators that was initiated in [RäSc99] and pushed forward in [LVV06]. The basic ideas of the approach are the same as in the theory of absorption semigroups; see [Voi86] and [Voi88].

Throughout the section let (Ω, A, μ) be a measure space. We do not require the measure μ to be σ-finite, which raises a few technical issues. In particular, L_{∞}(μ) will denote the space of all locally measurable functions from Ω to 𝕂 that are bounded outside a local null set, and functions are identified if they coincide locally a.e.

We are going to study perturbation of positive propagators on L_{p}(μ) by locally measurable potentials V : J ×Ω → ℝ. Here, J ×Ω is endowed with the σ-algebra generated by B_{fin} ⊗ A_{fin}, where B_{fin} denotes the system of all Lebesgue measurable subsets of J of finite measure, and A_{fin} ⊆ A is the subsystem of all μ-measurable sets of finite measure. On this σ-algebra, the product measure of Lebesgue measure and the measure μ is uniquely defined. If A ∈ A_{fin} then V is measurable on J × A, so V (t, ·) is measurable on A for all t ∈ J. It follows that V (t, ·) is locally measurable for all t ∈ J.

The chapter is organised as follows. In Section 2.1 we develop the abstract theory of absorption propagators, and we present a new version of the notions of admissible and regular potentials. Several of the results are straightforward generalisations of the corresponding results for C_{0}-semigroups, but some of the proofs are substantially simplified. We point out the following two new aspects: We study perturbation of strongly measurable propagators (as opposed to strongly continuous propagators), and we introduce a larger class of perturbations that includes highly oscillating potentials (see Example 2.14). Section 2.2 deals with the connection between admissibility and the Miyadera perturbation theorem. In Section 2.3 we present the backward Miyadera condition that can be used to obtain local L_{∞}-boundedness of perturbed propagators.

In Section 2.4 we introduce the notion of logarithmically convex operator-valued functions, and we prove an interpolation inequality that is similar to the Stein interpolation theorem. The concept of logarithmic convexity is used in Section 2.5 to prove interpolation inequalities for absorption propagators and for the corresponding Dyson-Phillips series; the latter lead to a deeper understanding of the relation between the Miyadera condition and admissibility. The Trotter product formula is a crucial ingredient in the proofs, so from this point on we will only consider perturbations of strongly continuous propagators. In Section 2.6 we apply the results of Section 2.5 to the interpolation of admissibility in different weighted L_{p}-spaces.

2.1 Abstract theory of absorption propagators

Let 1 ⩽ p < ∞, and let U be a positive locally bounded strongly measurable propagator on L_{p}(μ) with parameter interval J ⊆ ℝ. (A positive operator-valued function is, by definition, a function taking its values in the positive operators on L_{p}(μ).) We are going to apply the perturbation results of Section 1.2 to bounded potentials and to extend the definition of a perturbed propagator to suitable unbounded real-valued potentials.

Let V : J × Ω → ℝ be locally measurable and bounded. As explained in the introduction to the chapter, V (t) := V (t, ·) is locally measurable (and bounded) and can hence be considered as a bounded multiplication operator on L_{p}(μ), for all t ∈ J. We define a strongly measurable function B: J → L(L_{p}(μ)) by

B(t) := −V (t) (t ∈ J).  (2.1)

With this B we define

U_{V} := U_{B},

in the sense of Section 1.2. We choose the negative sign in order to stay compatible with the notation usually chosen for Schrödinger operators; positive V corresponds to absorption, negative V to excitation.

2.1 Remarks. (a) It is important to notice that for V ∈ L_{∞}(J × Ω), the perturbed propagator U_{V} is defined independently of the chosen representative of V. Indeed, let V, Ṽ V : J × Ω → ℝ be locally measurable and bounded, and assume that V = Ṽ V locally a.e. Then V (·)U(·, s)f = Ṽ V (·)U(·, s)f a.e. on J_{⩾s}, for all s ∈ J and all f ∈ L_{p}(μ) (cf. part (b) below), so by uniqueness in the first Duhamel formula (1.2) we obtain that U_{V} = UṼ. Nevertheless it can happen that V (t) ≠ Ṽ V (t) (as elements of L_{∞}(μ)) for all t ∈ J:

As in Example 1.3 let J := Ω := [0, 1], and let μ be the counting measure on Ω. We define V : J × Ω → ℝ as the indicator function of the diagonal {(t, t) ∈ J × Ω; 0 ⩽ t ⩽ 1}. Then V is locally measurable and V = 0 locally a.e., but V (t) = 𝟙_{{t}} ≠ 0 for all t ∈ J.

(b) If h: J → L_{p}(μ) is measurable then there exists a σ-finite set Ω_{h} ⊆ Ω such that h(t) = 𝟙_{Ω_{h}}h(t) a.e. for a.e. t ∈ J. Indeed, let h_{n}: J → L_{p}(μ) be simple functions such that h_{n} → h a.e. as n → ∞. Then Ω_{h} := ⋃{[h_{n}(t) ≠ 0]; n ∈ ℕ, t ∈ J} is σ-finite, and for all t ∈ J such that h_{n}(t) → h(t) in L_{p}(μ) we obtain that h(t) = 𝟙_{Ω_{h}}h(t) a.e. Here, for a function f: Ω → 𝕂, we denote [f ≠ 0] := {x ∈ Ω; f(x) ≠ 0}.

2.2 Remark. Recall from Theorem 1.20(b) that (U_{V_{1}})_{V_{2}} = U_{V_{1}+V_{2}} for V_{1}, V_{2} ∈ L_{∞}(J ×Ω). Moreover, if V = c for some c ∈ ℝ then U_{V} (t, s) = e^{−c(t−s)}U(t, s) for all (t, s) ∈ D_{J}, by Remark 1.12(c). As a consequence we obtain that

U_{V} (t, s) = e^{−c(t−s)}U_{V−c}(t, s) ((t, s) ∈ D_{J})

for all c ∈ ℝ and all V ∈ L_{∞}(J × Ω).

The next proposition, which is a complete analogue of the corresponding statement for C_{0}-semigroups [Voi88; Prop. 1.3], is the corner stone to the whole approach. For strongly continuous propagators, part (a) has already been proved in [RäSc99; Prop. 2.3(c)], part (b) in [LVV06; Prop. 2.1(b)].

2.3 Proposition. Let U_{1} and U_{2} be positive locally bounded strongly measurable propagators on L_{p}(μ) with parameter interval J, and let V, V_{1}, V_{2} ∈ L_{∞}(J × Ω).

(a) If V_{1} ⩽ V_{2} then U_{V_{1}} ⩾ U_{V_{2}} ⩾ 0.

(b) If U_{1} ⩽ U_{2} then (U_{1})_{V} ⩽ (U_{2})_{V}. If additionally V ⩾ 0 then (U_{2})_{V} −(U_{1})_{V} ⩽ U_{2} −U_{1}.

Proof. (a) By Remark 2.2 we can assume without loss of generality that V_{1} ⩽ V_{2} ⩽ 0. Then for the constituents of the Dyson-Phillips series for U_{V_{1}} and U_{V_{2}} (see Remark 1.12(a)) we obtain by induction that U_{k,−V_{1}} ⩾ U_{k,−V_{2}} ⩾ 0 for all k ∈ ℕ_{0}. Summation over k yields the assertion.

(b) The first assertion follows as in (a) from Remark 2.2 and the Dyson-Phillips series expansions of (U_{1})_{V} and (U_{2})_{V}. The second, in the form U_{1} − (U_{1})_{V} ⩽ U_{2} − (U_{2})_{V}, then follows from the Duhamel formula (1.2).

We are now going to define U_{V} for locally measurable V : J × Ω → ℝ that are possibly unbounded. Our approach differs from the classical theory of absorption semigroups ([Voi86], [Voi88]) and from the approach in [LVV06; Sec. 2.1], where V is approximated by means of cut-offs V^{(n)} := (V ∧ n) ∨ (−n). We use this approximation only for the cases V ⩾ 0 and V ⩽ 0. For the link with the classical theory see Remark 2.22(c) below.

2.4 Definition. (a) If V ⩾ 0 then U_{V ∧n}(t, s) ⩾ U_{V ∧(n+1)}(t, s) ⩾ 0 for all n ∈ ℕ and all (t, s) ∈ D_{J}, by Proposition 2.3(a), so the dominated convergence theorem implies that

U_{V} (t, s) := s-lim_{n→∞} U_{V ∧n}(t, s)  (2.2)

exists for all (t, s) ∈ D_{J}. Clearly, this defines a locally bounded strongly measurable propagator U_{V}.

(b) If V ⩽ 0 then (U_{V ∨(−n)}(t, s)) is an increasing sequence of positive operators, for all (t, s) ∈ D_{J}, and we say that V is weakly U-admissible if

U_{V} (t, s) := s-lim_{n→∞} U_{V ∨(−n)}(t, s)  (2.3)

exists for all (t, s) ∈ D_{J} and defines a locally bounded function U_{V} : D_{J} → L(X). Again, U_{V} is a strongly measurable propagator in this case.

(c) In the general case let V^{+} and V^{−} denote the positive and negative parts of V, respectively, i.e., V^{+} = V ∨ 0 and V^{−} = (−V)^{+}. We say that V is weakly U-admissible if −V^{−} is weakly U_{V^{+}}-admissible. (In particular, any V ⩾ 0 is weakly U-admissible.) Then we set

U_{V} := (U_{V^{+}})_{−V^{−}}.  (2.4)

By Remark 2.2, this is consistent with the definition of U_{V} in the case of bounded V.

(d) If U is strongly continuous then we say that V is U-admissible if U_{V} is strongly continuous. In [LVV06; Def. 2.2], parallel to [Voi88; Def. 2.5], it was required more restrictively for U-admissibility of V that V^{+} and −V^{−} are U-admissible (cf. Remark 2.18(b) below).

2.5 Remarks. (a) Observe that from (2.3) and (2.4) it follows that (2.3) holds for any weakly U-admissible potential V, not only for negative V.

(b) Let V be weakly U-admissible, and let Ṽ V = V locally a.e. Then it follows from Remark 2.1(a) that Ṽ V is weakly U-admissible, and UṼ = U_{V}.

We now show that the inequalities stated in Proposition 2.3 for bounded potentials carry over to weakly admissible potentials.

2.6 Proposition. Let U_{1} and U_{2} be positive locally bounded strongly measurable propagators on L_{p}(μ) with parameter interval J, and let V, V_{1}, V_{2} be locally measurable potentials.

(a) If V_{1} is weakly U-admissible and V_{1} ⩽ V_{2}, then V_{2} is weakly U-admissible and U_{V_{1}} ⩾ U_{V_{2}} ⩾ 0.

(b) If V is weakly U_{2}-admissible and U_{1} ⩽ U_{2}, then V is weakly U_{1}-admissible and (U_{1})_{V} ⩽ (U_{2})_{V}. If additionally V ⩾ 0 then (U_{2})_{V} − (U_{1})_{V} ⩽ U_{2} − U_{1}.

Proof. The assertions are clear from (2.2), (2.3) and Proposition 2.3 if the potentials are of one sign. As a consequence we obtain the first assertion of (b) in the general case: From (U_{1})_{V^{+}} ⩽ (U_{2})_{V^{+}} and the weak (U_{2})_{V^{+}}-admissibility of −V^{−} it follows that −V^{−} is weakly (U_{1})_{V^{+}}-admissible and

(U_{1})_{V} = ((U_{1})_{V^{+}})_{−V^{−}} ⩽ ((U_{2})_{V^{+}})_{−V^{−}} = (U_{2})_{V} .

Using part (b) we also obtain (a) in the general case: For all n ∈ ℕ, the inequalities U_{V_{1}^{+}} ⩾ U_{V_{2}^{+}} and V_{1}^{−} ∧ n ⩾ V_{2}^{−} ∧ n imply that

U_{V_{1}} ⩾ (U_{V_{1}^{+}})_{−V_{1}^{−}∧n} ⩾ (U_{V_{2}^{+}})_{−V_{1}^{−}∧n} ⩾ (U_{V_{2}^{+}})_{−V_{2}^{−}∧n} ⩾ 0,

and the assertions of (a) follow.

In the next chapter we will study propagators that consist of integral operators. The next result implies that then the corresponding absorption propagators consist of integral operators, too.

2.7 Proposition. Let V be weakly U-admissible. Let t, s ∈ J satisfy t > s, and assume that U(t, s) is an integral operator. Then U_{V} (t, s) is an integral operator.

Proof. For n ∈ ℕ let V_{n} := V ∨ (−n). Then 0 ⩽ U_{V_{n}}(t, s) ⩽ e^{n(t−s)}U(t, s) by Proposition 2.6(a) and Remark 2.2, so U_{V_{n}}(t, s) is an integral operator by a theorem of Bukhvalov and Schep, [AbAl02; Thm. 5.9]. By the monotone convergence theorem, this implies the assertion since U_{V_{n}}(t, s) ↑ U_{V} (t, s) by Remark 2.5(a).

The major issues for the remainder of the section will be to investigate the following two questions. Firstly, to what extent does the definition of U_{V} depend on the decomposition of V into a difference of positive potentials (see Theorem 2.17 and Remark 2.18(a) below), and secondly, when does convergence a.e. of a sequence of weakly U-admissible potentials imply convergence of the corresponding perturbed propagators (Theorem 2.21 below)?

For this we first show, for perturbations of one sign, that one can use different approximations than in (2.2) and (2.3), and that one can iterate perturbations. In the context of C_{0}-semigroups, the former problem has already been addressed in [Voi86; Props. A.1 and A.2], the latter in [Man01; Prop. 4.1.35]. For a sequence (U_{n}) of propagators on L_{p}(μ) we write U_{n} → U strongly or U = s-lim_{n→∞} U_{n} if U_{n}(t, s) → U(t, s) strongly for all (t, s) ∈ D_{J}.

2.8 Proposition. (a) Let V, V_{n}: J × Ω → ℝ be locally measurable with V ⩽ V_{n} ⩽ 0 for all n ∈ ℕ, and assume that V_{n} → V locally a.e. as n → ∞. Then the following are equivalent:

(i) V is weakly U-admissible; (ii) the V_{n} are weakly U-admissible, and s-lim_{n→∞} U_{V_{n}} exists and is locally bounded.

If V is weakly U-admissible then U_{V_{n}} → U_{V} strongly.

(b) Let V_{1}, V_{2} : J ×Ω → (−∞, 0] be locally measurable, and assume that V_{1} is weakly Uadmissible. Then V_{1} +V_{2} is weakly U-admissible if and only if V_{2} is weakly U_{V_{1}}-admissible, and U_{V_{1}+V_{2}} = (U_{V_{1}})_{V_{2}} in this case.

Proof. (a) First observe that by Remark 2.5(b) we can assume without loss of generality that V_{n} → V pointwise: Just replace each of the V_{n} with V on the local null set on which the convergence does not hold.

Assume that (ii) holds, and let m ∈ ℕ. For n ∈ ℕ let V_{n}^{m} := V_{n} ∨ (−m), and let V^{m} := V ∨ (−m). Then V_{n}^{m}(t, ·) → V^{m}(t, ·) strongly as n → ∞, for all t ∈ J, where we consider V_{n}^{m}(t, ·) and V^{m}(t, ·) as multiplication operators in L(L_{p}(μ)). By Proposition 1.23 we thus obtain that

U_{V^{m}} = s-lim_{n→∞} U_{V_{n}^{m}} ⩽ s-lim_{n→∞} U_{V_{n}}.  (2.5)

Since the right-hand side is locally bounded, it follows that V is weakly U-admissible.

Now assume that (i) holds; then the V_{n} are weakly U-admissible by Proposition 2.6(a). To complete the proof of (a), we show that U_{V_{n}} → U_{V} strongly. Let (t, s) ∈ D_{J} and f ∈ L_{p}(μ). Let ε > 0. Then there exists m ∈ ℕ such that

||U_{V} (t, s)f − U_{V^{m}}(t, s)f||_{p} ⩽ ε.  (2.6)

Moreover, by (2.5) we obtain N ∈ ℕ such that

||U_{V^{m}}(t, s)f − U_{V_{n}^{m}}(t, s)f||_{p} ⩽ ε (n ⩾ N).  (2.7)

From U_{V} (t, s)f ⩾ U_{V_{n}}(t, s)f ⩾ U_{V_{n}^{m}}(t, s)f we conclude that

||U_{V} (t, s)f − U_{V_{n}}(t, s)f||_{p} ⩽ ||U_{V} (t, s)f − U_{V_{n}^{m}}(t, s)f||_{p} ⩽ 2ε

for all n ⩾ N. Therefore, U_{V_{n}}(t, s)f → U_{V} (t, s)f as n → ∞.

(b) First assume that V_{2} is bounded. Then we only have to show the identity U_{V_{1}+V_{2}} = (U_{V_{1}})_{V_{2}}. By Remark 2.2 we have U_{V_{1}∨(−n)+V_{2}} = (U_{V_{1}∨(−n)})_{V_{2}} for all n ∈ ℕ. Moreover, by part (a) we obtain that U_{V_{1}∨(−n)+V_{2}} → U_{V_{1}+V_{2}} strongly as n → ∞, and from Proposition 1.23 it follows that (U_{V_{1}∨(−n)})_{V_{2}} → (U_{V_{1}})_{V_{2}} strongly as n → ∞. This implies the desired identity.

If V_{2} is unbounded then by the above we have U_{V_{1}+V_{2}∨(−n)} = (U_{V_{1}})_{V_{2}∨(−n)} for all n ∈ ℕ. Moreover, s-lim_{n→∞}(U_{V_{1}})_{V_{2}∨(−n)} exists (and equals (U_{V_{1}})_{V_{2}}) if and only if V_{2} is weakly U_{V_{1}}- admissible, and by part (a), s-lim_{n→∞} U_{V_{1}+V_{2}∨(−n)} exists (and equals U_{V_{1}+V_{2}}) if and only if V_{1} + V_{2} is weakly U-admissible. This completes the proof of (b).

2.9 Remark. In the situation of Proposition 2.8(a) assume that U is strongly continuous and that V is U-admissible, i.e., U_{V} is strongly continuous. We show that then the strong convergence U_{V_{n}} → U_{V} is uniform on compact subsets of D_{J}. Assume without loss of generality that J is compact. Observe that the left-hand side of (2.6) is monotone in m if f ⩾ 0, and that it is a continuous function of (t, s) ∈ D_{J} since U_{V^{m}} is strongly continuous by Theorem 1.8. From Dini’s theorem we thus infer that m can be chosen independently of (t, s) ∈ D_{J} in (2.6). By Proposition 1.23 we obtain the same for the choice of N in (2.7), and the asserted uniform strong convergence follows.

For positive perturbations we obtain an analogous result; only the weak admissibility is not an issue in this case, which makes the formulation simpler. The proof is also analogous (but simpler) and therefore omitted. As in Remark 2.9 one sees that the convergence in part (a) is uniform on compact subsets of D_{J} if U is strongly continuous and V is U-admissible.

2.10 Proposition. (a) (cf. [RäSc99; Lemma 3.1(c)]) Let V, V_{n}: J × Ω → ℝ be locally measurable with 0 ⩽ V_{n} ⩽ V for all n ∈ ℕ, and assume that V_{n} → V locally a.e. as n → ∞. Then U_{V_{n}} → U_{V} strongly.

(b) Let V_{1}, V_{2} : J × Ω → [0, ∞) be locally measurable. Then U_{V_{1}+V_{2}} = (U_{V_{1}})_{V_{2}}.

The next lemma says in particular that one can interchange the order of perturbations in (2.4) if −V^{−} is weakly U-admissible.

2.11 Lemma. (cf. [LVV06; Prop. 2.3(b)]) Let V_{±} ⩾ 0, and assume that −V_{−} is weakly U-admissible. Then −V_{−} is weakly U_{V^{+}}-admissible, and (U_{V_{+}})_{−V_{−}} = (U_{−V_{−}})_{V_{+}}.

Proof. Let n ∈ ℕ. By Proposition 2.3(b) we have

0 ⩽ (U_{−V_{−}})_{V_{+}∧m} − (U_{−V_{−}∧n})_{V_{+}∧m} ⩽ U_{−V_{−}} − U_{−V_{−}∧n}  (2.8)

for all m ∈ ℕ. Moreover, (U_{−V_{−}∧n})_{V_{+}∧m} = (U_{V_{+}∧m})_{−V_{−}∧n} → (U_{V_{+}})_{−V ∧n} strongly as m → ∞, by Remark 2.2 and Proposition 1.23. Thus, letting m → ∞ in (2.8) yields

0 ⩽ (U_{−V_{−}})_{V_{+}} − (U_{V_{+}})_{−V_{−}∧n} ⩽ U_{−V_{−}} − U_{−V_{−}∧n}.  (2.9)

This implies the assertion since U_{−V_{−}∧n} → U_{−V_{−}} strongly as n → ∞.

2.12 Remark. Let V be weakly U-admissible; we show that then −V is weakly U_{V} - admissible and

(U_{V})_{−V} = (U_{V^{+}})_{−V^{+}} = s-lim_{n→∞} U_{V^{+}−V^{+}∧n} ⩽ U.

If V ⩾ 0 then for n ∈ ℕ we obtain by Proposition 2.10(b) that U_{V} = (U_{V −V ∧n})_{V ∧n}, so from Remark 2.2 it follows that (U_{V})_{−V ∧n} = U_{V −V ∧n} ⩽ U. This implies the claim in the case V ⩾ 0. In the general case, −V^{−} is weakly U_{V^{+}}-admissible, so

(U_{V})_{(−V)^{+}} = ((U_{V^{+}})_{−V^{−}})_{V^{−}} = U_{V^{+}}

by Proposition 2.13 below. It follows that −(−V)^{−} = −V^{+} is weakly (U_{V})_{(−V)^{+}}-admissible (i.e., −V is weakly U_{V} -admissible) and (U_{V})_{−V} = (U_{V^{+}})_{−V^{+}}.

The above observation leads to the notion of regularity that is crucial for the subsequent results. We say that V is U-regular if V is weakly U-admissible and (U_{V})_{−V} = U. This is in analogy with [Voi88; Sec. 3], where regularity is defined for positive potentials in the context of C_{0}-semigroups. Observe that, by Remark 2.12, V is U-regular if and only if V is weakly U-admissible and V^{+} is U-regular. We point out that in [Voi88], for V being U-regular it was also required that V is U-admissible. We do not know if, in the case of strongly continuous propagators, U-regularity of V implies U-admissibility.

2.13 Proposition. (cf. [Voi88; Prop. 3.3(b)]) Let V ⩾ 0 be such that −V is weakly U-admissible. Then (U_{V})_{−V} = (U_{−V})_{V} = U, i.e., V and −V are U-regular.

Proof. The first identity is a special case Lemma 2.11. For the second identity we use (2.8) with n = m: Since (U_{−V ∧n})_{V ∧n} = U by Remark 2.2, it follows from (2.8) that (U_{−V})_{V ∧n} → U strongly as n → ∞, i.e., (U_{−V})_{V} = U.

2.14 Example. Let n ⩾ 2, and let U be the heat propagator on L_{1}(ℝ^{n}) with parameter interval ℝ, i.e., U(t, s) = e^{(t−s)Δ} for all (t, s) ∈ D_{ℝ}. For the oscillating potential V defined by

V (t, x) := e^{1/|x|} sin e^{1/|x|}

it follows from [Stu92a; Cor. 5.5] that cV is U-admissible for all c ∈ ℝ and hence also Uregular, by Proposition 2.13. We point out that in this example V is admissible although −V^{−} is far from being (weakly) admissible.

Next we compare regularity with respect to different propagators. The corresponding results for C_{0}-semigroups are shown in [LiMa97; Cor. 1.15] and [Voi88; Prop. 3.4].

2.15 Lemma. (a) Let Ũ U be a strongly measurable propagator on L_{p}(μ) with parameter interval J, and assume that 0 ⩽ Ũ U ⩽ U. Then any U-regular potential V is also Ũ U-regular.

(b) Let V_{±} ⩾ 0 be such that −V_{−} is weakly U-admissible. Then V_{+} is U-regular if and only if V_{+} is U_{−V_{−}}-regular.

Proof. (a) By Proposition 2.6(b), V is weakly Ũ U-admissible, so we only have to show that V^{+} is Ũ U-regular. In other words, we can assume without loss of generality that V ⩾ 0. Then by Proposition 2.6 we have

0 ⩽ Ũ U − Ũ U_{V −V ∧n} ⩽ U − U_{V −V ∧n},

and from Remark 2.12 it follows that 0 ⩽ Ũ U − (Ũ U_{V})_{−V} ⩽ U − (U_{V})_{−V} = 0.

(b) Assume that V_{+} is U-regular. Then by Lemma 2.11 and Proposition 2.8(b) we obtain that

((U_{−V_{−}})_{V_{+}})_{−V_{+}} = ((U_{V_{+}})_{−V_{−}})_{−V_{+}} = ((U_{V_{+}})_{−V_{+}})_{−V_{−}} = U_{−V_{−}},

so V_{+} is U_{−V_{−}}-regular. The other implication follows from part (a).

2.16 Corollary. (cf. [Voi88; Prop. 3.3(a)]) The set {V ⩾ 0; V is U-regular} is a solid convex cone.

Proof. Let V_{1}, V_{2} ⩾ 0 be U-regular. Then V_{2} is U_{V_{1}}-regular by Lemma 2.15(a), so from Propositions 2.8(b) and 2.10(b) we deduce that (U_{V_{1}+V_{2}})_{−V_{1}−V_{2}} = (U_{V_{1}})_{−V_{1}} = U.

Let now V_{1} ⩾ 0 be U-regular, and let 0 ⩽ V_{2} ⩽ V_{1}. Then V_{2} − V_{2} ∧ n ⩽ V_{1} − V_{1} ∧ n and hence U_{V_{1}−V_{1}∧n} ⩽ U_{V_{2}−V_{2}∧n} ⩽ U for all n ∈ ℕ, so it follows from Remark 2.12 that V_{2} is U-regular.

We can now characterise which decompositions V = V_{+} − V_{−} lead to U_{V} = (U_{V_{+}})_{−V_{−}}.

2.17 Theorem. Let V_{±} ⩾ 0 and V := V_{+} − V_{−}.

(a) If V is weakly U-admissible then −V_{−} is weakly U_{V_{+}}-admissible.

(b) The following are equivalent:

(i) V is weakly U-admissible and U_{V} = (U_{V_{+}})_{−V_{−}}; (ii) −V_{−} is weakly U_{V_{+}}-admissible and V_{+} ∧ V_{−} is U_{V^{+}}-regular.

Proof. Let W := V_{+} ∧ V_{−}; then V_{+} = V^{+} + W and V_{−} = V^{−} + W.

(a) By Proposition 2.10(b) and Lemma 2.11 we obtain that

(U_{V_{+}})_{−V^{−}} = ((U_{V^{+}})_{W} )_{−V^{−}} = ((U_{V^{+}})_{−V^{−}})_{W} = (U_{V})_{W} .

In particular, −W is weakly (U_{V_{+}})_{−V^{−}}-admissible. From Proposition 2.8(b) we conclude that −V_{−} is weakly U_{V_{+}}-admissible and (U_{V_{+}})_{−V_{−}} = ((U_{V_{+}})_{−V^{−}})_{−W} = ((U_{V})_{W} )_{−W}.

(b) Assume that (i) holds. Then by the above, −V_{−} is weakly U_{V_{+}}-admissible and U_{V} = ((U_{V})_{W} )_{−W}. Therefore, W is U_{V} -regular and hence U_{V^{+}}-regular by Lemma 2.15(a).

Conversely, if (ii) holds then by Proposition 2.10(b) we obtain that

U_{V^{+}} = ((U_{V^{+}})_{W} )_{−W} = (U_{V_{+}})_{−W} .

Since −V_{−} = −V^{−} − W is weakly U_{V_{+}}-admissible, we infer by Proposition 2.8(b) that −V^{−} is weakly (U_{V_{+}})_{−W} -admissible and

(U_{V_{+}})_{−V_{−}} = ((U_{V_{+}})_{−W} )_{−V^{−}} = (U_{V^{+}})_{−V^{−}} = U_{V} .

2.18 Remarks. (a) Let V_{±} ⩾ 0 be such that −V_{−} is weakly U_{V_{+}}-admissible, and let V = V_{+} − V_{−}. Then by Proposition 2.13, Corollary 2.16 and Lemma 2.15(a) we obtain that

−V_{−} is weakly U-admissible ⟹ V_{−} is U-regular
⟹ V_{+} ∧ V_{−} is U-regular ⟹ V_{+} ∧ V_{−} is U_{V^{+}}-regular,

and by Theorem 2.17, each of these properties implies that V is weakly U-admissible and U_{V} = (U_{V_{+}})_{−V_{−}}. If −V_{−} is weakly U-admissible then by Lemma 2.11 we also have U_{V} = (U_{−V_{−}})_{V_{+}}.

(b) Assume that U is strongly continuous and that V_{+} and −V_{−} are U-admissible. Then by (2.9) and Remark 2.9 one obtains that (U_{V_{+}})_{−V_{−}∧n} → (U_{−V_{−}})_{V_{+}} strongly, uniformly on compact subsets of D_{J}. Thus, (U_{−V_{−}})_{V_{+}} = U_{V} is strongly continuous, i.e., V is Uadmissible.

Next we show that the two notions of U-regularity that were introduced in [Voi86; Def. 2.12] and [Voi88; Def. 3.1] are equivalent.

2.19 Proposition. (cf. [Man01; Satz 4.1.54]) Let V ⩾ 0. Then U_{εV} → (U_{V})_{−V} strongly as ε → 0. In particular, V is U-regular if and only if U_{εV} → U strongly as ε → 0.

Proof. Let ε ∈ (0, 1). Then −εV is weakly U_{εV} -admissible by Remark 2.12. From Proposition 2.13 and Corollary 2.16 it follows that (1−ε)V is U_{εV} -regular. By Propositions 2.10(b) and 2.8(a) we conclude that

U_{εV} = ((U_{εV})_{(1−ε)V} )_{−(1−ε)V} = (U_{V})_{−(1−ε)V} → (U_{V})_{−V}

strongly as ε → 0.

For later use we note the following auxiliary result.

2.20 Lemma. Let V be U-regular. Then V^{+} and V^{−} are U-regular.

Proof. Remark 2.12 yields the U-regularity of V^{+}. Since −V^{−} is weakly U_{V^{+}}-admissible, V^{−} is U_{V^{+}}-regular by Proposition 2.13. By Lemma 2.15(b) it follows that V^{−} is (U_{V^{+}})_{−V^{+}}- regular and hence U-regular.

We conclude the section with an analogue of the non-monotone convergence result for semigroups [Voi88; Thm. 3.5].

2.21 Theorem. Let (V_{n}) be a sequence of potentials satisfying V_{n} → V locally a.e. as n → ∞ and −V_{−} ⩽ V_{n} ⩽ V +V_{+} for all n ∈ ℕ, where V_{±} ⩾ 0, −V_{−} is weakly U-admissible and V_{+} is U_{V} -regular. Then U_{V_{n}} → U_{V} strongly as n → ∞.

(Note that V and the V_{n} are weakly U-admissible since V_{n} ⩾ −V_{−} for all n ∈ ℕ and hence V ⩾ −V_{−}.)

2.22 Remarks. (a) In [Voi88; Thm. 3.5], in the context of C_{0}-semigroups, it was assumed more restrictively that V_{+} is U-regular. This implies U_{V} -regularity of V_{+} by Lemma 2.15. If V is not U-regular then V_{+} = cV^{+}, with c > 0, is an example of a U_{V} -regular potential that is not U-regular.

(b) The assumption −V_{−} ⩽ V_{n} ⩽ V + V_{+} can be replaced with the seemingly weaker assumption −V_{−} ⩽ V_{n} ⩽ V^{+} + V_{+}. Indeed, as in Remark 2.18(a) we obtain from the weak U-admissibility of −V_{−} that V^{−} is U_{V} -regular. Thus Ṽ V_{+} := V_{+} + V^{−} is U_{V} -regular by Corollary 2.16, and V + Ṽ V_{+} = V^{+} + V_{+}.

(c) If −V^{−} is weakly U-admissible and V_{n} = V^{(n)} := (V ∧ n) ∨ (−n), then the assumptions of Theorem 2.21 are satisfied with V_{−} = V^{−} and V_{+} = 0. It follows that in this case, our definition of the absorption propagator coincides with the classical definition

U_{V} = s-lim_{n→∞} U_{V^{(n)}}.

(d) Assume that U is strongly continuous, that −V_{−} and V are U-admissible and that V_{+} is U_{V} -admissible. Then a close inspection of the proof below shows that the strong convergence U_{V_{n}} → U_{V} is uniform on compact subsets of D_{J}; cf. Remark 2.9.

Proof of Theorem 2.21. In a first step we prove the assertion for the case V = V_{−} = 0. Then V_{n} ⩾ 0 and 0 ⩽ W_{n} := V_{+} − V_{n} ⩽ V_{+} for all n ∈ ℕ. Moreover, the W_{n} are U-regular by Corollary 2.16, and W_{n} → V_{+} locally a.e. as n → ∞. Therefore, by Remark 2.18(a) and Proposition 2.8(a) we obtain that

U_{V_{n}} = (U_{V_{+}})_{−W_{n}} → (U_{V_{+}})_{−V_{+}} = U (n → ∞).

In a second step we prove the assertion for the case V_{−} = 0. Then V, V_{n} ⩾ 0 for all n ∈ ℕ. For n ∈ ℕ and W_{n} := V_{n} − V_{n} ∧ V it follows from Propositions 2.10(b) and 2.6(b) that

0 ⩽ U_{V_{n}} − (U_{V})_{W_{n}} = (U_{V_{n}∧V})_{W_{n}} − (U_{V})_{W_{n}} ⩽ U_{V_{n}∧V} − U_{V} .  (2.10)

Observe that U_{V_{n}∧V} → U_{V} strongly as n → ∞ by Proposition 2.10(a). Moreover, W_{n} → 0 locally a.e. as n → ∞, 0 ⩽ W_{n} ⩽ V_{+} for all n ∈ ℕ and V_{+} is U_{V} -regular, so by the first step we obtain that (U_{V})_{W_{n}} → U_{V} strongly as n → ∞. From (2.10) we thus infer that U_{V_{n}} → U_{V} strongly as n → ∞.

To conclude the proof, we show that the general case can be reduced to the case V_{−} = 0. Let Ṽ V := V + V_{−} and Ṽ V_{n} := V_{n} + V_{−} for all n ∈ ℕ. Then Ṽ V_{n} → Ṽ V locally a.e. as n → ∞ and 0 ⩽ Ṽ V_{n} ⩽ Ṽ V + V_{+} for all n ∈ ℕ. Moreover, (U_{−V_{−}})Ṽ_{n} = U_{V_{n}} for all n ∈ ℕ and (U_{−V_{−}})Ṽ = U_{V} by Remark 2.18(a); in particular, V_{+} is (U_{−V_{−}})Ṽ-regular. Thus, (U_{−V_{−}})Ṽ_{n} → (U_{−V_{−}})Ṽ strongly as n → ∞ by the second step, and the proof is complete.

2.2 Miyadera class for potentials

Let (Ω, μ) be a measure space, let 1 ⩽ p < ∞, and let U be a locally bounded positive strongly measurable propagator on L_{p}(μ) with parameter interval J ⊆ ℝ. Let V : J ×Ω → ℝ be locally measurable. Then V (t) = V (t, ·) is locally measurable for all t ∈ J and hence defines a (closed) multiplication operator in L_{p}(μ). Thus, V is a (closed) Miyadera perturbation of U if and only if there exist α > 0 and γ ⩾ 0 such that

∫_{s}^{t} ||V (τ)U(τ, s)f||_{p} dτ ⩽ γ||f||_{p}  (2.11)

for all (t, s) ∈ D_{J} with t−s ⩽ α and all f ∈ L_{p}(μ). (Recall from Lemma 1.17 that X_{s} = X for closed Miyadera perturbations; moreover, by the monotone convergence theorem one sees that V (·)U(·, s)f is measurable on J_{⩾s} if it is a.e. defined.) Note that it suffices to require (2.11) for f ⩾ 0 since ||V (τ)U(τ, s)f||_{p} ⩽ ||V (τ)U(τ, s)|f|||_{p} for all (τ, s) ∈ D_{J}.

In the following theorem, which extends [LVV06; Thm. 2.8], we formulate a sufficient condition under which a potential V ⩾ 0 is U-regular, and U-admissible in case that U is strongly continuous. The latter generalises the strong continuity assertion in [RäSc99; Thm. 3.3].

2.23 Theorem. Let V ⩾ 0. Assume that for each s ∈ J there exists a dense sublattice X_{s} of L_{p}(μ) such that τ ↦ U(t, τ)V (τ)U(τ, s)f ∈ L_{p}(μ) is a.e. defined and integrable on (s, t), for all t ∈ J_{>s} and all f ∈ X_{s}. (This assumption is satisfied with X_{s} = L_{p}(μ) if V is a Miyadera perturbation of U.) Then V is U-regular, and

U_{V} (t, s)f = U(t, s)f − ∫_{s}^{t} U_{V} (t, τ)V (τ)U(τ, s)f dτ  (2.12)
= U(t, s)f − ∫_{s}^{t} U(t, τ)V (τ)U_{V} (τ, s)f dτ  (2.13)

for all (t, s) ∈ D_{J} and all f ∈ X_{s}. Moreover, if U is strongly continuous then V is U-admissible.

Proof. Let (t, s) ∈ D_{J} and f ∈ X_{s}. Then |f| ∈ X_{s}. For n ∈ ℕ and V_{n} := V ∧ n we have U_{V_{n}} ⩽ U and hence

|U_{V_{n}}(t, τ)V_{n}(τ)U(τ, s)f| ⩽ U(t, τ)V (τ)U(τ, s)|f|,
|U(t, τ)V_{n}(τ)U_{V_{n}}(τ, s)f| ⩽ U(t, τ)V (τ)U(τ, s)|f|

for all τ ∈ (s, t). Thus, by the dominated convergence theorem we can pass to the limit in

U_{V_{n}}(t, s)f = U(t, s)f − ∫_{s}^{t} U_{V_{n}}(t, τ)V_{n}(τ)U(τ, s)f dτ
= U(t, s)f − ∫_{s}^{t} U(t, τ)V_{n}(τ)U_{V_{n}}(τ, s)f dτ

to obtain (2.12) and (2.13).

The assumption of the theorem is also satisfied for εV in place of V, and U_{εV} ⩽ U for all ε > 0. By (2.12) we thus obtain that

U_{εV} (t, s)f = U(t, s)f − ε ∫_{s}^{t} U_{εV} (t, τ)V (τ)U(τ, s)f dτ → U(t, s)f

as ε → 0, for all (t, s) ∈ D_{J} and all f ∈ X_{s}, so Proposition 2.19 implies that V is U-regular. Finally, if U is strongly continuous, then from Proposition 1.10 we conclude that U_{V} is strongly continuous, i.e., V is U-admissible.

In the case of a small Miyadera perturbation V of U, a perturbed propagator can be defined either by Theorem 1.8 or by using absorption propagators. The following result shows that these two approaches lead to the same object.

2.24 Theorem. Let V be a Miyadera perturbation of U, with V^{−} Miyadera small. Then V is U-regular, (2.12) and (2.13) hold for all (t, s) ∈ D_{J} and all f ∈ L_{p}(μ), and U_{V} is the unique locally bounded strongly measurable propagator satisfying either of these Duhamel formulas. Moreover, if U is strongly continuous then V is U-admissible.

Proof. We only have to show that V is weakly U-admissible, that V is a Miyadera perturbation of U_{V}, and that the two Duhamel formulas hold. Then the U-regularity of V^{+} and hence of V follows from Theorem 2.23, the uniqueness assertion results from Corollary 1.19, and, as in the proof of Theorem 2.23, Proposition 1.10 implies that V is U-admissible if U is strongly continuous.

Assume first that V ⩽ 0. Then V is Miyadera small, so by Theorems 1.8 and 1.20(a) there exists a locally bounded strongly measurable propagator Ũ U_{V} satisfying (2.12) and (2.13) with Ũ U_{V} in place of U_{V}, for all (t, s) ∈ D_{J} and all f ∈ L_{p}(μ). Moreover, V is a Miyadera perturbation of Ũ U_{V} by Theorem 1.20(b). From Corollary 1.19 we infer that U is the unique locally bounded strongly measurable propagator satisfying

U(t, s)f = Ũ U_{V} (t, s)f − ∫_{s}^{t} U(t, τ)(−V)(τ)Ũ U_{V} (τ, s)f dτ ((t, s) ∈ D_{J}, f ∈ L_{p}(μ)).

By Theorem 2.23, the latter also holds with (Ũ U_{V})_{−V} in place of U, so we obtain that (Ũ U_{V})_{−V} = U. Therefore, V is weakly U-admissible, and U_{V} = Ũ U_{V} since −V is Ũ U_{V} -regular by Theorem 2.23. This proves the theorem in the case V ⩽ 0.

If V ⩾ 0 then V is a Miyadera perturbation of U_{V} since U_{V} ⩽ U; the case V ⩾ 0 is thus already covered by Theorem 2.23. In the general case, the above implies that −V^{−} and hence V is weakly U-admissible. Moreover, it follows from Theorem 1.20(b) that V is a Miyadera perturbation of U_{−V^{−}} and hence of U_{V} since U_{V} ⩽ U_{−V^{−}}. By Proposition 1.18 we conclude from the cases V ⩾ 0 and V ⩽ 0 that U_{V} = (U_{V^{+}})_{−V^{−}} satisfies (2.12) and (2.13) for all (t, s) ∈ D_{J} and all f ∈ L_{p}(μ), and the proof is complete.

We conclude this section by investigating under what conditions a weakly U-admissible potential is a Miyadera perturbation of U or of U_{V}. The results are limited to the case that p = 1 and that V is of one sign, and they demonstrate that Miyadera perturbations are particularly interesting in this case. Part (b) of the following proposition is a propagator version of [Voi86; Lemma 4.1]. We use the notation D_{α} := {(t, s) ∈ D_{J}; t − s ⩽ α}, for α ∈ (0, ∞].

2.25 Proposition. Let p = 1 and c > 0, and let V ⩾ 0. Assume that there exist M ⩾ 1 and α ∈ (0, ∞] such that ||U(t, s)|| ⩽ M for all (t, s) ∈ D_{α}.

(a) Assume that ||U_{V} (t, s)f||_{1} ⩾ c||f||_{1} for all (t, s) ∈ D_{α} and all 0 ⩽ f ∈ L_{1}(μ). Then V is a Miyadera perturbation of U with constants (α, \frac{M}{c} − 1).

(b) Assume that ||U(t, s)f||_{1} ⩾ c||f||_{1} for all (t, s) ∈ D_{α} and all 0 ⩽ f ∈ L_{1}(μ). Then V is a Miyadera perturbation of U_{V} with constants (α, \frac{M}{c}).

Proof. Let 0 ⩽ f ∈ L_{1}(μ) and (t, s) ∈ D_{α}, and for n ∈ ℕ let V_{n} := V ∧ n.

(a) For n ∈ ℕ we have U_{V} ⩽ U_{V_{n}}, so we can estimate

c ∫_{s}^{t} ||V_{n}(τ)U(τ, s)f||_{1} dτ ⩽ ∫_{s}^{t} ||U_{V_{n}}(t, τ)V_{n}(τ)U(τ, s)f||_{1} dτ

= ||∫_{s}^{t} U_{V_{n}}(t, τ)V_{n}(τ)U(τ, s)f dτ||_{1} = ||U(t, s)f − U_{V_{n}}(t, s)f||_{1}.

Letting n → ∞ we infer that

c ∫_{s}^{t} ||V (τ)U(τ, s)f||_{1} dτ ⩽ ||U(t, s)f||_{1} − ||U_{V} (t, s)f||_{1} ⩽ M||f||_{1} − c||f||_{1},

and the assertion follows.

(b) As above we obtain that

c ∫_{s}^{t} ||V_{n}(τ)U_{V} (τ, s)f||_{1} dτ ⩽ ∫_{s}^{t} ||U(t, τ)V_{n}(τ)U_{V_{n}}(τ, s)f||_{1} dτ
= ||U(t, s)f − U_{V_{n}}(t, s)f||_{1} ⩽ M||f||_{1}

for all n ∈ ℕ, and the assertion follows for n → ∞.

Recall from Proposition 2.13 that (U_{V})_{−V} = U if V ⩽ 0 is weakly U-admissible. Thus, applying Proposition 2.25 with U_{V} and −V in place of U and V, we immediately obtain the following result.

2.26 Corollary. Let p = 1 and c > 0, and let V ⩽ 0 be weakly U-admissible. Assume that there exist M ⩾ 1 and α ∈ (0, ∞] such that ||U_{V} (t, s)|| ⩽ M for all (t, s) ∈ D_{α}.

(a) Assume that ||U(t, s)f||_{1} ⩾ c||f||_{1} for all (t, s) ∈ D_{α} and all 0 ⩽ f ∈ L_{1}(μ). Then V is a Miyadera perturbation of U_{V} with constants (α, \frac{M}{c} − 1).

(b) Assume that ||U_{V} (t, s)f||_{1} ⩾ c||f||_{1} for all (t, s) ∈ D_{α} and all 0 ⩽ f ∈ L_{1}(μ). Then V is a Miyadera perturbation of U with constants (α, \frac{M}{c}).

2.27 Remarks. (a) Let p = 1 and assume that U is stochastic, i.e., ||U(t, s)f||_{1} = ||f||_{1} for all (t, s) ∈ D_{J} and all 0 ⩽ f ∈ L_{1}(μ). Let V ⩽ 0 be weakly U-admissible and assume that U_{V} is exponentially bounded. Then M_{α} := sup{||U_{V} (t, s)||; (t, s) ∈ D_{α}} < ∞ for all α > 0, and by Corollary 2.26(a), V is a Miyadera perturbation of U_{V} with constants (α, M_{α} − 1). In the context of C_{0}-semigroups, an analogous result has already been shown in [Voi86; Prop. 4.6].

Since U ⩽ U_{V}, the above implies that V is also a Miyadera perturbation of U with constants (α, M_{α} −1). The constant M_{α} −1 will be considerably improved in Remark 2.44(b) below, but for this we will have to assume that U is strongly continuous.

(b) Proposition 2.25(b) implies in particular that any V ⩾ 0 is a Miyadera perturbation of U_{V} with constants (∞, 1) if U is stochastic.

(c) It follows from Corollary 2.26(a) that there are no “borderline” weakly U-admissible potentials V ⩽ 0 if U is stochastic. More precisely, if K is a compact subinterval of J then U_{V} |_{D_{K}} is bounded, so there exists ε_{K} > 0 such that ε_{K}V |_{K×Ω} is a small Miyadera perturbation of U_{V} |_{D_{K}}. Using Theorem 2.24, we infer that there exists φ: J → (0, ∞) such that V_{φ}(t, x) := φ(t)V (t, x) defines a weakly U_{V} -admissible potential, and from Proposition 2.10(b) we obtain that V + V_{φ} is weakly U-admissible.

2.3 The backward Miyadera condition

Let (Ω, A, μ) be a measure space, and let 1 ⩽ p < ∞. In the case p = 1 we assume that μ is localisable, which is equivalent to L_{1}(μ)′ = L_{∞}(μ). Let U be a locally bounded positive strongly measurable propagator on L_{p}(μ) with parameter interval J ⊆ ℝ, and let V : J × Ω → ℝ be locally measurable.

2.28 Remark. Let A ∈ A be σ-finite, let t ∈ J, and let 0 ⩽ f ∈ L_{p′}(μ). Then the function

J_{⩽t} ∋ τ ↦ φ(τ) := ||𝟙_{A}V (τ)U(t, τ)′f||_{1} ∈ [0, ∞]

is measurable. Indeed, let (A_{n}) be a sequence of sets of finite measure such that 𝟙_{A_{n}} ↑ 𝟙_{A} a.e. Then g_{n}(τ) := |V (τ)| ∧ (n𝟙_{A_{n}}) defines a measurable function on g_{n}: J_{⩽t} → L_{p}(μ), for all n ∈ ℕ. It follows that

J_{⩽t} ∋ τ ↦ ||g_{n}(τ)U(t, τ)′f||_{1} = ∫ f U(t, τ)g_{n}(τ) dμ

is measurable, so the monotone convergence theorem implies that φ is measurable.

2.29 Proposition. Let γ ⩾ 0, and let t, s ∈ J satisfy t > s. Then the following conditions on V are equivalent.

(i) Let h: (s, t) → L_{p}(μ) be such that τ ↦ |V (τ)|h(τ) ∈ L_{p}(μ) is integrable on (s, t) and 0 ⩽ h(τ) ⩽ 1 a.e. for all τ ∈ (s, t). Then

||∫_{s}^{t} U(t, τ)|V (τ)|h(τ) dτ||_{∞} ⩽ γ.

(ii) For all 0 ⩽ g ∈ L_{p}(μ) one has

||∫_{s}^{t} U(t, τ)(|V (τ)| ∧ g) dτ||_{∞} ⩽ γ.

(As in (i), the integral on the left-hand side is an L_{p}-valued Bochner integral.) (iii) For all 0 ⩽ f ∈ L_{1}(μ) ∩ L_{p′}(μ) and all σ-finite sets A ∈ A one has

∫_{s}^{t} ||𝟙_{A}V (τ)U(t, τ)′f||_{1} dτ ⩽ γ||f||_{1}.

Proof. Without loss of generality assume that V ⩾ 0. Let h: (s, t) → L_{p}(μ) be as in condition (i). Then, for all 0 ⩽ f ∈ L_{1}(μ) ∩ L_{p′}(μ), the function φ_{f}: (s, t) → ℝ,

φ_{f}(τ) := ||V (τ)h(τ)U(t, τ)′f||_{1} = ∫_{Ω} f U(t, τ)V (τ)h(τ) dμ

is integrable, and

∫_{s}^{t} φ_{f}(τ) dτ = ∫_{Ω} f ∫_{s}^{t} U(t, τ)V (τ)h(τ) dτ dμ.

It follows that the inequality in (i) holds if and only if

∫_{s}^{t} ||V (τ)h(τ)U(t, τ)′f||_{1} dτ ⩽ γ||f||_{1} (0 ⩽ f ∈ L_{1}(μ) ∩ L_{p′}(μ)).  (2.14)

In the same way we obtain that the inequality in (ii) is equivalent to

∫_{s}^{t} ||(V (τ) ∧ g)U(t, τ)′f||_{1} dτ ⩽ γ||f||_{1} (0 ⩽ f ∈ L_{1}(μ) ∩ L_{p′}(μ))  (2.15)

(use the integrability of (s, t) ∋ τ ↦ V (τ) ∧ g ∈ L_{p}(μ)).

Now we derive the equivalence of conditions (i) to (iii) from the monotone convergence theorem.

(i) ⇒ (iii). Let A ∈ A be σ-finite, and let (A_{n}) be a sequence of sets of finite measure such that 𝟙_{A_{n}} ↑ 𝟙_{A} a.e. For n ∈ ℕ we define h_{n}: (s, t) → L_{p}(μ) by h_{n}(τ) := (nV (τ)^{−1}) ∧ 𝟙_{A_{n}}. Then V (τ)h_{n}(τ) = n ∧ (𝟙_{A_{n}}V (τ)) ↑ 𝟙_{A}V (τ) as n → ∞, for all τ ∈ (s, t), and τ ↦ V (τ)h_{n}(τ) ∈ L_{p}(μ) is integrable for all n ∈ ℕ. Thus, (iii) follows from (2.14).

(iii) ⇒ (ii). Let 0 ⩽ g ∈ L_{p}(μ). Then A := [g > 0] = {x ∈ Ω; g(x) > 0} is σ-finite and V (τ) ∧ g ⩽ 𝟙_{A}V (τ) for all τ ∈ (s, t), so (2.15) follows from (iii).

(ii) ⇒ (i). Let h: (s, t) → L_{p}(μ) be as in condition (i). By Remark 2.1(b) there exists a σ-finite set Ω_{h} ⊆ Ω such that V (τ)h(τ) = 𝟙_{Ω_{h}}V (τ)h(τ) a.e. for a.e. τ ∈ (s, t). Choose 0 ⩽ g ∈ L_{p}(μ) such that [g > 0] = Ω_{h}. Then V (τ) ∧ (ng) ↑ 𝟙_{Ω_{h}}V (τ) ⩾ V (τ)h(τ) a.e. as n → ∞, for a.e. τ ∈ (s, t), and hence (2.15) implies (2.14).

For p, q ∈ [1, ∞] and a linear operator B in L_{1}(μ) + L_{∞}(μ), we denote the norm of B as an operator from L_{p}(μ) to L_{q}(μ) by

||B||_{p→q} := sup{||Bf||_{q}; f ∈ L_{p}(μ) ∩ D(B), ||f||_{p} ⩽ 1} ∈ [0, ∞].

In the following we assume that the propagator U is locally L_{∞}-bounded, i.e., for all compact subintervals K of J there exists M_{K} ⩾ 1 such that ||U(t, s)||_{∞→∞} ⩽ M_{K} for all (t, s) ∈ D_{K}.

We say that V is a backward Miyadera perturbation of U (with constants (α, γ) ∈ (0, ∞] × [0, ∞)) if the conditions of Proposition 2.29 are satisfied for all (t, s) ∈ D_{J} with t − s ⩽ α. This is a Desch-Schappacher type condition; in the case of the heat equation one easily shows that it coincides with the backward Kato condition mentioned in the introduction (see also Remark 3.21(d)). Observe that for exponentially bounded U any V ∈ L_{∞}(J × Ω) is an infinitesimally small backward Miyadera perturbation (which is defined in the obvious way).

2.30 Theorem. Assume that V is weakly U-admissible and that V^{−} is a backward Miyadera perturbation of U with constants (α, γ), γ < 1. Then U_{V} is locally L_{∞}-bounded. If ||U(t, s)||_{∞→∞} ⩽ M for all (t, s) ∈ D_{J} then

||U_{V} (t, s)||_{∞→∞} ⩽ \frac{M}{1−γ} e^{ω(t−s)} ((t, s) ∈ D_{J}),

where ω = \frac{1}{α} ln(1 + \frac{γ}{1−γ} M).

Proof. We only have to show the second assertion since for the first assertion we can assume without loss of generality that J is compact. Moreover, U_{V^{+}} ⩽ U and hence V^{−} is a backward Miyadera perturbation of U_{V^{+}} with constants (α, γ), so we can assume without loss of generality that V ⩽ 0. Then V_{n} := V ∨ (−n) is a backward Miyadera perturbation of U with constants (α, γ), for each n ∈ ℕ. Since the operators U_{V_{n}}(t, s) are positive and U_{V_{n}}(t, s)f ↑ U_{V} (t, s)f as n → ∞, for all (t, s) ∈ D_{J} and all 0 ⩽ f ∈ L_{p}(μ)∩L_{∞}(μ), we only have to estimate ||U_{V} (t, s)f||_{∞} for f ⩾ 0, and we can assume without loss of generality that V is bounded.

By Proposition 2.3(a) we have ||U_{V} (t, s)||_{∞→∞} ⩽ Me^{||V ||_{∞}(t−s)} for all (t, s) ∈ D_{J}. Given f ∈ L_{p}(μ) such that 0 ⩽ f ⩽ 1 a.e., we are going to prove that

c_{j} := sup\{||U_{V} (t, s)f||_{∞}; (t, s) ∈ D_{J}, t − s ⩽ jα\} ⩽ \frac{M}{1−γ} (1 + \frac{γ}{1−γ} M)^{j−1}  (2.16)

for all j ∈ ℕ. As in the proof of Proposition 1.14, the assertion then follows by choosing j such that (j − 1)α ⩽ t − s ⩽ jα. Let n ∈ ℕ and assume that (2.16) holds for all j < n. Let (t, s) ∈ D_{J} with t − s ⩽ nα, and let t_{j} := s + \frac{j}{n}(t − s) for j = 0, …, n. Applying condition (i) of Proposition 2.29 with h = U_{V} (·, s)f: (t_{j−1}, t_{j}) → L_{p}(μ), 0 ⩽ h(τ) ⩽ c_{j} a.e. for all τ ∈ (t_{j−1}, t_{j}), we obtain that

||∫_{t_{j−1}}^{t_{j}} U(t_{j}, τ)V (τ)U_{V} (τ, s)f dτ||_{∞} ⩽ γc_{j}

for j = 1, …, n. From the second Duhamel formula (2.13) we infer that

||U_{V} (t, s)f||_{∞} ⩽ ||U(t, s)f||_{∞} + ∑_{j=1}^{n} ||∫_{t_{j−1}}^{t_{j}} U(t, t_{j})U(t_{j}, τ)V (τ)U_{V} (τ, s)f dτ||_{∞}

⩽ M + M ∑_{j=1}^{n−1} γc_{j} + γc_{n},

and as in the proof of Proposition 1.14 we conclude that (2.16) holds for j = n.

2.31 Remarks. (a) The local L_{∞}-boundedness of U implies that U(t, s)′|_{L_{1}(μ)∩L_{p′}(μ)} extends to a bounded operator U′(s, t) on L_{1}(μ), for each (t, s) ∈ D_{J}. We thus obtain a locally bounded backward propagator U′ on L_{1}(μ) with parameter interval J, i.e., a function

U′: {(s, t) ∈ J × J; s ⩽ t} → L(L_{1}(μ))

that satisfies

U′(t, t) = I, U′(r, t) = U′(r, s) U′(s, t) (r ⩽ s ⩽ t in J).

(We refer to [Gul02; Def. 3] for this definition.) Note that taking the adjoint of a propagator is connected with time reversal. Moreover, observe that U′ is (separately and jointly) weakly measurable. Therefore, U′ is a strongly measurable backward propagator if L_{1}(μ) is separable. If one only assumes μ to be σ-finite then U′ need not be strongly measurable; see Example 2.32(a) below.

(b) If μ is σ-finite then condition (iii) in Proposition 2.29 simplifies to

∫_{s}^{t} ||V (τ)U′(τ, t)f||_{1} dτ ⩽ γ||f||_{1} (0 ⩽ f ∈ L_{1}(μ) ∩ L_{p′}(μ)).  (2.17)

It follows from Remark 2.1(b) that the same is true if U′(·, t) is strongly measurable. In Example 2.32(b) below we will show that one cannot omit the indicator function 𝟙_{A} in the general case. We point out that in [LVV06; Sec. 3.1], strong continuity of U′ was required for the backward Miyadera condition.

(c) Assume that U′ is strongly measurable. As observed in part (b), V is a backward Miyadera perturbation of U with constants (α, γ) if and only if (2.17) holds for all (t, s) ∈ D_{J} with t − s ⩽ α. This can also be expressed as V being a Miyadera perturbation of the backward propagator U′ with constants (α, γ) (cf. (2.11)), or equivalently, as (−J) × Ω ∋ (t, x) ↦ V (−t, x) being a Miyadera perturbation of the associated (forward) propagator

D_{−J} ∋ (t, s) ↦ U′(−t, −s).

If γ < 1 then from the above it is clear how the perturbed backward propagator (U′)_{V} is defined. Following the last step in the proof of [LVV06; Prop. 3.6] one can show that (U′)_{V} = (U_{V})′ if additionally V is a small Miyadera perturbation of U. For a weakly U-admissible potential V it follows that U_{V} is locally L_{∞}-bounded if and only if V is weakly U′-admissible, and again (U′)_{V} = (U_{V})′ in this case.

2.32 Example. (a) Let Ω := [0, 1]^{[0,1]}, and let μ be the canonical product probability measure on Ω. Then L_{1}(μ) is not separable since any function y ↦ f(y) in L_{1}(μ) depends only on countably many of the variables y_{s}, s ∈ [0, 1]. Let J := [0, 1]. We define a propagator U: D_{J} → L(L_{1}(μ)) by U(t, t) := I for all t ∈ J and

U(t, s)f := ∫_{Ω} 2y_{s}f(y) dμ(y) · 𝟙_{Ω} (0 ⩽ s < t ⩽ 1, f ∈ L_{1}(μ)).

Then U(t, s)𝟙_{Ω} = 𝟙_{Ω} for all (t, s) ∈ D_{J}, and it follows that U is a propagator. Moreover, U(t, s)f = ∫_{Ω} f(y) dμ(y) · 𝟙_{Ω} =: P f if t > s and f ∈ L_{1}(μ) does not depend on the variable y_{s}, so for t ∈ J we obtain that U(t, s)f = P f for a.e. s ∈ J_{⩽t}. This implies that U is a strongly measurable propagator. Note, however, that the function (s, y) ↦ 2y_{s} is not measurable on J × Ω since it depends on all the variables y_{s}.

Now observe that ||U(t, s)||_{∞→∞} ⩽ 1 for all (t, s) ∈ D_{J} and that U′ is given by U′(t, t) = I for all t ∈ J and

U′(s, t)f(x) = 2x_{s} ∫_{Ω} f(y) dμ(y) (0 ⩽ s < t ⩽ 1, f ∈ L_{1}(μ), x ∈ Ω).

Therefore, if t > 0 and ∫_{Ω} f(y) dy ≠ 0, then U′(·, t)f is not separably valued and hence not measurable.

(b) As in Remark 2.1(a) let J := Ω := [0, 1], let μ be the counting measure on Ω, and let V : J × Ω → ℝ be the indicator function of the diagonal {(t, t) ∈ J × Ω; 0 ⩽ t ⩽ 1}. We define U: D_{J} → L(L_{1}(μ)) by U(t, t) := I for all t ∈ J and

U(t, s)f := (f(s) + f(0))δ_{0} (0 ⩽ s < t ⩽ 1, f ∈ L_{1}(μ)).

One easily checks that U is a propagator on L_{1}(μ). Moreover, ||U(t, s)||_{∞→∞} ⩽ 2 for all (t, s) ∈ D_{J}.

Since V = 0 locally a.e., V is a backward Miyadera perturbation of U with constants (∞, 0); in fact U_{V} = U by Remark 2.1(a). Let now t ∈ J and 0 ⩽ f ∈ L_{1}(μ) ∩ L_{∞}(μ). Then V (τ)U(t, τ)′f = V (τ)f(0)(δ_{0} + δ_{τ}) = f(0)δ_{τ} for all 0 < τ < t, so

∫_{0}^{t} ||V (τ)U(t, τ)′f||_{1} dτ = ∫_{0}^{t} |f(0)| dτ = t|f(0)|.

This shows that in general one cannot omit the indicator function 𝟙_{A} in condition (iii) of Proposition 2.29.

2.4 Logarithmically convex functions

The ideas presented in this section play a central role for this thesis; they are the basis for the interpolation inequalities of the next section. The following definition is motivated by the paper [Haa07].

Let X be an ordered vector space, i.e., X is a (real or complex) vector space endowed with a proper convex cone X_{+} of positive elements, where X_{+} being proper means that X_{+} ∩(−X_{+}) = {0}. Let D be a non-empty subset of ℝ. We say that a function f: D → X is logarithmically convex if

f(ξ_{θ}) ⩽ (1 − θ)r^{−θ}f(ξ_{0}) + θr^{1−θ}f(ξ_{1}) (r > 0)  (2.18)

for all ξ_{0}, ξ_{1} ∈ D and all θ ∈ (0, 1) with ξ_{θ} := (1 − θ)ξ_{0} + θξ_{1} ∈ D. By choosing ξ_{θ} = ξ_{0} = ξ_{1} and r ≠ 1 (so that (1 − θ)r^{−θ} + θr^{1−θ} > 1) we see that a logarithmically convex function f takes its values in X_{+}.

In the following let (Ω, μ) be a measure space. The next lemma implies in particular that f: D → [0, ∞) is logarithmically convex if and only if ln ∘f is convex, where we use the convention ln 0 := −∞.

2.33 Lemma. Let M(μ) be the ordered vector space of all scalar-valued locally measurable functions on Ω, where functions are identified if they coincide locally a.e., and M(μ)_{+} = {f ∈ M(μ); f ⩾ 0 locally a.e.}.

(a) A function f: D → M(μ)_{+} is logarithmically convex if and only if

f(ξ_{θ}) ⩽ f(ξ_{0})^{1−θ}f(ξ_{1})^{θ}

locally a.e. for all ξ_{0}, ξ_{1} ∈ D and all θ ∈ (0, 1) with ξ_{θ} := (1 − θ)ξ_{0} + θξ_{1} ∈ D.

(b) For h ∈ M(μ) the function ξ ↦ e^{ξh} is logarithmically convex. If f, g: D → M(μ)_{+} are logarithmically convex, then ξ ↦ f(ξ)g(ξ) is logarithmically convex, too.

Proof. (a) follows from Young’s inequality: For a, b ⩾ 0 and θ ∈ (0, 1) we have

a^{1−θ}b^{θ} = (r^{−θ}a)^{1−θ}(r^{1−θ}b)^{θ} ⩽ (1 − θ)r^{−θ}a + θr^{1−θ}b (r > 0)

and

a^{1−θ}b^{θ} = inf{(1 − θ)r^{−θ}a + θr^{1−θ}b; 0 < r ∈ ℚ}.

(b) is an immediate consequence of part (a).

The next result, though being elementary, is important for the kernel estimates in the next chapter. We assume that L_{p}(μ) and L(L_{p}(μ)) are endowed with their natural orderings, in particular, L(L_{p}(μ))_{+} consists of the positive operators on L_{p}(μ).

2.34 Proposition. Let p ∈ [1, ∞), and let T: D → L(L_{p}(μ)). Then the following are equivalent:

(i) T is logarithmically convex; (ii) ξ ↦ T (ξ)f is logarithmically convex for all f ∈ L_{p}(μ)_{+}; (iii) ξ ↦ ⟨T (ξ)f, g⟩ is logarithmically convex for all f ∈ L_{p}(μ)_{+}, g ∈ L_{p′}(μ)_{+}.

Assume, in addition, that μ is σ-finite and that the operators T (ξ) are integral operators with integral kernels k_{ξ}. Then (i)–(iii) hold if and only if D ∋ ξ ↦ k_{ξ} ∈ M(μ ⊗ μ) is logarithmically convex.

Proof. The equivalences are immediate from the following two facts: A function h ∈ L_{p}(μ) is in L_{p}(μ)_{+} if and only if ⟨h, g⟩ ⩾ 0 for all g ∈ L_{p′}(μ)_{+}, and an integral operator is positive if and only if its integral kernel is ⩾ 0 a.e.

2.35 Remarks. (a) We find it worth emphasizing that the conclusion from the logarithmic convexity of T to the logarithmic convexity of ξ ↦ k_{ξ} is entirely elementary. In the literature, δ-sequences (f_{n}), (g_{n}) and continuity of the kernel or Lebesgue points of the kernel are sometimes used instead; see, e.g., [Sim82; proof of Thm. B.6.7] and [LiSo03; proof of Prop. 2.8].

(b) Clearly, if f: D → X is logarithmically convex and T: X → X is a positive operator, then T ∘ f is logarithmically convex. In the case X = L_{p}(μ) we can prove the following more general result.

2.36 Lemma. Let p ∈ [1, ∞), and let f: D → L_{p}(μ) and T: D → L(L_{p}(μ)) be logarithmically convex. Then D ∋ ξ ↦ T (ξ)f(ξ) ∈ L_{p}(μ) is logarithmically convex, too.

Proof. Let ξ_{0}, ξ_{1} ∈ D and θ ∈ (0, 1) with ξ_{θ} := (1 − θ)ξ_{0} + θξ_{1} ∈ D. Let g := f(ξ_{0}) and h := f(ξ_{1}). Then f(ξ_{θ}) ⩽ g^{1−θ}h^{θ} by Lemma 2.33(a), so the assertion follows if we show that

T (ξ_{θ})(g^{1−θ}h^{θ}) ⩽ (1 − θ)r^{−θ}T (ξ_{0})g + θr^{1−θ}T (ξ_{1})h (r > 0).  (2.19)

For the proof of this inequality we can assume without loss of generality that g and h are simple functions,

g = ∑_{j=1}^{m} a_{j}𝟙_{A_{j}}, h = ∑_{j=1}^{m} b_{j}𝟙_{A_{j}},

with m ∈ ℕ, a_{j}, b_{j} ⩾ 0 and pairwise disjoint measurable sets A_{j} of finite measure (j = 1, …, m). By Proposition 2.34, (i)⇒(ii), and Lemma 2.33(a), the logarithmic convexity of T implies that

a_{j}^{1−θ}b_{j}^{θ}T (ξ_{θ})𝟙_{A_{j}} ⩽ (a_{j}T (ξ_{0})𝟙_{A_{j}})^{1−θ}(b_{j}T (ξ_{1})𝟙_{A_{j}})^{θ}
⩽ (1 − θ)r^{−θ}a_{j}T (ξ_{0})𝟙_{A_{j}} + θr^{1−θ}b_{j}T (ξ_{1})𝟙_{A_{j}}

for all r > 0 and j = 1, …, m. Summation over j yields (2.19), and the proof is complete.

The following consequence of Lemma 2.36 will be applied in Section 3.1 to show ultracontractivity of perturbed propagators. It replaces the use of the Stein interpolation theorem in [LVV06]. As before, if B is a linear operator in L_{1}(μ) + L_{∞}(μ), then ||B||_{p→q} denotes the norm of B considered as as operator from L_{p}(μ) to L_{q}(μ).

2.37 Corollary. Let p ∈ [1, ∞), and let T: D → L(L_{p}(μ)) be logarithmically convex. Let ξ_{0}, ξ_{1} ∈ D and p_{0}, p_{1}, q_{0}, q_{1} ∈ [1, ∞], and assume that ||T (ξ_{j})||_{p_{j}→q_{j}} < ∞ for j = 0, 1. Let θ ∈ (0, 1) be such that ξ_{θ} := (1 − θ)ξ_{0} + θξ_{1} ∈ D, and define p_{θ}, q_{θ} ∈ [1, ∞] by

\frac{1}{p_{θ}} = \frac{1 − θ}{p_{0}} + \frac{θ}{p_{1}} ,  \frac{1}{q_{θ}} = \frac{1 − θ}{q_{0}} + \frac{θ}{q_{1}} .  (2.20)

Then

||T (ξ_{θ})||_{p_{θ}→q_{θ}} ⩽ ||T (ξ_{0})||_{p_{0}→q_{0}}^{1−θ}||T (ξ_{1})||_{p_{1}→q_{1}}^{θ}.

If, in addition, ρ_{1}, ρ_{2} : D → M(μ)_{+} are logarithmically convex then

||ρ_{1}(ξ_{θ})T (ξ_{θ})ρ_{2}(ξ_{θ})||_{p_{θ}→q_{θ}} ⩽ ||ρ_{1}(ξ_{0})T (ξ_{0})ρ_{2}(ξ_{0})||_{p_{0}→q_{0}}^{1−θ}||ρ_{1}(ξ_{1})T (ξ_{1})ρ_{2}(ξ_{1})||_{p_{1}→q_{1}}^{θ},

where ρ_{j}(ξ) is considered as a multiplication operator, for j = 1, 2 and ξ ∈ D.

Proof. Let g ∈ L_{1}(μ)_{+} be bounded, μ(spt g) < ∞ and ||g||_{1} ⩽ 1. Define f: [0, 1] → L_{p}(μ) by f(τ) := g^{1/p_{τ}} 𝟙_{spt g}, where p_{τ} is defined as in (2.20). (The indicator function is only needed for the case p_{τ} = ∞.) Then f is logarithmically convex, so from Lemma 2.36 we infer that

T (ξ_{θ})f(θ) ⩽ (T (ξ_{0})f(0))^{1−θ}(T (ξ_{1})f(1))^{θ}.

By Hölder’s inequality and the estimate ||f(j)||_{p_{j}} ⩽ 1 for j = 0, 1 it follows that

||T (ξ_{θ})f(θ)||_{q_{θ}} ⩽ ||T (ξ_{0})f(0)||_{q_{0}}^{1−θ}||T (ξ_{1})f(1)||_{q_{1}}^{θ} ⩽ ||T (ξ_{0})||_{p_{0}→q_{0}}^{1−θ}||T (ξ_{1})||_{p_{1}→q_{1}}^{θ} =: c.

In the case p_{θ} < ∞ we thus have shown ||T (ξ_{θ})h||_{q_{θ}} ⩽ c for all bounded functions h ∈ L_{p}(μ)_{+} with μ(spt h) < ∞ and ||h||_{p_{θ}} ⩽ 1; in the case p_{θ} = ∞ we have shown ||T (ξ_{θ})𝟙_{A}||_{q_{θ}} ⩽ c for all measurable sets A with μ(A) < ∞. This implies the first assertion since T (ξ_{θ}) is a positive operator.

For the proof of the second assertion let

A_{n} := {x ∈ Ω; ρ_{k}(ξ_{j})(x) ⩽ n (k = 1, 2, j = 0, 1)}

for all n ∈ ℕ. Then [0, 1] ∋ θ ↦ 𝟙_{A_{n}}ρ_{1}(ξ_{θ})T (ξ_{θ})ρ_{2}(ξ_{θ})𝟙_{A_{n}} ∈ L(L_{p}(μ)) is logarithmically convex for all n ∈ ℕ, by Lemma 2.36 and Proposition 2.34, and 𝟙_{A_{n}} ↑ 𝟙_{Ω} as n → ∞. Thus, by the monotone convergence theorem, the second assertion follows from the first one.

2.5. INTERPOLATION INEQUALITIES FOR ABSORPTION PROPAGATORS 49

2.5 Interpolation inequalities for absorption propagators

In this section we assume that U is a positive strongly continuous propagator on L_{p}(μ) with parameter interval J ⊆ ℝ, where (Ω, μ) is a measure space and p ∈ [1, ∞). We do not know if the subsequent results are true without the strong continuity assumption. The following is the first main result of the section; for a discussion of its assumptions see Remark 2.41(a) below.

2.38 Theorem. Let V_{0}, V_{1} : J × Ω → ℝ be weakly U-admissible potentials, and assume that V_{0}^{+} ∧ V_{1}^{−} and V_{0}^{−} ∧ V_{1}^{+} are U-regular. Then V_{θ} := (1 − θ)V_{0} + θV_{1} is weakly Uadmissible, for every θ ∈ (0, 1), and [0, 1] ∋ θ ↦ U_{V_{θ}} is logarithmically convex.

By Propositions 2.7 and 2.34 we immediately obtain the following consequence of Theorem 2.38, which will be crucial for the kernel estimates in the next chapter.

2.39 Corollary. In addition to the assumptions of Theorem 2.38 suppose that U(t, s) is an integral operator, for some t, s ∈ J with t > s. Then U_{V_{θ}}(t, s) is an integral operator with kernel p_{t,s}^{V_{θ}}, for every θ ∈ [0, 1], and [0, 1] ∋ θ ↦ p_{t,s}^{V_{θ}} is logarithmically convex.

2.40 Remark. One of the first results on interpolation of integral kernels is proved in [HeSl78; Thm. 8.2], for translation invariant ultracontractive positive selfadjoint C_{0}-semi-groups. There the perturbation V_{1} is allowed to contain a form small distributional part, and V_{0} = 0. The proof is based on the Trotter product formula, as our proof below, but it additionally involves kernel techniques.

Several different methods have since been used to prove versions of the above results. For Schrödinger semigroups it is well-known that the Feynman-Kac formula and Hölder’s inequality in path space can be used to obtain interpolation inequalities; see, e.g., [Sim82; Lemma B.4.1]. In the rather general framework of perturbation of positive C_{0}-semi-groups on a Banach lattice X by operators from the centre of X, logarithmic convexity of θ ↦ U_{V_{θ}} is implicit in [Voi88; top of p. 121]; the proof is based on the three lines theorem. In [ArDe06; Thm. 2.4], the case of positive C_{0}-semigroups on L_{p}-spaces and negative admissible perturbations is treated by means of the Gelfand-Naimark theorem.

The method of [HeSl78] is generalised to the non-autonomous situation in [MiSe03; Sec. 2], under the assumption that the propagator U has local infinitesimal generators that are sectorial, that V_{0} = 0, and that −V_{1}^{−} is a small Miyadera perturbation of U. General positive strongly continuous propagators are investigated in [LVV06; Prop. 3.8]; the proof given there is based on the three lines theorem as in [Voi88], and again the V_{j}^{−} are assumed to be Miyadera small.

All the above results require at least some smallness of the perturbations or admissibility of the negative parts, and mostly V_{0} = 0 is assumed. Much more general perturbations are allowed in [Stu93; Lemma 2.20], which deals with Schrödinger semigroups on Riemannian manifolds with (possibly highly oscillating) signed smooth measures. (We refer to [Stu93; top of p. 326] for an explanation where the smoothness assumption comes into play.) The assumption that the measures are smooth corresponds to our regularity assumption. Our definitions of weak admissibility and of regularity in Section 2.1 were motivated by the desire to cover, in the context of potentials, the same generality in the perturbations.

Proof of Theorem 2.38. First assume that V_{0} and V_{1} are bounded. Then we only have to show that [0, 1] ∋ θ ↦ U_{V_{θ}}(t, s) ∈ L(L_{p}(μ)) is logarithmically convex for all (t, s) ∈ D_{J}, without loss of generality t > s. Let t_{0}, …, t_{n} ∈ [s, t] satisfy s = t_{0} < t_{1} < ⋯ < t_{n} = t, and for k = 1, …, n let

V_{θ,k} := ∫_{t_{k−1}}^{t_{k}} V_{θ}(τ) dτ = ∫_{t_{k−1}}^{t_{k}} V_{0}(τ) dτ + θ ∫_{t_{k−1}}^{t_{k}}(V_{1} − V_{0})(τ) dτ (θ ∈ [0, 1]).

Recall from Remark 2.35(b) that logarithmic convexity is stable under composition with positive operators. By Lemma 2.33(b) we thus obtain that

[0, 1] ∋ θ ↦ e^{V_{θ,n}}U(t_{n}, t_{n−1})e^{V_{θ,n−1}}U(t_{n−1}, t_{n−2}) ⋯ e^{V_{θ,1}}U(t_{1}, t_{0})

is logarithmically convex, so the assertion follows from the Trotter product formula (Theorem 1.26) since logarithmic convexity is also stable under strong convergence.

Next assume that V_{0} ⩾ c and V_{1} ⩾ c for some c ∈ ℝ. Let θ ∈ [0, 1]. Then V_{θ}^{n} := (1 − θ)(V_{0} ∧ n) + θ(V_{1} ∧ n) → V_{θ} as n → ∞, and c ⩽ V_{θ}^{n} ⩽ V_{θ} for all n ∈ ℕ. Therefore, U_{V_{θ}^{n}} → U_{V_{θ}} strongly as n → ∞ by Theorem 2.21, and the assertion follows from the first step.

For the proof of the general case let V_{θ,±} := (1 − θ)V_{0}^{±} + θV_{1}^{±},

V_{θ,−}^{n} := (1 − θ)(V_{0}^{−} ∧ n) + θ(V_{1}^{−} ∧ n) and V_{θ}^{n} := V_{θ,+} − V_{θ,−}^{n}

for all θ ∈ [0, 1] and all n ∈ ℕ. Then for θ ∈ [0, 1] we infer from Theorem 2.17(b) and the second step that

(U_{V_{θ,+}})_{−V_{θ,−}^{n}} = U_{V_{θ}^{n}} ⩽ (1 − θ)U_{V_{0}^{n}} + θU_{V_{1}^{n}} ⩽ (1 − θ)U_{V_{0}} + θU_{V_{1}}

for all n ∈ ℕ. By Proposition 2.8(a) it follows that −V_{θ,−} is weakly U_{V_{θ,+}}-admissible since V_{θ,−}^{n} ↑ V_{θ,−} as n → ∞ and (1 − θ)U_{V_{0}} + θU_{V_{1}} is locally bounded. Since V_{θ} = V_{θ,+} − V_{θ,−}, and the potential

V_{θ,+} ∧ V_{θ,−} = (θV_{1}^{+}) ∧ ((1 − θ)V_{0}^{−}) + (θV_{1}^{−}) ∧ ((1 − θ)V_{0}^{+})

is U-regular by Corollary 2.16, we conclude from Remark 2.18(a) that V_{θ} is weakly U-

admissible and U_{V_{θ}} = (U_{V_{θ,+}})_{−V_{θ,−}} = s-lim_{n→∞}(U_{V_{θ,+}})_{−V_{θ,−}^{n}}. Finally, θ ↦ (U_{V_{θ,+}})_{−V_{θ,−}^{n}} = U_{V_{θ}^{n}} is

logarithmically convex by the second step, so the proof is complete.

2.41 Remarks. (a) The assumption that V_{0}^{+} ∧V_{1}^{−} and V_{0}^{−} ∧V_{1}^{+} are U-regular is a rather technical condition. It is trivially satisfied if V_{0} = 0, so as a special case of Theorem 2.38 we obtain: If V is weakly U-admissible then θV is weakly U-admissible for all θ ∈ [0, 1], and [0, 1] ∋ θ ↦ U_{V_{θ}} is logarithmically convex.

For the lower bounds in the next chapter we will need another application of Theorem 2.38: There we assume that V_{1} = V is U-regular and that V_{0} := −εV is weakly U-admissible, for some ε > 0 (see Theorem 3.12); then it follows from Lemma 2.20 that the assumptions on V_{0} and V_{1} are satisfied.

(b) We use part (a) to show that Proposition 2.19 is valid for any weakly U-admissible potential V (not just for V ⩾ 0) if U is strongly continuous. For ε > 0 we have

U_{εV^{+}} ⩽ U_{εV} ⩽ (1 − ε)U + εU_{V} ,  (2.21)

where the second inequality is due to Theorem 2.38. If V is U-regular then both the righthand side and the left-hand side of (2.21) converge to U strongly, by Proposition 2.19, so U_{εV} → U strongly as ε → 0.

Assume now that V is weakly U-admissible, and let U_{0,V} := (U_{V})_{−V} = (U_{V^{+}})_{−V^{+}}. Then it follows from Proposition 2.13 and Lemma 2.15(b) that V^{+} and hence also V is U_{0,V} -regular. Moreover, using Theorem 2.17 one shows that (U_{0,V})_{εV^{+}} = U_{εV^{+}} for all ε > 0, so by the above we conclude that U_{εV} = (U_{0,V})_{εV} → U_{0,V} = (U_{V})_{−V} as ε → 0.

Whereas versions of Theorem 2.38 have long been known (see Remark 2.40), our second main result on logarithmic convexity and its consequences seem to be new. As in Remark 1.12(a) we denote by U_{k} = U_{k,−V} the constituents of the Dyson-Phillips series for U and V if V is a Miyadera perturbation of U; recall that the minus sign is due to (2.1).

2.42 Theorem. Let V ⩽ 0 be a Miyadera perturbation of U. Then ℕ_{0} ∋ k ↦ k!U_{k} is logarithmically convex.

By Proposition 2.34 and Lemma 2.33(a), the logarithmic convexity of k ↦ k!U_{k} can be expressed as follows: If (t, s) ∈ D_{J} and f ∈ L_{p}(μ)_{+} then

(ℓ!U_{ℓ}(t, s)f)^{m−k} ⩽ (k!U_{k}(t, s)f)^{m−ℓ}(m!U_{m}(t, s)f)^{ℓ−k}  (0 ⩽ k ⩽ ℓ ⩽ m). (2.22)

The proof of Theorem 2.42 will be given at the end of the section. We first present an application that is particularly interesting for the perturbation of stochastic propagators on L_{1}(μ); see Remark 2.44(b) below.

2.43 Corollary. Let V ⩽ 0 be a small Miyadera perturbation of U, and let (t, s) ∈ D_{J}.

(a) Let f ∈ L_{p}(μ)_{+} and Ω_{0} := [U(t, s)f = 0]. Then U_{1}(t, s)f = U_{V} (t, s)f = 0 a.e. on Ω_{0} and

\frac{U_{1}(t, s)f}{U(t, s)f} ⩽ ln \frac{U_{V} (t, s)f}{U(t, s)f}

a.e. on Ω \ Ω_{0}.

(b) If p = 1 then

\frac{||U_{1}(t, s)f||_{1}}{||U(t, s)f||_{1}} ⩽ ln \frac{||U_{V} (t, s)f||_{1}}{||U(t, s)f||_{1}}

for all f ∈ L_{1}(μ)_{+} with ||U(t, s)f||_{1} > 0.

Proof. (a) By Theorem 2.42, k ↦ k!U_{k}(t, s)f is logarithmically convex. By (2.22) it follows that U_{k}(t, s)f = 0 a.e. on Ω_{0} for all k ∈ ℕ and hence also U_{V} (t, s)f = 0 a.e. on Ω_{0}. Moreover,

(U_{1}(t, s)f)^{k} ⩽ (U(t, s)f)^{k−1} · k!U_{k}(t, s)f  (2.23)

for all k ∈ ℕ and thus

\frac{1}{k!} (\frac{U_{1}(t, s)f}{U(t, s)f} )^{k} ⩽ \frac{U_{k}(t, s)f}{U(t, s)f}

a.e. on Ω \ Ω_{0}. The latter inequality is trivially true for k = 0, so we conclude that

exp(\frac{U_{1}(t, s)f}{U(t, s)f} ) ⩽ \frac{U_{V} (t, s)f}{U(t, s)f}

a.e. on Ω \ Ω_{0}.

(b) By Hölder’s inequality it follows from (2.23) that

||U_{1}(t, s)f||_{1}^{k} ⩽ ||U(t, s)f||_{1}^{k−1} · k!||U_{k}(t, s)f||_{1}.

Since ||U_{V} (t, s)f||_{1} = || ∑_{k=0}^{∞} U_{k}(t, s)f||_{1} = ∑_{k=0}^{∞} ||U_{k}(t, s)f||_{1}, we obtain as in (a) that

exp(\frac{||U_{1}(t, s)f||_{1}}{||U(t, s)f||_{1}} ) ⩽ \frac{||U_{V} (t, s)f||_{1}}{||U(t, s)f||_{1}} ,

and the proof is complete.

2.44 Remarks. (a) The estimate in Corollary 2.43(a) is sharp as one sees in the trivial example that U(t, s) = I for all (t, s) ∈ D_{J}: Then

U_{1}(t, s)f = − ∫_{s}^{t} V (τ) dτ · f and U_{V} (t, s)f = exp(− ∫_{s}^{t} V (τ) dτ) · f

for all (t, s) ∈ D_{J} and all f ∈ L_{p}(μ).

(b) In the situation of Corollary 2.43(b) assume that U is stochastic (as defined in Remark 2.27(a)). Let (t, s) ∈ D_{J}. Then for f ∈ L_{1}(μ)_{+} with ||f||_{1} = 1 we obtain by Corollary 2.43(b) that ||U_{1}(t, s)f||_{1} ⩽ ln ||U_{V} (t, s)f||_{1} ⩽ ln ||U_{V} (t, s)||, so

||U_{1}(t, s)|| ⩽ ln ||U_{V} (t, s)||.  (2.24)

Moreover,

||U_{1}(t, s)f||_{1} = ∫_{s}^{t} ||U(t, τ)V (τ)U(τ, s)f||_{1} dτ = ∫_{s}^{t} ||V (τ)U(τ, s)f||_{1} dτ

for all f ∈ L_{1}(μ)_{+}. Given α > 0, it follows that

γ_{α} := sup{||U_{1}(t, s)||; (t, s) ∈ D_{J}, t − s ⩽ α}

is the smallest constant such that V is a Miyadera perturbation of U with constants (α, γ_{α}). With M_{α} := sup{||U_{V} (t, s)||; (t, s) ∈ D_{J}, t − s ⩽ α} we infer from (2.24) that

γ_{α} ⩽ ln M_{α}.

This estimate is considerably better that the estimate γ_{α} ⩽ M_{α} − 1 from Remark 2.27(a). On the other hand, by Proposition 1.14 we know that

M_{α} ⩽ \frac{1}{1 − γ_{α}}

if γ_{α} < 1, so we obtain the two-sided estimate 1 − \frac{1}{M_{α}} ⩽ γ_{α} ⩽ ln M_{α}.

(c) We want to show that an estimate analogous to the one in part (b) also applies to backward Miyadera perturbations of Markovian propagators. For simplicity we assume that the measure μ is σ-finite. We will say that an operator B ∈ L(L_{p}(μ)) extrapolates to a bounded operator on L_{q}(μ), for some q ∈ [1, ∞), if B|_{L_{p}(μ)∩L_{q}(μ)} extends to an operator in L(L_{q}(μ)). We will also use this notion for operator-valued functions and for the case q = ∞ and weakly^{∗} continuous extensions; the space of weakly^{∗} continuous operators on L_{∞}(μ) will be denoted by L_{w^{∗}}(L_{∞}(μ)).

We start with the following observation. Let B be a positive operator on L_{p}(μ), and assume that ||B||_{∞→∞} < ∞. Then B extrapolates to an operator B_{∞} ∈ L_{w^{∗}}(L_{∞}(μ)), and if L_{p}(μ)_{+} ∋ f_{n} ↑ 𝟙 a.e. then Bf_{n} ↑ B_{∞}𝟙 a.e. Indeed, B′ extrapolates to a bounded operator B̃ B′ on L_{1}(μ), and B_{∞} := (B̃ B′)′ is a weakly^{∗} continuous extension of B|_{L_{p}(μ)∩L_{∞}(μ)}. Moreover, f_{n} ↑ 𝟙 a.e. implies that f_{n} → 𝟙 weakly^{∗}, so Bf_{n} → B_{∞}𝟙 weakly^{∗}, and from the positivity of B it follows that Bf_{n} ↑ B_{∞}𝟙 a.e.

If the propagator U is locally L_{∞}-bounded then by the above, U extrapolates to a locally bounded propagator Ũ U: D_{J} → L_{w^{∗}}(L_{∞}(μ)). We assume that U is Markovian, i.e., U is L_{∞}-contractive and Ũ U(t, s)𝟙 = 𝟙 for all (t, s) ∈ D_{J}. Let V ∈ L_{∞}(J × Ω), V ⩽ 0. Then U_{1} and U_{V} are locally L_{∞}-bounded and hence extrapolate to functions Ũ U_{1} and Ũ U_{V} from D_{J} to L_{w^{∗}}(L_{∞}(μ)). By the above observation we infer from Corollary 2.43(a) that Ũ U_{1}(t, s)𝟙 ⩽ ln Ũ U_{V} (t, s)𝟙 a.e. for all (t, s) ∈ D_{J}. Now it is easy to see that

∫_{s}^{t} U(t, τ)(|V (τ)| ∧ g) dτ = ∫_{s}^{t} U(t, τ)(|V (τ)| ∧ g)Ũ U(τ, s)𝟙 dτ ⩽ Ũ U_{1}(t, s)𝟙

a.e. for all (t, s) ∈ D_{J} and all g ∈ L_{1}(μ), so we obtain the following.

Let V ⩽ 0 be weakly U-admissible, let α > 0, and assume that

M_{α} := sup{||U_{V} (t, s)||_{∞→∞}; (t, s) ∈ D_{J}, t − s ⩽ α} < ∞.

Then from the above, applied with V_{n} := V ∨ (−n) in place of V, we conclude that

∫_{s}^{t} U(t, τ)(|V_{n}(τ)| ∧ g) dτ ⩽ ln Ũ U_{V_{n}}(t, s)𝟙 ⩽ ln ||U_{V} (t, s)||_{∞→∞} ⩽ ln M_{α}

a.e. for all n ∈ ℕ, g ∈ L_{1}(μ) and all (t, s) ∈ D_{J} with t − s ⩽ α, so V is a backward Miyadera perturbation of U with constants (α, ln M_{α}).

We point out that the above argument requires strong continuity of U but not of the (stochastic) backward propagator U′ defined in Remark 2.31(a).

(d) If U is only strongly measurable then one can use the observation in part (c) to show the following analogues of Proposition 2.25(a) and of Corollary 2.26(a). Let α ∈ (0, ∞] and M ⩾ c > 0. If V ⩾ 0 and c𝟙 ⩽ Ũ U_{V} (t, s)𝟙 ⩽ Ũ U(t, s)𝟙 ⩽ M𝟙 for all (t, s) ∈ D_{J} with t − s ⩽ α, then V is a backward Miyadera perturbation of U with constants (α, \frac{M}{c} − 1). On the other hand, if V ⩽ 0 is weakly U-admissible and c𝟙 ⩽ Ũ U(t, s)𝟙 ⩽ Ũ U_{V} (t, s)𝟙 ⩽ M𝟙 for all (t, s) ∈ D_{J} with t − s ⩽ α, then V is a backward Miyadera perturbation of U_{V} and hence also of U with constants (α, \frac{M}{c} − 1).

We now turn to the proof of Theorem 2.42. The crucial ingredient is the following result on logarithmically convex sequences in [0, ∞). Observe that a sequence x: ℕ_{0} → [0, ∞) is logarithmically convex if and only if x_{k}^{2} ⩽ x_{k−1}x_{k+1} for all k ∈ ℕ.

2.45 Proposition. Let x: ℕ_{0} → [0, ∞) be logarithmically convex. Then

ℕ_{0} ∋ k ⟼ ∑_{j=0}^{k} \binom{k}{j}x_{j} ∈ [0, ∞)

is logarithmically convex, too.

Proof. Let ℓ ∈ ℕ. We have to show that

L := (∑_{j=0}^{ℓ} \binom{ℓ}{j}x_{j})^{2} ⩽ (∑_{j=0}^{ℓ−1} \binom{ℓ − 1}{j}x_{j})(∑_{j=0}^{ℓ+1} \binom{ℓ + 1}{j}x_{j}) =: R.

For k = 0, …, 2ℓ we are going to prove that

L_{k} := ∑_{j=0}^{k} \binom{ℓ}{j}\binom{ℓ}{k − j}x_{j}x_{k−j} ⩽ ∑_{j=0}^{k} \binom{ℓ − 1}{j}\binom{ℓ + 1}{k − j}x_{j}x_{k−j} =: R_{k}  (2.25)

(where \binom{ℓ}{j} = 0 if j > ℓ); then the assertion follows since L = ∑_{k=0}^{2ℓ} L_{k} and R = ∑_{k=0}^{2ℓ} R_{k}. Observe that for y ∈ ℝ we have

∑_{k=0}^{2ℓ} y^{k} ∑_{j=0}^{k} \binom{ℓ}{j}\binom{ℓ}{k − j} = (∑_{j=0}^{ℓ} \binom{ℓ}{j}y^{j})^{2} = (1 + y)^{2ℓ}

= (∑_{j=0}^{ℓ−1} \binom{ℓ − 1}{j}y^{j})(∑_{j=0}^{ℓ+1} \binom{ℓ + 1}{j}y^{j}) = ∑_{k=0}^{2ℓ} y^{k} ∑_{j=0}^{k} \binom{ℓ − 1}{j}\binom{ℓ + 1}{k − j}.

For a fixed k ∈ {0, …, 2ℓ} we thus obtain, with

α_{j} := \binom{ℓ}{j}\binom{ℓ}{k − j} − \binom{ℓ − 1}{j}\binom{ℓ + 1}{k − j} (j = 0, …, k),

that ∑_{j=0}^{k} α_{j} = 0. With this notation, (2.25) simplifies to ∑_{j=0}^{k} α_{j}x_{j}x_{k−j} ⩽ 0. The latter is equivalent to ∑_{j=0}^{k} β_{j}x_{j}x_{k−j} ⩽ 0, where β_{j} := α_{j} +α_{k−j}. From the logarithmic convexity of x it follows that j ↦ x_{j}x_{k−j} is decreasing on [0, \frac{k}{2}] ∩ ℕ_{0} and increasing on [\frac{k}{2}, k] ∩ ℕ_{0}. Below we will show that there exists j_{0} ∈ [0, \frac{k}{2}] ∩ ℕ_{0} such that β_{j} ⩾ 0 for j = j_{0}, …, k − j_{0} and β_{j} ⩽ 0 otherwise. Combining these properties with ∑_{j=0}^{k} β_{j} = 0, we then conclude that ∑_{j=0}^{k} β_{j}x_{j}x_{k−j} ⩽ 0, which proves (2.25) and hence the assertion.

Let now j ∈ {0, …, k}, and let i := k − j. Assume that i, j ⩽ ℓ + 1; otherwise β_{j} = α_{j} + α_{i} = 0. The inequality β_{j} ⩾ 0 holds if and only if

2\binom{ℓ}{j}\binom{ℓ}{i} ⩾ \binom{ℓ − 1}{j}\binom{ℓ + 1}{i} + \binom{ℓ − 1}{i}\binom{ℓ + 1}{j}.

Since \binom{ℓ−1}{j} = \frac{ℓ−j}{ℓ} \binom{ℓ}{j} = \frac{ℓ−j}{ℓ} \frac{ℓ+1−j}{ℓ+1} \binom{ℓ+1}{j} and \binom{ℓ+1}{j}, \binom{ℓ+1}{i} ≠ 0, the latter is equivalent to

2 \frac{ℓ + 1 − j}{ℓ + 1} \frac{ℓ + 1 − i}{ℓ + 1} ⩾ \frac{ℓ − j}{ℓ} \frac{ℓ + 1 − j}{ℓ + 1} + \frac{ℓ − i}{ℓ} \frac{ℓ + 1 − i}{ℓ + 1}

and thus to

a_{j} := \frac{2ℓ}{ℓ+1}(ℓ + 1 − j)(ℓ + 1 − i) ⩾ (ℓ − j)(ℓ + 1 − j) + (ℓ − i)(ℓ + 1 − i) =: b_{j}.

Note that j ↦ a_{j} is concave on {0, …, k} since i = k − j. Moreover,

b_{j} = 2ℓ(ℓ + 1) − (2ℓ + 1)j + j^{2} − (2ℓ + 1)i + i^{2} = 2ℓ(ℓ + 1) − (2ℓ + 1)k + j^{2} + (k − j)^{2},

so j ↦ b_{j} is convex on \{0, …, k\}. Therefore, there exists j_{0} ∈ [0, \frac{k}{2}] ∩ ℕ_{0} such that a_{j} ⩾ b_{j} for j = j_{0}, …, k − j_{0} and a_{j} < b_{j} otherwise. Then β_{j} ⩾ 0 for j = j_{0}, …, k − j_{0} and β_{j} ⩽ 0 otherwise, and the proof is complete.

2.46 Remark. Let x: ℕ_{0} → [0, ∞) be logarithmically convex, and let q ⩾ 0. Then k ↦ q^{k}x_{k} is logarithmically convex. By Proposition 2.45 we thus obtain that

k ⟼ ∑_{j=0}^{k} \binom{k}{j}q^{k−j}x_{j} =: z_{k}

is logarithmically convex. (For the case q = 0 note that then z_{k} = x_{k}.) This might lead one to the conjecture that more generally k ↦ ∑_{j=0}^{k} \binom{k}{j}x_{j}y_{k−j} is logarithmically convex if y: ℕ_{0} → [0, ∞) is a second logarithmically convex sequence. We do not know whether this is true, but the idea of the proof of Proposition 2.45 does not work: One can show, e.g., that the analogue of the estimate (2.25) does not hold for ℓ = k = 2.

Proof of Theorem 2.42. Let (t, s) ∈ D_{J} and f ∈ L_{p}(μ)_{+}; we show logarithmic convexity of k ↦ k!U_{k,−V} (t, s)f. Using the monotone convergence theorem, one easily proves by induction that U_{k,(−V)∧n}(t, s) → U_{k,−V} (t, s) strongly as n → ∞, for all k ∈ ℕ_{0}, so we can assume without loss of generality that V is bounded. Then by Remark 1.25 we can assume without loss of generality that U and V are “discrete” on [s, t], i.e., there exist m ∈ ℕ and s = t_{0} < t_{1} < ⋯ < t_{m} = t such that

U(τ, σ) = I and V (τ) = V (t_{n})

for all (τ, σ) ∈ D_{(t_{n−1},t_{n}]} and n = 1, …, m.

We show by induction on n that k ↦ k!U_{k}(t_{n}, t_{0})f is logarithmically convex for n = 0, …, m. For n = 0 this is trivial since U_{k}(t_{0}, t_{0}) = 0 for all k ∈ ℕ. Assume that logarithmic convexity is valid for some n ∈ {0, …, m − 1}. Then

k ⟼ g_{k} := k!U(t_{n+1}, t_{n})U_{k}(t_{n}, t_{0})f

is logarithmically convex by Remark 2.35(b). As explained in Remark 1.25(b), we have U_{k}(t_{n+1}, t_{n}) = \frac{1}{k!}W^{k}U(t_{n+1}, t_{n}) for all k ∈ ℕ_{0}, where W := −(t_{n+1} − t_{n})V (t_{n+1}) ⩾ 0. By Proposition 1.13 we infer that

k!U_{k}(t_{n+1}, t_{0})f = k! ∑_{j=0}^{k} U_{k−j}(t_{n+1}, t_{n})U_{j}(t_{n}, t_{0})f = ∑_{j=0}^{k} \frac{k!}{(k − j)!j!}W^{k−j}g_{j},

so from Remark 2.46 it follows that k ↦ k!U_{k}(t_{n+1}, t_{0})f is logarithmically convex.

2.6 Consistent propagators on weighted L_{p}-spaces

Let (Ω, μ) be a measure space. Let ρ: Ω → (0, ∞) be locally measurable, and let p ∈ [1, ∞). Then the operator of multiplication with ρ^{1/p} is an isometry ρ^{1/p}: L_{p}(ρμ) → L_{p}(μ). Let U be a locally bounded positive strongly measurable propagator on L_{p}(ρμ) with parameter interval J. Then Ũ U(t, s) := ρ^{1/p}U(t, s)ρ^{−1/p} defines a locally bounded positive strongly measurable propagator on L_{p}(μ).

2.47 Lemma. A potential V is weakly U-admissible if and only if it is weakly Ũ U-admis-sible, and Ũ U_{V} (t, s) = ρ^{1/p}U_{V} (t, s)ρ^{−1/p} for all (t, s) ∈ D_{J} in this case.

Proof. Since ρ^{1/p}V ρ^{−1/p} = V, the assertion for bounded V follows from uniqueness in Duhamel’s formula. Then for general V one uses the definition of absorption propagators to complete the proof.

For j = 0, 1 let now ρ_{j}: Ω → (0, ∞) be locally measurable, let p_{j} ∈ [1, ∞), and let U_{j} be a positive strongly continuous propagator on L_{p_{j}}(ρ_{j}μ) with parameter interval J. We assume that U_{0} and U_{1} are consistent, i.e., U_{0}(t, s)f = U_{1}(t, s)f for all (t, s) ∈ D_{J} and all f ∈ L_{p_{0}}(ρ_{0}μ) ∩ L_{p_{1}}(ρ_{1}μ). Observe that L_{p_{0}}(ρ_{0}μ) ∩ L_{p_{1}}(ρ_{1}μ) is dense in L_{p_{j}}(ρ_{j}μ), for j = 0, 1, so U_{0} and U_{1} determine each other uniquely.

2.48 Lemma. (cf. [Voi86; Prop. 3.1]) Let V be weakly U_{0}- and U_{1}-admissible. (a) Then (U_{0})_{V} and (U_{1})_{V} are consistent.

(b) The potential V is U_{0}-regular if and only if it is U_{1}-regular.

Proof. (a) For bounded V one obtains by induction that the constituents of the Dyson-Phillips series for (U_{0})_{V} and (U_{1})_{V} are consistent, which implies the assertion in this case. The general case follows since strong limits of consistent propagators are again consistent.

(b) It follows from part (a) that ((U_{0})_{V} )_{−V} and ((U_{1})_{V} )_{−V} are consistent. This yields the assertion.

2.6. CONSISTENT PROPAGATORS ON WEIGHTED L_{P} -SPACES 57

For θ ∈ (0, 1) define p_{θ} ∈ [1, ∞) by \frac{1}{p_{θ}} = \frac{1−θ}{p_{0}} + \frac{θ}{p_{1}} , and let ρ_{θ} := ρ_{0}^{(1−θ)p_{θ}/p_{0}}ρ_{1}^{θp_{θ}/p_{1}}. We conclude the chapter with a result on interpolation of weak admissibility in the spaces L_{p_{θ}}(ρ_{θ}μ). In the application in Section 3.2 we will only use the case p_{0} = p_{1}; note that then ρ_{θ} = ρ_{0}^{1−θ}ρ_{1}^{θ}.

2.49 Proposition. For j = 0, 1 let V_{j} be weakly U_{j}-admissible. Assume that V_{0}^{+} ∧ V_{1}^{−} and V_{0}^{−} ∧ V_{1}^{+} are U_{0}-regular. Then U_{0} extrapolates to a strongly continuous propagator U_{θ} on L_{p_{θ}}(ρ_{θ}μ), for each θ ∈ (0, 1), the potential V_{θ} := (1−θ)V_{0} +θV_{1} is weakly U_{θ}-admissible, and

||(U_{θ})_{V_{θ}}(t, s)|| ⩽ ||(U_{0})_{V_{0}}(t, s)||^{1−θ}||(U_{1})_{V_{1}}(t, s)||^{θ}  ((t, s) ∈ D_{J}).  (2.26)

Proof. First assume that V_{0} and V_{1} are bounded from below. Then [0, 1] ∋ θ ↦ (U_{0})_{V_{θ}} is logarithmically convex by Theorem 2.38. Moreover, [0, 1] ∋ θ ↦ ρ_{θ}^{1/p_{θ}} ∈ M(μ)_{+} is logarithmically convex. Given θ ∈ (0, 1), we thus we obtain by Corollary 2.37 that

||ρ_{θ}^{1/p_{θ}}(U_{0})_{V_{θ}}(t, s)ρ_{θ}^{−1/p_{θ}}||_{p_{θ}→p_{θ}} ⩽ ||ρ_{0}^{1/p_{0}}(U_{0})_{V_{0}}(t, s)ρ_{0}^{−1/p_{0}}||_{p_{0}→p_{0}}^{1−θ}||ρ_{1}^{1/p_{1}}(U_{0})_{V_{1}}(t, s)ρ_{1}^{−1/p_{1}}||_{p_{1}→p_{1}}^{θ}
= ||(U_{0})_{V_{0}}(t, s)||^{1−θ}||(U_{1})_{V_{1}}(t, s)||^{θ}

for all (t, s) ∈ D_{J}, where we have used that (U_{0})_{V_{1}} and (U_{1})_{V_{1}} are consistent by Lemma 2.48(a). By considering V_{θ} = V_{0} = V_{1} = 0, we obtain in particular that U_{0} extrapolates to a propagator U_{θ} on L_{p_{θ}}(ρ_{θ}μ); the strong continuity of U_{θ} is easily proved by means of the inequality ||g||_{L_{p_{θ}} (ρ_{θ}μ)} ⩽ ||g||_{L_{p_{0}}(ρ_{0}μ)}^{1−θ}||g||_{L_{p_{1}}(ρ_{1}μ)}^{θ} that follows from Hölder’s inequality and is valid for all g ∈ L_{p_{0}}(ρ_{0}μ) ∩ L_{p_{1}}(ρ_{1}μ). This completes the proof in the case that V_{0} and V_{1} are bounded from below since (U_{0})_{V_{θ}} and (U_{θ})_{V_{θ}} are consistent by Lemma 2.48(a).

In the case of general V_{0}, V_{1} we proceed as in the proof of Theorem 2.38: Let θ ∈ (0, 1) and V_{θ,±} := (1 − θ)V_{0}^{±} + θV_{1}^{±}. Using that the right-hand side of (2.26) defines a locally bounded function on D_{J}, we infer from the first part of the proof that −V_{θ,−} is weakly (U_{θ})_{V_{θ,+}}-admissible and that

||((U_{θ})_{V_{θ,+}})_{−V_{θ,−}}(t, s)|| ⩽ ||(U_{0})_{V_{0}}(t, s)||^{1−θ}||(U_{1})_{V_{1}}(t, s)||^{θ}

for all (t, s) ∈ D_{J}. As in the proof of Theorem 2.38 we obtain that V_{θ,+} ∧ V_{θ,−} is U_{0}-reg-ular, and hence U_{θ}-regular by Lemma 2.48(b). Since V_{θ} = V_{θ,+} − V_{θ,−}, we conclude from Remark 2.18(a) that V_{θ} is weakly U-admissible and (U_{θ})_{V_{θ}} = ((U_{θ})_{V_{θ,+}})_{−V_{θ,−}}.

Bounds for the integral kernels of propagators

In this chapter we apply the results of the previous two chapters and prove stability of kernel estimates for strongly continuous propagators under perturbation by weakly admissible potentials.

Section 3.1, which deals with rather general kernel estimates for ultracontractive propagators, is adapted from [LVV06; Sec. 3]. The main difference is that the assumptions on the perturbation V are formulated in terms of weak admissibility and regularity, not in terms of Miyadera conditions. Moreover, we avoid the technical assumption of strong continuity of the backward propagator from Remark 2.31(a).

In Section 3.2 we investigate Gaussian type estimates with boundary terms, still in the setting of ultracontractive propagators. The main issue here is stability of the boundary terms under perturbations. In Section 3.3 we drop the ultracontractivity assumption and prove stability of a type of kernel estimate that includes heat kernel estimates on complete Riemannian manifolds with Ricci curvature bounded below.

3.1 Stability of kernel bounds for ultracontractive propagators

Throughout this section we assume that (Ω, μ) is a σ-finite measure space and that

(A1) U is a positive strongly continuous propagator on L_{1}(μ) with parameter interval J ⊆ ℝ, and there is a constant L ⩾ 1 such that

||U(t, s)||_{1→1} ⩽ L ((t, s) ∈ D_{J}).

3.1 Remark. If U is only exponentially bounded then U can be rescaled to a propagator (t, s) ↦ e^{−ω(t−s)}U(t, s) satisfying the bound in (A1). Observe that a potential V is (weakly) U-admissible if and only if V is (weakly) admissible with respect to the rescaled propagator. Moreover, a small Miyadera perturbation of U is also a small Miyadera perturbation of the rescaled propagator. (Make α in the definition smaller if necessary.) Thus, all the qualitative assertions of the subsequent results remain true in the more general case of exponentially bounded U.

We will also use the ultracontractivity assumption that (A2) there exist constants K, ν > 0 and A ∈ ℝ such that

||U(t, s)||_{1→∞} ⩽ K(t − s)^{−ν}e^{A(t−s)} ((t, s) ∈ D′ J),

where D′ J := {(t, s) ∈ J × J; t > s}.

3.2 Remarks. (a) In assumption (A2), only the case A ⩽ 0 is of actual interest. In fact, if (A1) holds and ||U(t, s)||_{1→∞} ⩽ K(t − s)^{−ν} for all (t, s) ∈ D′ J with t − s ⩽ 1, then for (t, s) ∈ D′ J with t − s > 1 we have

||U(t, s)||_{1→∞} ⩽ ||U(t, t − 1)||_{1→∞}||U(t − 1, s)||_{1→1} ⩽ KL.

With C := KL and f(t) := t^{−ν} ∨ 1 we thus obtain that

(A2’) ||U(t, s)||_{1→∞} ⩽ Cf(t − s) for all (t, s) ∈ D′ J

(which is slightly weaker than (A2) with A = 0, but much stronger than (A2) with A > 0). More generally, let us briefly discuss the assumption (A2’) with

f(t) := t^{−ν_{0}} (0 < t ⩽ 1), f(t) := t^{−ν_{1}} (t > 1),  (3.1)

where ν_{0}, ν_{1} ⩾ 0. (Above we had ν_{0} = ν, ν_{1} = 0.) For heat propagators on manifolds, the different t-exponents for t ⩽ 1 and t > 1 are important; e.g., for a compact complete Riemannian manifold one has to take ν_{1} = 0. Our subsequent results can be adapted to this more general setting, but we confine ourselves to assumption (A2) for the sake of simplicity.

(b) By the Dunford-Pettis theorem, assumption (A2) implies that U(t, s) is an integral operator, for each (t, s) ∈ D′ J, with a positive kernel p_{t,s} bounded by K(t − s)^{−ν}e^{A(t−s)}. Here, p_{t,s} being a kernel of U(t, s) means that

U(t, s)f(x) = ∫_{Ω} p_{t,s}(x, y)f(y) dμ(y)

for all f ∈ L_{1}(μ) and a.e. x ∈ Ω.

We start with the following result that requires only assumptions (A1) and (A2).

3.3 Proposition. Assume (A1) and (A2). Let p ∈ (1, ∞), and let V be a potential such that pV is weakly U-admissible. Assume that U_{pV} is exponentially bounded. Then V is weakly U-admissible, and there exist c > 0 and ω ∈ ℝ such that

||U_{V} (t, s)||_{1→p} ⩽ c(t − s)^{−ν/p′}e^{ω(t−s)}  ((t, s) ∈ D′ J).  (3.2)

If V^{−} is a small Miyadera perturbation of U with constants (α, γ), γ < 1, then for all 1 ⩽ p < \frac{1}{γ} there exists c > 0 such that (3.2) holds with ω = \frac{1}{pα} ln(1 + \frac{pγL}{1−pγ} ) + (1 − \frac{1}{p})A.

Proof. By Theorem 2.38 and Corollary 2.37 we obtain that V is weakly U-admissible and that

||U_{V} (t, s)||_{1→p} ⩽ ||U(t, s)||_{1→∞}^{1−1/p}||U_{pV} (t, s)||_{1→1}^{1/p}  (3.3)

for all (t, s) ∈ D′ J. The estimate (3.2) thus follows from (A2) and the exponential boundedness of U_{pV}.

Let now V^{−} be a small Miyadera perturbation of U with constants (α, γ), γ < 1, and let 1 ⩽ p < \frac{1}{γ}. Then (pV)^{−} is a small Miyadera perturbation of U with constants (α, pγ), so Proposition 1.14 yields

||U_{pV} (t, s)||_{1→1} ⩽ \frac{L}{1−pγ} e^{ω_{p}(t−s)} ((t, s) ∈ D_{J}),

where ω_{p} = \frac{1}{α} ln(1 + \frac{pγ}{1−pγ} L). By (3.3) this implies the second assertion.

In order to obtain L_{1}–L_{∞}-estimates, we will need in addition that the propagator U is L_{∞}-bounded,

(A3) ||U(t, s)||_{∞→∞} ⩽ L for all (t, s) ∈ D_{J}.

3.4 Remarks. (a) Let assumptions (A1) and (A3) be satisfied. Then ||U(t, s)||_{p→p} ⩽ L for all 1 ⩽ p ⩽ ∞ and all (t, s) ∈ D_{J} by Riesz-Thorin interpolation. Thus, U extrapolates to a consistent family of bounded strongly continuous propagators U_{p} on L_{p}(μ), 1 ⩽ p < ∞ (for f ∈ L_{1}(μ) ∩ L_{∞}(μ), the continuity of (t, s) ↦ U_{p}(t, s)f can be obtained from the inequality ||g||_{p} ⩽ ||g||_{1}^{1/p}||g||_{∞}^{1−1/p}, valid for all g ∈ L_{1}(μ) ∩ L_{∞}(μ)).

(b) In addition, let V be a potential such that V^{−} is a small Miyadera perturbation of U, with bound γ < 1. For 1 ⩽ p < \frac{1}{γ} and (t, s) ∈ D_{J} we obtain by Theorem 2.38 and Corollary 2.37 that

||U_{V} (t, s)||_{p→p} ⩽ ||U_{pV} (t, s)||_{1→1}^{1/p}||U(t, s)||_{∞→∞}^{1/p′}.

Thus, U_{V} extrapolates to a consistent family of exponentially bounded strongly continuous propagators U_{p,V} on L_{p}(μ), 1 ⩽ p < \frac{1}{γ}. It follows from Lemma 2.48(a) that V is U_{p}- admissible and U_{p,V} = (U_{p})_{V}, for 1 ⩽ p < \frac{1}{γ}. If additionally (A2) holds, then by Riesz-Thorin interpolation the above inequality together with (3.2) implies the estimates

||U_{p,V} (t, s)||_{p→q} ⩽ C(t − s)^{−ν( \frac{1}{p} − \frac{1}{q})}e^{ω(t−s)} ((t, s) ∈ D′ J, 1 ⩽ p ⩽ q < \frac{1}{γ}),  (3.4)

where C > 0 and ω ∈ ℝ are constants depending on q.

In particular, if V^{−} is infinitesimally Miyadera small then the above holds with \frac{1}{γ} replaced by ∞. This observation generalises [Gul02; Thm. 2]. In [Gul05; Thm. 3(b)], an example is presented showing that (3.4) need not hold for p = q = ∞, for infinitesimally Miyadera small V.

The following extrapolation lemma is a propagator version of [Cou90; Lemme 1], which deals with semigroups. The analogous proof is included for the reader’s convenience.

3.5 Lemma. Let W be a propagator on L_{1}(μ) with parameter interval J. Let p ∈ (1, ∞) and ν > 0, and assume that there exist c, M > 0 and ω, ω_{∞} ∈ ℝ such that

||W (t, s)||_{1→p} ⩽ c(t − s)^{−ν/p′}e^{ω(t−s)}, ||W (t, s)||_{∞→∞} ⩽ Me^{ω_{∞}(t−s)} ((t, s) ∈ D′ J).

Then there exists c_{1} > 0 such that

||W (t, s)||_{1→∞} ⩽ c_{1}(t − s)^{−ν}e^{ω_{1}(t−s)}  ((t, s) ∈ D′ J),  (3.5)

where ω_{1} = max{ω, ω_{∞}}.

Proof. By rescaling W we can assume without loss of generality that ω, ω_{∞} ⩽ 0. Let t ∈ J, T > 0 and f ∈ L_{1}(μ) ∩ L_{∞}(μ) with ||f||_{1} ⩽ 1. Then

φ(s) := ||W (t, s)′f||_{∞} ⩽ ||W (t, s)′||_{p′→∞}||f||_{p′} ⩽ c(t − s)^{−ν/p′}||f||_{p′}

for all s ∈ J_{<t}, by the assumption on ||W (t, s)||_{1→p}, so there exists c_{0} ⩾ 0 (depending on the quantities fixed above) such that

φ(s) ⩽ c_{0}(t − s)^{−ν} (s ∈ [t − T, t) ∩ J).

We choose the minimal c_{0} making this estimate valid. Let r ∈ [t − T, t) ∩ J, and let s := \frac{r+t}{2}. For g := W (t, s)′f we can estimate

||g||_{p′} ⩽ ||g||_{1}^{1/p′}||g||_{∞}^{1/p} ⩽ ||W (t, s)′||_{1→1}^{1/p′}φ(s)^{1/p} ⩽ M^{1/p′}(c_{0}(t − s)^{−ν})^{1/p}.

Moreover, ||W (s, r)′||_{p′→∞} ⩽ c(s − r)^{−ν/p′}. Since t − s = s − r = \frac{1}{2}(t − r), we infer that

φ(r) = ||W (s, r)′W (t, s)′f||_{∞} ⩽ ||W (s, r)′||_{p′→∞}||g||_{p′} ⩽ cM^{1/p′}c_{0}^{1/p}2^{ν}(t − r)^{−ν}.

From the choice of c_{0} it follows that c_{0} ⩽ cM^{1/p′}c_{0}^{1/p}2^{ν} and hence c_{0} ⩽ c^{p′}M2^{νp′}. We thus have shown

||W (t, s)′f||_{∞} ⩽ c^{p′}M2^{νp′}(t − s)^{−ν}

for all (t, s) ∈ D′ J and all f ∈ L_{1}(μ)∩L_{∞}(μ) with ||f||_{1} ⩽ 1, which implies the assertion.

3.6 Remark. In the situation of Lemma 3.5 assume that ω ≠ ω_{∞}. Then by a suitable modification of the above proof one can show that for any ω_{1} > min{ω, ω_{∞}} there exists c_{1} > 0 such that (3.5) holds. The following simple argument yields a better estimate for large t − s.

By Lemma 3.5 and the Riesz-Thorin interpolation theorem we obtain C > 0 such that ||W (t, t − 1)||_{p→∞} ⩽ C for all t ∈ J with t − 1 ∈ J. It follows that there exists C̃ C > 0 such that for (t, s) ∈ D_{J} with t − s ⩾ 2 we have

||W (t, s)||_{1→∞} ⩽ ||W (t, t − 1)||_{p→∞}||W (t − 1, s)||_{1→p}
⩽ Cc(t − 1 − s)^{−ν/p′}e^{ω(t−s−1)} ⩽ C̃ C(t − s)^{−ν/p′}e^{ω(t−s)}.

If ω_{∞} < ω then the estimate

||W (t, s)||_{1→∞} ⩽ ||W (t, s + 1)||_{∞→∞}||W (s + 1, s)||_{1→∞} ⩽ C_{1}Me^{−ω_{∞}}e^{ω_{∞}(t−s)}

is better for large t − s, where C_{1} := sup{||W (s + 1, s)||_{1→∞}; s ∈ J, s + 1 ∈ J}. In the same way, if ω_{1} ∈ ℝ is such that ||W (t, s)||_{1→1} ⩽ Me^{ω_{1}(t−s)} for all (t, s) ∈ D_{J}, then ||W (t, s)||_{1→∞} ⩽ C_{1}Me^{−ω_{1}}e^{ω_{1}(t−s)} for t − s ⩾ 1.

As an immediate consequence of Proposition 3.3 and Lemma 3.5 we obtain the following result which strengthens [LVV06; Prop. 3.6]. Besides the weaker qualitative assumptions, the difference is that the condition on the L_{∞}-bound is posed on the propagator U_{V} itself, not on U_{pV} for some p > 1 as the condition on the L_{1}-bound. This is parallel to [ArDe06; Thm. 3.3], where also a variant of [Cou90; Lemme 1] is used.

3.7 Proposition. Assume (A1) to (A3). Let p ∈ (1, ∞), and let V be a potential such that pV is weakly U-admissible. Assume that U_{pV} is exponentially bounded and that U_{V} is exponentially L_{∞}-bounded. Then there exist c > 0 and ω ∈ ℝ such that

||U_{V} (t, s)||_{1→∞} ⩽ c(t − s)^{−ν}e^{ω(t−s)}  ((t, s) ∈ D′ J).  (3.6)

3.8 Remark. Assume that V^{−} is a small Miyadera perturbation and a small backward Miyadera perturbation of U, both with constants (α, γ), γ < 1. Then one can use Proposition 3.3 and an analogous estimate for ||U_{V} (t, s)||_{p′→∞} to show as in [LVV06; Prop. 3.6] that for θ ∈ (γ, 1) there exists c > 0 such that (3.6) holds with ω = \frac{θ}{α} ln(1+ \frac{γL}{θ−γ} )+(1−θ)A.

More strongly than in (A2) we now assume that the integral kernels p_{t,s} of the operators U(t, s) satisfy the Gaussian type upper bound

p_{t,s}(x, y) ⩽ K(t − s)^{−ν}e^{A(t−s)−ψ_{t,s}(x,y)} ((t, s) ∈ D′ J, x, y ∈ Ω),  (3.7)

with constants K, ν > 0, A ∈ ℝ and measurable functions ψ_{t,s}: Ω × Ω → [0, ∞).

3.9 Remark. If Ω = ℝ^{n}, ν = \frac{n}{2}, A = 0 and ψ_{t,s}(x, y) = a\frac{d(x,y)^{2}}{t−s} for some a > 0, then (3.7) becomes the classical Gaussian bound. By the results of [Aro68], this bound holds for the fundamental solution of (0.1).

Let now Ω be a complete n-dimensional Riemannian manifold with Ricci curvature bounded below, μ the Riemannian volume and d the Riemannian distance. Assume that inf_{x∈Ω} μ(B(x, 1)) > 0. Then it follows from [Dav93; Thm. 3 and bottom of p. 3] that a modified form of the above Gaussian bound also holds for the heat kernel on Ω: The term (t − s)^{−n/2} has to be replaced by (t − s)^{−n/2} ∨ 1 (cf. Remark 3.2(a)), and one can choose A = −λ_{0} and any 0 < a < \frac{1}{4}, where λ_{0} ⩾ 0 is the bottom of the spectrum of the Laplace-Beltrami operator in L_{2}(Ω, μ). In particular, for a negatively curved manifold one can have A < 0. However, without a uniform lower bound on the volume of unit balls, not even the ultracontractivity bound (A2) is valid. We will show in Section 3.3 how our method can be adapted to cover this more general case, too.

After the above preparations we easily obtain the following result on stability of upper kernel bounds.

3.10 Theorem. Let U satisfy (A1), (A3) and the Gaussian type upper bound (3.7). Let p ∈ (1, ∞), and let V be a potential such that pV is weakly U-admissible. Assume that U_{pV} is exponentially bounded and exponentially L_{∞}-bounded. Then the operators U_{V} (t, s) are integral operators, and for \frac{1}{p} < β < 1 there exist c > 0 and ω ∈ ℝ such that the kernels p_{t,s}^{V} satisfy

p_{t,s}^{V} ⩽ c(t − s)^{−ν}e^{ω(t−s)−(1−β)ψ_{t,s}} ((t, s) ∈ D′ J).

Proof. We have \frac{1}{β} < p, so by Proposition 3.7 there exist c_{1} > 0 and ω_{1} ∈ ℝ such that

||U_{V/β}(t, s)||_{1→∞} ⩽ c_{1}(t − s)^{−ν}e^{ω_{1}(t−s)} ((t, s) ∈ D′ J).

From Corollary 2.39 we obtain that p_{t,s}^{V} ⩽ (p_{t,s}^{V/β})^{β}(p_{t,s})^{1−β}. Thus, by (3.7) the assertion follows with ω = βω_{1} + (1 − β)A.

In the following, a Miyadera perturbation V of U will also be called a forward Miyadera perturbation.

3.11 Remarks. (a) Assume that V^{−} is a small forward and backward Miyadera perturbation of U, both with constants (α, γ), γ < 1. Then the assumptions on V in Theorem 3.10 are satisfied for all p ∈ (1, \frac{1}{γ}), by Theorem 2.24, Proposition 1.14 and Theorem 2.30. In this situation one can use Remark 3.8 to show that for γ < θ < β < 1 there exists c > 0 such that the kernel estimate of Theorem 3.10 holds with ω = \frac{θ}{α} ln(1 + \frac{γL}{θ−γ} ) + (1 − θ)A; cf. [LVV06; Thm. 3.10].

Assume that Ω = ℝ^{n}, that U is stochastic and Markovian and that U satisfies a classical Gaussian upper bound as in Remark 3.9. This is the case, e.g., if U corresponds to the Cauchy problem for (0.1). Then for V ⩽ 0 the conditions on V in the previous paragraph are close to necessary for the assertion of Theorem 3.10. Indeed, if U_{V} satisfies a Gaussian upper bound, then U_{V} is exponentially bounded and exponentially L_{∞}-bounded, so from Corollary 2.26(a) and Remark 2.44(c) it follows that V is a (not necessarily small) forward and backward Miyadera perturbation of U.

(b) If V^{−} is an infinitesimally small forward and backward Miyadera perturbation of U then one can choose any β > 0 in Theorem 3.10, and we obtain sharp upper bounds for t − s ⩽ 1. In applications to second-order parabolic equations this observation shows the difference between the non-autonomous Kato class NK and the enlarged non-autonomous Kato class N̂K.

(c) Assume that (3.7) holds with A = 0; we call this a global upper bound. For questions of long time behaviour it is important to know under what conditions the global upper bound is stable under perturbation by a potential V. By part (a) this is the case if V^{−} is a forward and backward Miyadera perturbation of U with constants (∞, γ), γ < 1.

With the same methods as above we obtain the following result about stability of the lower bound of the integral kernels, in which regularity of the perturbation V plays a crucial role.

3.12 Theorem. Let U satisfy (A1) to (A3) and the Gaussian type lower bound

p_{t,s}(x, y) ⩾ K_{1}(t − s)^{−ν}e^{−A_{1}(t−s)−ψ_{t,s}(x,y)}  ((t, s) ∈ D′ J, x, y ∈ Ω),  (3.8)

with some K_{1} > 0, A_{1} ∈ ℝ and measurable functions ψ_{t,s}: Ω × Ω → [0, ∞). Let V be a U-regular potential. Let ε > 0, and assume that −εV is weakly U-admissible and that U_{−εV} is exponentially bounded and exponentially L_{∞}-bounded. Then for β > \frac{1}{ε} there are constants c > 0 and ω ∈ ℝ such that the kernels p_{t,s}^{V} of the operators U_{V} (t, s) satisfy

p_{t,s}^{V} ⩾ c(t − s)^{−ν}e^{−ω(t−s)−(1+β)ψ_{t,s}} ((t, s) ∈ D′ J).

Proof. We have \frac{1}{β} < ε, so by Proposition 3.7 there exist c_{1} > 0 and ω_{1} ∈ ℝ such that

||U_{−V/β}(t, s)||_{1→∞} ⩽ c_{1}(t − s)^{−ν}e^{ω_{1}(t−s)} ((t, s) ∈ D′ J).  (3.9)

Recall from Lemma 2.20 that V^{+} and V^{−} are U-regular. Therefore, V_{0}^{+} ∧V^{−} and V_{0}^{−} ∧V^{+} are U-regular by Corollary 2.16, and Corollary 2.39 yields

p_{t,s} ⩽ (p_{t,s}^{−V/β})^{\frac{β}{1+β}} (p_{t,s}^{V})^{\frac{1}{1+β}} , so p_{t,s}^{V} ⩾ (p_{t,s}^{−V/β})^{−β}(p_{t,s})^{1+β}

for all (t, s) ∈ D′ J. The asserted lower bound thus follows from (3.8) and (3.9), with ω = βω_{1} + (1 + β)A_{1}.

Observations similar to those in Remark 3.11 also apply to the lower bounds.

3.13 Remarks. (a) If V is weakly U-admissible and V^{+} is a forward and backward Miyadera perturbation of U, both with constants (α, γ), γ ⩾ 0, then the assumptions on V in Theorem 3.12 are satisfied for all ε ∈ (0, \frac{1}{γ}), and one can use Remark 3.8 to show that for β > θ > γ there exists c > 0 such that the kernel estimate of Theorem 3.12 holds with ω = \frac{θ}{α} ln(1 + \frac{γL}{θ−γ} ) + (β − θ)A + (1 + β)A_{1}; cf. [LVV06; Thm. 3.12].

Assume that Ω = ℝ^{n} and that U satisfies a classical Gaussian lower bound (cf. Remark 3.9). Then for V ⩾ 0 the above conditions on V are necessary for the validity of a Gaussian lower bound for U_{V}. Indeed, one easily sees that there exists c ∈ (0, 1] such that ||U_{V} (t, s)f||_{1} ⩾ c||f||_{1} and Ũ U_{V} (t, s)𝟙 ⩾ c𝟙 for all (t, s) ∈ D_{J} with t − s ⩽ 1 and all 0 ⩽ f ∈ L_{1}(μ), where Ũ U_{V} is defined as in Remark 2.44(c). From Proposition 2.25(a) and Remark 2.44(d) it thus follows that V is a forward and backward Miyadera perturbation of U. We point out that this argument does not require U to be stochastic or Markovian.

(b) If V is weakly U-admissible and V^{+} is an infinitesimally small forward and backward Miyadera perturbation of U, then one can choose any β > 0 in Theorem 3.12, and we obtain sharp lower bounds for t − s ⩽ 1.

(c) Assume that A = 0 in the ultracontractivity bound (A2) and A_{1} = 0 in (3.8). Then by part (a), the global lower bound is stable under perturbation by the potential V if V^{+} is a forward and backward Miyadera perturbation of U with constants (∞, γ), with any γ ⩾ 0.

3.2 Stability of the boundary behaviour

Let Ω be a bounded connected open subset of ℝ^{n} with C^{1,α}-boundary, let T > 0, and let U be the propagator with parameter interval [0, T) associated with the uniformly parabolic equation (0.4) with Dini continuous coefficients. Recall from the introduction that by [Cho06] there exist c, a > 0 such that the kernels p_{t,s} of U(t, s) satisfy the Gaussian type upper bound

p_{t,s}(x, y) ⩽ c(t − s)^{−n/2}(\frac{d(x)d(y)}{t − s} ∧ 1) · exp(−a\frac{|x − y|^{2}}{t − s} )  (3.10)

for all (t, s) ∈ D_{J} and a.e. x, y ∈ Ω, where d(x) denotes the distance from x to the boundary of Ω, and also an analogous lower bound.

The propagator U is ultracontractive, so one can apply Theorem 3.10 in this situation; however, this does not yield stability of the kernel estimate: It leads to the boundary term \frac{d(x)d(y)}{t−s} ∧ 1 being raised to the power 1 − β. In fact, one can show that without stronger assumptions on the perturbation V, this change of the boundary behaviour of the kernel is possible. It will turn out that for stability of (3.10) one has to assume weak admissibility of V in a suitable weighted space. We will now study this type of kernel estimate in a more general framework.

For the remainder of the section let (Ω, μ) be a σ-finite measure space and d: Ω × Ω → [0, ∞) a measurable semi-metric on Ω (see also Remark 3.14(a) below). We assume that the volume growth condition

μ(B(x, r)) ⩽ Cr^{n}e^{κr}  (x ∈ Ω, r > 0)  (3.11)

is satisfied for some C, n > 0 and κ ⩾ 0, where B(x, r) denotes the open ball with centre x and radius r with respect to d. Moreover, let U be a positive strongly measurable propagator on L_{1}(μ) with parameter interval J ⊆ ℝ. We assume that for each (t, s) ∈ D′ J the operator U(t, s) is an integral operator whose kernel p_{t,s} satisfies the Gaussian type estimate

p_{t,s}(x, y) ⩽ c_{0}(t − s)^{−n/2} ∏_{j=1}^{m}(\frac{d_{j}(x)d_{j}(y)}{t − s} ∧ 1)^{α_{j}} · exp(−a\frac{d(x, y)^{2}}{t − s} )  (3.12)

for a.e. x, y ∈ Ω, where n is the constant from (3.11), m ∈ ℕ, c_{0}, a, α_{j} > 0 for j = 1, …, m, and d_{j}: Ω → (0, ∞) is a d-Lipschitz continuous function with Lipschitz constant 1 for each j = 1, …, m.

3.14 Remarks. (a) The measurability of the semi-metric d is meant with respect to the product σ-algebra on Ω × Ω, so it implies that d(x, ·) is measurable on Ω for all x ∈ Ω. Therefore, the open balls B(x, r) in (3.11) are measurable sets. The setting we chose here is more flexible than that of metric measure spaces, i.e., metric spaces with a measure on the Borel σ-algebra. It allows, e.g., to take a bounded open set Ω ⊆ ℝ^{n} with Lebesgue measure and the discrete metric. In this (admittedly pathological) example, the Borel σ-algebra is the power set of Ω, so one does not obtain a metric measure space.

(b) For the validity of (3.11) it is necessary that μ is σ-finite.

(c) Condition (3.11) is chosen in such a way that it comprises both the volume growth of ℝ^{n} and exponential volume growth. For the case of exponential volume growth it would be more natural to assume μ(B(x, r)) ⩽ Ce^{κr} for r > 1, but we want to keep the notational effort to a minimum.

(d) We have assumed for simplicity that the d_{j} are everywhere strictly positive. The case that d_{j} > 0 a.e. can easily be accommodated by replacing Ω with ⋂_{j=1}^{m}[d_{j} > 0].

(e) We observe that the upper bound in (3.12) is not changed essentially if the terms \frac{d_{j}(x)d_{j}(y)}{t−s} ∧ 1 are replaced with (\frac{d_{j}(x)}{\sqrt{t−s}} ∧ 1)(\frac{d_{j}(y)}{\sqrt{t−s}} ∧ 1). Indeed, for a, b ⩾ 0 one easily shows the estimate

(a ∧ 1)(b ∧ 1) ⩽ (ab) ∧ 1 ⩽ (1 + |a − b|)(a ∧ 1)(b ∧ 1)

(in the case a < 1 < b use that ab ∧ 1 ⩽ a(b − a + 1)), and for a = \frac{d_{j}(x)}{\sqrt{t−s}} and b = \frac{d_{j}(y)}{\sqrt{t−s}} one obtains that |a − b| ⩽ \frac{d(x,y)}{\sqrt{t−s}} ⩽ c_{ε} exp(ε\frac{d(x,y)^{2}}{t−s} ).

3.15 Example. Let n ⩾ 2, let Ω be an open subset of ℝ^{n}, and let T be the C_{0}-semigroup on L_{1}(Ω) associated with the heat equation ∂_{t}u = Δu on [0, ∞) × Ω with homogeneous Dirichlet boundary condition at ∂Ω.

(a) Let Ω := (0, ∞)^{n}. Then it is easy to see that, for t > 0, the kernel k_{t} of T (t) is given by

k_{t}(x, y) = (4πt)^{−n/2} ∏_{j=1}^{n}(1 − e^{−x_{j}y_{j}/t}) · exp(−\frac{|x − y|^{2}}{4t} ) (x, y ∈ Ω)

(cf. [Dav89; Examples 4.1.1 and 4.1.2]). Since \frac{1}{2}(r ∧ 1) ⩽ 1 − e^{−r} ⩽ r ∧ 1 for all r > 0, the kernels k_{t} satisfy (an autonomous version of) the kernel bound (3.12) and an analogous lower bound, with m = n, α_{j} = 1 and d_{j}(x) = x_{j} for j = 1, …, n and x ∈ Ω.

(b) Let Ω be a conical domain, i.e., Ω = {rξ; r > 0, ξ ∈ Σ}, where Σ is a proper open subset of the unit sphere S_{n−1}. We assume that Σ has C^{2}-boundary; then the ground state φ_{0} ⩾ 0 of the spherical Dirichlet Laplacian on Σ satisfies c^{−1}d_{Σ} ⩽ φ_{0} ⩽ cd_{Σ} for some c ⩾ 1, where d_{Σ}(ξ) denotes the distance of ξ ∈ Σ from ∂Σ; see, e.g., [Var99; formula (0.2.1)]. If λ_{0} > 0 denotes the ground state energy and α is the positive solution of α(α +n−2) = λ_{0}, then h(x) := |x|^{α}φ_{0}(\frac{x}{|x|}) defines a positive harmonic function on Ω that vanished at ∂Ω, the so-called réduite of Ω; cf. [Var99; p. 335]. By the above, there exists C ⩾ 1 such that

C^{−1}|x|^{α−1}d(x) ⩽ h(x) ⩽ C|x|^{α−1}d(x) (x ∈ Ω),

where d(x) denotes the distance from x to ∂Ω.

We further suppose that α ⩾ 1 (which is satisfied, e.g., if Ω is convex) and that e_{n} ∈ Ω and −e_{n} ∉ \overline{Ω}, where e_{n} denotes the n-th unit vector. The latter implies that ∂Ω is the graph of a Lipschitz function Φ: ℝ^{n−1} → ℝ. Under these assumptions it is straightforward to show the estimate

C_{1}^{−1}(\frac{|x|}{\sqrt{t}} ∧ 1)^{α−1}(\frac{d(x)}{\sqrt{t}} ∧ 1) ⩽ \frac{h(x)}{h(x + \sqrt{t}e_{n})} ⩽ C_{1}(\frac{|x|}{\sqrt{t}} ∧ 1)^{α−1}(\frac{d(x)}{\sqrt{t}} ∧ 1)

for all x ∈ Ω and all t > 0, with a constant C_{1} ⩾ 1. From [GySa09; Cor. 6.14 and Sec. 6.5.2] we now conclude, taking into account Remark 3.14(e), that the kernel k_{t} of T (t) satisfies (an autonomous version of) the kernel bound (3.12) and an analogous lower bound, for all t > 0. Here one takes m = 2, α_{1} = α − 1, d_{1}(x) = |x|, α_{2} = 1 and d_{2}(x) = d(x), and d(x, y) is the intrinsic geodesic distance of x, y ∈ Ω (d(x, y) = |x − y| if Ω is convex).

We are going to study under what conditions the kernel estimate (3.12) is stable under perturbation by a potential V : J ×Ω → ℝ; for a precise statement see Theorem 3.20 below. As a preparation we show that U extrapolates to a scale of weighted L_{1}-spaces. In the following, a measurable function ρ: Ω → (0, ∞) will be called a weight.

3.16 Proposition. Let a and the α_{j} be as in (3.12), and let ρ be a weight satisfying

\frac{ρ(x)}{ρ(y)} ⩽ c ∏_{j=1}^{m}(\frac{d_{j}(x)}{d_{j}(y)} ∨ \frac{d_{j}(y)}{d_{j}(x)})^{α_{j}}  (x, y ∈ Ω)  (3.13)

for some c ⩾ 1. Then for all a_{1} < a there exists M ⩾ 1 such that

||ρU(t, s)ρ^{−1}||_{p→p} ⩽ Me^{ω(t−s)} ((t, s) ∈ D_{J}, 1 ⩽ p ⩽ ∞)

holds with ω = \frac{κ^{2}}{4a_{1}} . Moreover, U extrapolates to an exponentially bounded strongly measurable propagator U_{ρ} on L_{1}(ρμ), and U_{ρ} is bounded if κ = 0. If U is strongly continuous then so is U_{ρ}.

The above proposition implies in particular that U is exponentially bounded, and bounded if κ = 0. In the proof of the proposition we will use the following lemma in which the volume growth condition (3.11) is essentially reformulated as an integral condition.

3.17 Lemma. Let y ∈ Ω and n > 0. If

μ(B(y, r)) ⩽ c_{1}r^{n}e^{κr}  (r > 0)  (3.14)

for some c_{1} > 0 and κ ⩾ 0, then for all θ ∈ (0, 1) there exists c_{2} > 0 such that

∫_{Ω} e^{−d(x,y)^{2}/t} dμ(x) ⩽ c_{2}t^{n/2}e^{ωt}  (t > 0)  (3.15)

holds with ω = \frac{κ^{2}}{4θ}, in particular, ω = 0 if κ = 0. Conversely, if (3.15) holds for some c_{2} > 0 and ω ⩾ 0, then (3.14) holds with c_{1} = ec_{2} and κ = 2\sqrt{ω}.

Proof. Assume that (3.14) holds, without loss of generality with c_{1} = 1, and let m(r) := r^{n}e^{κr} for all r ⩾ 0. Then for any decreasing function f: [0, ∞) → [0, ∞) we can estimate

∫_{Ω} f(d(x, y)) dμ(x) ⩽ ∫_{0}^{∞} f(r) dm(r).

(This is clear if f = 𝟙_{[0,r)} for some r > 0; for general f it follows by a superposition argument.) Given t > 0 we thus obtain, substituting r = \sqrt{ts} and integrating by parts, that

∫_{Ω} e^{−d(x,y)^{2}/t} dμ(x) ⩽ ∫_{0}^{∞} e^{−r^{2}/t} dm(r) = ∫_{0}^{∞} e^{−s} dm(\sqrt{ts}) = ∫_{0}^{∞} m(\sqrt{ts})e^{−s} ds.

Let θ ∈ (0, 1). Then κ\sqrt{ts} ⩽ \frac{κ^{2}}{4θ} t + θs for all s > 0 and hence m(\sqrt{ts}) ⩽ (ts)^{n/2}e^{ωt+θs}, with ω = \frac{κ^{2}}{4θ}. We conclude that

∫_{Ω} e^{−d(x,y)^{2}/t} dμ(x) ⩽ t^{n/2}e^{ωt} ∫_{0}^{∞} s^{n/2}e^{−(1−θ)s} ds = Γ(\frac{n}{2} + 1)(1 − θ)^{−n/2−1}t^{n/2}e^{ωt}.

Assume now that (3.15) holds, and let r > 0. Then for all t > 0 we have

μ(B(y, r)) ⩽ ∫_{Ω} e^{(r^{2}−d(x,y)^{2})/t} dμ(x) ⩽ e^{r^{2}/t} · c_{2}t^{n/2}e^{ωt}.

In the case r < ω^{−1/2} we choose t = r^{2}. Then ωt = ωr^{2} ⩽ \sqrt{ω} r and hence

μ(B(y, r)) ⩽ ec_{2}r^{n}e^{\sqrt{ω}r}.

In the case ω > 0 and r ⩾ ω^{−1/2} we choose t = rω^{−1/2}. Then t ⩽ r^{2}, r^{2}/t + ωt = 2\sqrt{ω} r, and thus

μ(B(y, r)) ⩽ c_{2}r^{n}e^{2\sqrt{ω}r}.

3.18 Remarks. (a) Let k: Ω × Ω → [0, ∞) be measurable. It is easy to see that

Bf(x) := ∫_{Ω} k(x, y)f(y) dμ(y) (f ∈ L_{1}(μ))

defines a bounded operator on L_{1}(μ) if and only if

b := ess sup_{y∈Ω} ∫_{Ω} k(x, y) dμ(x) < ∞,

and that then ||B|| = b. Moreover,

||B||_{∞→∞} = ess sup_{x∈Ω} ∫_{Ω} k(x, y) dμ(y) (∈ [0, ∞]).

For θ ∈ (0, 1) we thus obtain by Lemma 3.17 that there exists c_{2} > 0 such that the following holds: If a, t > 0 and k(x, y) ⩽ t^{−n/2}e^{−ad(x,y)^{2}/t} for a.e. x, y ∈ Ω, then ||B|| ⩽ c_{2}a^{−n/2}e^{ωt} and ||B||_{∞→∞} ⩽ c_{2}a^{−n/2}e^{ωt}, with ω = \frac{κ^{2}}{4θa}.

(b) Let k and B be as above, and let ρ be a weight. If B extrapolates to a bounded operator B_{ρ} on the weighted space L_{1}(ρμ), then the kernel k_{ρ} of B_{ρ} is given by k_{ρ}(x, y) = k(x, y)ρ(y)^{−1}. It follows that

b_{ρ} := ||B: L_{1}(ρμ) → L_{1}(ρμ)|| = ess sup_{y∈Ω} ∫_{Ω} \frac{ρ(x)}{ρ(y)}k(x, y) dμ(x),

and as in (a) we obtain that b_{ρ} ⩽ c_{2}a^{−n/2}e^{ωt} if \frac{ρ(x)}{ρ(y)}k(x, y) ⩽ t^{−n/2}e^{−ad(x,y)^{2}/t} for a.e. x, y ∈ Ω and some a, t > 0.

Proof of Proposition 3.16. Let (t, s) ∈ D′ J. First observe that for all x, y ∈ Ω and j = 1, …, m we have

\frac{d_{j}(x)}{d_{j}(y)}(\frac{d_{j}(x)d_{j}(y)}{t − s} ∧ 1) = \frac{d_{j}(x)^{2}}{t − s} ∧ \frac{d_{j}(x)}{d_{j}(y)} ⩽ 3(1 + \frac{d(x, y)^{2}}{t − s} ).

Indeed, in the case d_{j}(x) ⩽ 3d_{j}(y) the inequality is clear, and in the case d_{j}(x) > 3d_{j}(y) the Lipschitz continuity of d_{j} yields d(x, y) ⩾ d_{j}(x)−d_{j}(y) ⩾ \frac{2}{3}d_{j}(x) and hence \frac{d_{j}(x)^{2}}{t−s} ⩽ \frac{9}{4} \frac{d(x,y)^{2}}{t−s}. Interchanging the roles of x and y we obtain that

(\frac{d_{j}(x)}{d_{j}(y)} ∨ \frac{d_{j}(y)}{d_{j}(x)})(\frac{d_{j}(x)d_{j}(y)}{t − s} ∧ 1) ⩽ 3(1 + \frac{d(x, y)^{2}}{t − s} ).

From (3.12) and (3.13) we conclude that for θ ∈ (0, 1) there exists c_{θ} > 0 such that

\begin{aligned}\frac{ρ(x)}{ρ(y)}p_{t,s}(x, y) & ⩽ c · c_{0}(t − s)^{−n/2} ∏_{j=1}^{m} 3^{α_{j}}(1 + \frac{d(x, y)^{2}}{t − s} )^{α_{j}} · exp(−a\frac{d(x, y)^{2}}{t − s} ) \\ & ⩽ c_{θ}(t − s)^{−n/2} exp(−θa\frac{d(x, y)^{2}}{t − s} ) \end{aligned} (3.16)

for a.e. x, y ∈ Ω and all (t, s) ∈ D′ J. The first assertion now follows from Remark 3.18(a) and the Riesz-Thorin interpolation theorem. In particular, U extrapolates to an exponentially bounded propagator U_{ρ} on L_{1}(ρμ) (cf. Remark 3.18(b)), and U_{ρ} is bounded if κ = 0.

Given f ∈ L_{1}(ρμ) ∩ L_{1}(μ) we now show that (t, s) ↦ U_{ρ}(t, s)f ∈ L_{1}(ρμ) is separately measurable; then it follows that U_{ρ} is a strongly measurable propagator. For (t, s) ∈ D_{J} we have U(t, s)f ∈ L_{1}(ρμ) and hence ρU(t, s)f ∈ L_{1}(μ), so (ρ ∧ n)U(t, s)f → ρU(t, s)f in L_{1}(μ) as n → ∞. Since U is separately strongly measurable, this implies the separate measurability of (t, s) ↦ ρU(t, s)f ∈ L_{1}(μ) and hence of (t, s) ↦ U(t, s)f ∈ L_{1}(ρμ).

Now assume that U is strongly continuous. We first show that U_{ρ} is strongly continuous in (t_{0}, t_{0}), for all t_{0} ∈ J. Observe that the set

F := {f ∈ L_{1}(μ); spt f is bounded, d_{j}^{±1} is bounded on spt f (j = 1, …, m)}

is dense in L_{1}(ρμ). It thus suffices to show that U(t, s)f → U(t_{0}, t_{0})f = f in L_{1}(ρμ) as (t, s) → (t_{0}, t_{0}), for all f ∈ F. Given f ∈ F, we obtain from the Lipschitz continuity of the d_{j} that there exists ε > 0 such that ρ is bounded on B := {x ∈ Ω; dist(x, spt f) ⩽ ε}. Therefore, the strong continuity of U implies that 𝟙_{B}U(t, s)f → 𝟙_{B}f = f in L_{1}(ρμ) as (t, s) → (t_{0}, t_{0}), and it remains to show that 𝟙_{Ω\B}U(t, s)f → 0 in L_{1}(ρμ).

For x ∈ Ω \ B and y ∈ spt f we have d(x, y) ⩾ ε and hence d(x, y)^{2} ⩾ \frac{1}{2}d(x, y)^{2} + \frac{1}{2}ε^{2}, so by (3.16) we obtain that

\frac{ρ(x)}{ρ(y)}𝟙_{Ω\B}(x)p_{t,s}(x, y)𝟙_{spt f}(y) ⩽ exp(−\frac{aε^{2}}{4(t − s)}) · c_{1/2}(t − s)^{−n/2} exp(−\frac{a}{4} \frac{d(x, y)^{2}}{t − s} ).

By Remark 3.18(b) we infer that there exist M > 0 and ω ⩾ 0 such that

||𝟙_{Ω\B}U(t, s)f||_{L_{1}(ρμ)} ⩽ exp(−\frac{aε^{2}}{4(t − s)}) · Me^{ω(t−s)}||f||_{L_{1}(ρμ)} → 0

as t − s → 0.

Finally, given ε ∈ (0, 1), we show that U_{ρ} is strongly continuous on

Dε J := \{(t, s) ∈ D_{J}; ε ⩽ t − s ⩽ \frac{1}{ε}\}.

Let f ∈ L_{1}(ρμ) ∩ L_{1}(μ). Then the continuity of Dε J ∋ (t, s) ↦ U(t, s)f in L_{1}(ρμ) can be deduced from the continuity in L_{1}(μ) by means of the dominated convergence theorem if there exists g ∈ L_{1}(ρμ) such that |U(t, s)f| ⩽ g for all (t, s) ∈ Dε J. By (3.16) we obtain that

\frac{ρ(x)}{ρ(y)}p_{t,s}(x, y) ⩽ c_{1/2}ε^{−n/2} exp(−\frac{εa}{2}d(x, y)^{2}) =: k(x, y)

for all (t, s) ∈ Dε J and a.e. x, y ∈ Ω, so

|U(t, s)f(x)| ⩽ ρ(x)^{−1} · ∫_{Ω} k(x, y)|ρf|(y) dμ(y) =: g(x)

for all (t, s) ∈ Dε J and a.e. x ∈ Ω. This completes the proof of the strong continuity of U_{ρ} since g ∈ L_{1}(ρμ) by Remark 3.18(a).

From now on we assume that U is strongly continuous. Moreover, let ρ_{0} := ∏_{j=1}^{m} d_{j}^{α_{j}} in the following. Then ρ_{0} is a weight that satisfies the assumption of Proposition 3.16.

3.19 Corollary. Let V be weakly U_{ρ_{0}}-admissible, and let ρ be a weight satisfying

\frac{ρ(x)}{ρ(y)} ⩽ c ∏_{j=1}^{m}(\frac{d_{j}(x)}{d_{j}(y)} ∨ 1)^{α_{j}} (x, y ∈ Ω)

for some c ⩾ 1. Then \frac{1}{2}V is weakly U_{ρ}-admissible. Moreover, if (U_{ρ_{0}})_{V} is exponentially bounded then so is (U_{ρ})_{\frac{1}{2} V}.

Proof. Observe that the weight ρ_{1} := ρ^{2}ρ_{0}^{−1} satisfies the assumption of Proposition 3.16. Indeed,

\frac{ρ_{1}(x)}{ρ_{1}(y)} ⩽ c^{2} ∏_{j=1}^{m}(\frac{d_{j}(x)^{2}}{d_{j}(y)^{2}} ∨ 1)^{α_{j}} · \frac{ρ_{0}(y)}{ρ_{0}(x)} = c^{2} ∏_{j=1}^{m}(\frac{d_{j}(x)}{d_{j}(y)} ∨ \frac{d_{j}(y)}{d_{j}(x)})^{α_{j}}

for all x, y ∈ Ω. The assertions thus follow from Proposition 2.49, applied with U_{j} = U_{ρ_{j}} for j = 0, 1, V_{0} = V, V_{1} = 0 and θ = \frac{1}{2}.

After these preparations we are ready to prove our main result on stability of the boundary behaviour of kernel estimates.

3.20 Theorem. Let V be a potential such that qV is weakly U_{ρ_{0}}-admissible, for some q > 2. Assume that (U_{ρ_{0}})_{qV} is exponentially bounded and that

||ρ_{0}^{−1}(U_{ρ_{0}})_{qV} (t, s)ρ_{0}||_{∞→∞} ⩽ Me^{ω(t−s)}  ((t, s) ∈ D_{J})  (3.17)

for some M ⩾ 1 and ω ∈ ℝ. Then for 0 < a_{2} < (1 − \frac{2}{q})a there exist c_{2} > 0 and ω_{2} ∈ ℝ such that the kernels p_{t,s}^{V} of the operators U_{V} (t, s) satisfy

p_{t,s}^{V}(x, y) ⩽ c_{2}(t − s)^{−n/2} ∏_{j=1}^{m}(\frac{d_{j}(x)d_{j}(y)}{t − s} ∧ 1)^{α_{j}} · exp(ω_{2}(t − s) − a_{2} \frac{d(x, y)^{2}}{t − s} )

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω.

Proof. Let F ⊆ {1, …, m} and ρ_{F} := ∏_{j∈F} d_{j}^{α_{j}}. Then ρ_{F} satisfies the assumption of Corollary 3.19, so \frac{q}{2}V is weakly U_{ρ_{F}} -admissible and (U_{ρ_{F}})_{\frac{q}{2} V} is exponentially bounded. The operator of multiplication with ρ_{F} is an isometry ρ_{F} : L_{1}(ρ_{F}^{2} μ) → L_{1}(ρ_{F} μ), so

U_{F} (t, s) := ρ_{F}^{−1}U_{ρ_{F}} (t, s)ρ_{F} ((t, s) ∈ D_{J})

defines a strongly continuous propagator U_{F} on L_{1}(ρ_{F}^{2} μ). The kernel p_{t,s}^{F} of U_{F} (t, s) is given by p_{t,s}^{F}(x, y) = ρ_{F} (x)^{−1}p_{t,s}(x, y)ρ_{F} (y)ρ_{F} (y)^{−2}, for all (t, s) ∈ D_{J} (cf. Remark 3.18(b)), so it satisfies

p_{t,s}^{F}(x, y) = ∏_{j∈F}(d_{j}(x)d_{j}(y))^{−α_{j}}p_{t,s}(x, y) ⩽ c_{0}(t − s)^{−n/2−∑_{j∈F} α_{j}} exp(−a\frac{d(x,y)^{2}}{t−s} )

for a.e. x, y ∈ Ω. In particular we obtain that U_{F} is ultracontractive,

||U_{F} (t, s): L_{1}(ρ_{F}^{2} μ) → L_{∞}(ρ_{F}^{2} μ)|| ⩽ c_{0}(t − s)^{−n/2−∑_{j∈F} α_{j}} ((t, s) ∈ D′ J).

Let p := \frac{q}{2}, and let (t, s) ∈ D_{J}. By Lemma 2.47, the weak U_{ρ_{F}} -admissibility of pV implies that pV is weakly U_{F} -admissible, and with (U_{ρ_{F}})_{pV} also (U_{F})_{pV} is exponentially bounded. Moreover,

(U_{F})_{pV} (t, s)f = ρ_{F}^{−1}(U_{ρ_{F}})_{pV} (t, s)ρ_{F} f = ρ_{F}^{−1}(U_{ρ_{0}})_{pV} (t, s)ρ_{F} f  (3.18)

for all f ∈ L_{1}(ρ_{F}^{2} μ) with ρ_{F} f ∈ L_{1}(ρ_{0}μ) since (U_{ρ_{F}})_{pV} and (U_{ρ_{0}})_{pV} are consistent by Lemma 2.48(a). Let ρ_{1} := ρ_{F}^{2} ρ_{0}^{−1}, and observe that the weight ρ_{1}^{−1} satisfies the assumption of Proposition 3.16. By Theorem 2.38 and Corollary 2.37 we obtain that

||ρ_{F}^{−1}(U_{ρ_{0}})_{pV} (t, s)ρ_{F} ||_{∞→∞} ⩽ ||ρ_{1}^{−1}U_{ρ_{0}}(t, s)ρ_{1}||_{∞→∞}^{1/2}||ρ_{0}^{−1}(U_{ρ_{0}})_{qV} (t, s)ρ_{0}||_{∞→∞}^{1/2}.

From (3.18), Proposition 3.16 and the assumption (3.17) we thus infer that (U_{F})_{pV} is exponentially L_{∞}-bounded.

Let now a_{2} < (1 − \frac{1}{p})a. Then by the above we can apply Theorem 3.10 to obtain c_{F} > 0 and ω_{F} ∈ ℝ such that, for all (t, s) ∈ D′ J, the kernel p_{t,s}^{F,V} of (U_{F})_{V} (t, s) satisfies

p_{t,s}^{F,V} (x, y) ⩽ c_{F} (t − s)^{−n/2−∑_{j∈F} α_{j}} exp(ω_{F} (t − s) − a_{2} \frac{d(x,y)^{2}}{t−s} )

for a.e. x, y ∈ Ω (see also Remark 3.1 for the case that U_{F} is not bounded). Moreover, (U_{F})_{V} (t, s) = ρ_{F}^{−1}(U_{ρ_{F}})_{V} (t, s)ρ_{F} for all (t, s) ∈ D_{J} by Lemma 2.47, and (U_{ρ_{F}})_{V} and U_{V} are consistent by Lemma 2.48(a), so we conclude that

p_{t,s}^{V}(x, y) = ρ_{F} (x)p_{t,s}^{F,V} (x, y)ρ_{F} (y)

⩽ c_{F} (t − s)^{−n/2} ∏_{j∈F}(\frac{d_{j}(x)d_{j}(y)}{t − s})^{α_{j}} exp(ω_{F} (t − s) − a_{2} \frac{d(x, y)^{2}}{t − s} ),

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω. The assertion thus follows by taking the minimum over F ⊆ {1, …, m}.

3.21 Remarks. (a) Observe that the assumptions of Theorem 3.20 are necessary except for the factor q > 2. Indeed, if U_{V} satisfies the asserted upper bound, then by Proposition 3.16 there exist M ⩾ 1 and ω ∈ ℝ such that ||ρ_{0}^{±1}U_{V} (t, s)ρ_{0}^{∓1}||_{p→p} ⩽ Me^{ω(t−s)} for all (t, s) ∈ D_{J} and p = 1, ∞. From this it follows that V is weakly U_{ρ_{0}}-admissible, that (U_{ρ_{0}})_{V} is exponentially bounded and that (3.17) is satisfied.

(b) The assumption that qV is weakly U_{ρ_{0}}-admissible for some q > 2, and that (U_{ρ_{0}})_{qV} is exponentially bounded (the “forward” condition), is in particular satisfied if V^{−} is a Miyadera perturbation of U_{ρ_{0}} with constants (α, γ), γ < \frac{1}{2}.

Let F_{m} := {1, …, m}; then ρ_{F_{m}} = ρ_{0}. Let the propagator U_{F_{m}} on L_{1}(ρ_{0}^{2}μ) be defined as in the proof of Theorem 3.20, and assume that V^{−} is a backward Miyadera perturbation of U_{F_{m}} with constants (α, γ), γ < \frac{1}{2}. Then (U_{F_{m}})_{qV} is exponentially L_{∞}-bounded for all q ∈ (2, \frac{1}{γ}), and this implies that (3.17) (the “backward” condition) is satisfied since (U_{F_{m}})_{qV} (t, s) = ρ_{0}^{−1}(U_{ρ_{0}})_{qV} (t, s)ρ_{0} for all (t, s) ∈ D_{J}.

(c) Assume that the backward propagator U′ defined in Remark 2.31(a) is strongly measurable. Then it is possible to reformulate condition (3.17) so that it becomes analogous to the forward condition. Since ||ρ_{0}U′(s, t)ρ_{0}^{−1}||_{1→1} = ||ρ_{0}^{−1}U(t, s)ρ_{0}||_{∞→∞} for all (t, s) ∈ D_{J}, it follows from Proposition 3.16 that U′ extrapolates to an exponentially bounded strongly measurable backward propagator (U′)_{ρ_{0}} on L_{1}(ρ_{0}μ). By Remark 2.31(c), Lemma 2.47 and Lemma 2.48(a) we obtain, for (t, s) ∈ D_{J}, that the estimate in (3.17) holds if and only if

||((U′)_{ρ_{0}})_{qV ∨(−n)}(s, t)|| = ||ρ_{0}U_{qV ∨(−n)}(t, s)′ρ_{0}^{−1}||_{1→1} ⩽ Me^{ω(t−s)}

for all n ∈ ℕ. Therefore, (3.17) is satisfied if and only if qV is (U′)_{ρ_{0}}-admissible and ((U′)_{ρ_{0}})_{qV} is exponentially bounded.

(d) In the following we explain how the Miyadera conditions occurring in (b) can be formulated analogously to the non-autonomous Kato class conditions discussed in the introduction, as inequalities for integrals involving V, ρ and the kernels p_{t,s}.

We assume that the kernels p_{t,s} can be chosen in such a way that (t, s, x, y) ↦ p_{t,s}(x, y) is measurable on D′ J × Ω × Ω, and we fix such a choice. As in [ArBu94; proof of Thm. 2.1] one can show that this is always possible if L_{1}(μ) is separable. (The argument given there requires only strong measurability of U; Example 2.32(a) demonstrates that then σ-finiteness of μ is not sufficient, as was claimed in [ArBu94].)

Let ρ be a weight satisfying the assumption of Proposition 3.16 (or, more generally, a weight such that U extrapolates to a locally bounded strongly measurable propagator U_{ρ} on L_{1}(ρμ)). Observe that a potential V is a Miyadera perturbation of U_{ρ} with constants (α, γ) if and only if the operators A_{t,s}^{n} ∈ L(L_{1}(μ)) that are defined by

A_{t,s}^{n}f := ∫_{s}^{t}(|V | ∧ n)(τ)U(τ, s)f dτ

satisfy ||A_{t,s}^{n}: L_{1}(ρμ) → L_{1}(ρμ)|| ⩽ γ for all n ∈ ℕ and all (t, s) ∈ D′ J with t − s ⩽ α. Moreover, the kernel k_{t,s}^{n} of A_{t,s}^{n} is given by

k_{t,s}^{n}(x, y) = ∫_{s}^{t}(|V | ∧ n)(τ, x)p_{τ,s}(x, y) dτ,

for n ∈ ℕ and (t, s) ∈ D′ J. Therefore, by Remark 3.18(b) and the monotone convergence theorem, V is a Miyadera perturbation of U_{ρ} with constants (α, γ) if and only if

γ_{α}^{+}(ρ, V) := sup_{0<t−s⩽α} ess sup_{x∈Ω} ∫_{s}^{t} ∫_{Ω} \frac{ρ(y)}{ρ(x)}p_{τ,s}(y, x)|V (τ, y)| dμ(y) dτ ⩽ γ.

Let Ũ U_{ρ} be the strongly measurable propagator on L_{1}(ρ^{2}μ) defined by Ũ U_{ρ}(t, s) := ρ^{−1}U_{ρ}(t, s)ρ. By condition (ii) of Proposition 2.29 and the monotone convergence theorem we obtain that V is a backward Miyadera perturbation of Ũ U_{ρ} with constants (α, γ) if and only if

γ_{α}^{−}(ρ, V) := sup_{0<t−s⩽α} ess sup_{x∈Ω} ∫_{s}^{t} ∫_{Ω} \frac{ρ(y)}{ρ(x)}p_{t,τ}(x, y)|V (τ, y)| dμ(y) dτ ⩽ γ.

Note that the only difference to the definition of γ_{α}^{+}(ρ, V) is that there the term p_{τ,s}(y, x) occurs in the integral.

Now assume that μ(B(x, r)) > 0 for all x ∈ X and all r > 0. Observe that then any lower semicontinuous function h on (Ω, d) satisfies ess sup_{x∈Ω} h(x) = sup_{x∈Ω} h(x). Therefore, the ess sup in the definition of γ_{α}^{+}(ρ, V) can be replaced with a sup if the function

x ↦ ∫_{s}^{t} ∫_{Ω} \frac{ρ(y)}{ρ(x)}p_{τ,s}(y, x)|V (τ, y)| dμ(y) dτ

is lower semicontinuous on (Ω, d) for all (t, s) ∈ D′ J. By Fatou’s lemma, this is the case if p_{t,s}(y, ·) is lower semicontinuous for all (t, s) ∈ D′ J and a.e. y ∈ Ω. An analogous observation applies to the ess sup in the definition of γ_{α}^{−}(ρ, V).

For the last result of this section we assume, in addition to the Gaussian type upper bound (3.12), that the kernels p_{t,s} satisfy a matching lower bound

p_{t,s}(x, y) ⩾ K_{1}(t − s)^{−n/2} ∏_{j=1}^{m}(\frac{d_{j}(x)d_{j}(y)}{t − s} ∧ 1)^{α_{j}} · exp(−A_{1}(t − s) − a_{1} \frac{d(x, y)^{2}}{t − s} )

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω, where K_{1}, a_{1} > 0 and A_{1} ⩾ 0, and the other quantities are the same as in the upper bound.

3.22 Theorem. Let V be a U-regular potential. Let ε > 0, and assume that −εV is weakly U_{ρ_{0}}-admissible, that (U_{ρ_{0}})_{−εV} is exponentially bounded, and that

||ρ_{0}^{−1}(U_{ρ_{0}})_{−εV} (t, s)ρ_{0}||_{∞→∞} ⩽ Me^{ω(t−s)} ((t, s) ∈ D_{J})

for some M ⩾ 1 and ω ∈ ℝ. Then for a_{2} > (1 + \frac{2}{ε})a_{1} there exist c_{2} > 0 and ω_{2} ∈ ℝ such that

p_{t,s}^{V}(x, y) ⩾ c_{2}(t − s)^{−n/2} ∏_{j=1}^{m}(\frac{d_{j}(x)d_{j}(y)}{t − s} ∧ 1)^{α_{j}} · exp(−ω_{2}(t − s) − a_{2} \frac{d(x, y)^{2}}{t − s} )

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω.

Proof. Let a_{2} > (1+ \frac{2}{ε})a_{1}, and let β > \frac{2}{ε} be such that a_{2} = (1+β)a. Then q(−V/β) = −εV with q = εβ > 2, so by Theorem 3.20 there exist c_{1} > 0 and ω_{1} ∈ ℝ such that

p_{t,s}^{−V/β}(x, y) ⩽ c_{1}(t − s)^{−n/2} ∏_{j=1}^{m}(\frac{d_{j}(x)d_{j}(y)}{t − s} ∧ 1)^{α_{j}} · e^{ω_{1}(t−s)}

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω. Moreover, as in the proof of Theorem 3.12 we obtain that

p_{t,s}^{V} ⩾ (p_{t,s}^{−V/β})^{−β}(p_{t,s})^{1+β} ((t, s) ∈ D′ J).

The assertion thus follows from the lower bound on the kernels p_{t,s}.

3.3 Kernel estimates on manifolds

Throughout this section let (Ω, μ) be a σ-finite measure space, and let U be a positive strongly continuous propagator on L_{1}(μ) with parameter interval J ⊆ ℝ. We assume that the operators U(t, s) have integral kernels p_{t,s} satisfying

p_{t,s}(x, y) ⩽ h_{t−s}(x)h_{t−s}(y)e^{A(t−s)−ψ_{t,s}(x,y)}  (3.19)

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω, with A ∈ ℝ and measurable functions h_{t}: Ω → (0, ∞) and ψ_{t,s}: Ω × Ω → [0, ∞). We do not need a multiplicative constant c in the estimate since this can be incorporated into the functions h_{t}.

In Section 3.1, h_{t} was the constant function h_{t} = K^{1/2}t^{−ν/2}, for t > 0. For the heat kernel of a complete Riemannian manifold with non-negative Ricci curvature it is proved in [LiYa86] that the bound (3.19) and a matching lower bound hold with A = 0, h_{t}(x) = cμ(B(x, \sqrt{t}))^{−1/2} and ψ_{t,s}(x, y) = a\frac{d(x,y)^{2}}{t−s}, where μ is the Riemannian volume and d the Riemannian distance. This is the so-called Li-Yau estimate. Another possible choice for heat kernel estimates on Riemannian manifolds is h_{t}(x) = cμ(B(x, \sqrt{t} ∧ 1))^{−1/2}. Note that in the above mentioned cases, t ↦ h_{t}(x) is decreasing for all x ∈ Ω; this property of the h_{t} will be needed in some of the results below.

For our first result on stability of the kernel estimate (3.19) we only require the following asymmetric bound. For all k ∈ ℕ there exist c_{k} > 0 and r_{k} ∈ ℝ such that

p_{t,s}(x, y) ⩽ c_{k}h_{k(t−s)}(x)^{1+q}h_{k(t−s)}(y)^{1−q}e^{r_{k}(t−s)}  (3.20)

for all (t, s) ∈ D′ J, a.e. x, y ∈ Ω and q = ±(k − 1). This is analogous to assumption (IV) in [Stu93; Sec. 4], where similar results for Schrödinger semigroups with measures on manifolds are proved. (By [Stu93; Cor. 4.4], the heat kernel of a complete Riemannian manifold with Ricci curvature bounded below satisfies (3.20) with h_{t}(x) = cμ(B(x, \sqrt{t}))^{−1/2}; see also (3.30) below.) We point out that we do not require any quantitative assumptions on the measure μ such as volume growth conditions.

3.23 Theorem. Assume (3.19) and (3.20). Let γ ∈ (0, 1), and let V be a potential such that V/γ is weakly U-admissible. Assume that there exist M ⩾ 1 and ω_{γ} ∈ ℝ such that ||U_{V/γ}(t, s)||_{p→p} ⩽ Me^{ω_{γ}(t−s)} for all (t, s) ∈ D_{J} and p = 1, ∞. Then for β ∈ (γ, 1) there exist c > 0 and ω ∈ ℝ such that, for all (t, s) ∈ D′ J, the kernel p_{t,s}^{V} of U_{V} (t, s) satisfies

p_{t,s}^{V}(x, y) ⩽ ch_{t−s}(x)h_{t−s}(y)e^{ω(t−s)−(1−β)ψ_{t,s}(x,y)}  (x, y ∈ Ω).  (3.21)

More precisely, if k ∈ ℕ satisfies \frac{1}{k} ⩽ 1 − \frac{γ}{β}, then there exists c > 0 such that the above holds with ω = γω_{γ} + k′γ \frac{r_{k}}{k} + (1 − k′γ)A, where \frac{1}{k} + \frac{1}{k′} = 1.

Proof. Observe that the condition on k is equivalent to \frac{γ}{β} ⩽ \frac{1}{k′} and hence to β ⩾ k′γ. We can thus assume without loss of generality that β = k′γ. Then for (t, s) ∈ D′ J we obtain by Theorem 2.38, Corollary 2.37 and the bound (3.20) that

||h_{k(t−s)}^{1−2/k}U_{V/β}(t, s)h_{k(t−s)}^{−1}||_{1→k′} ⩽ ||h_{k(t−s)}^{k−2}U(t, s)h_{k(t−s)}^{−k}||_{1→∞}^{1/k}||U_{V/γ}(t, s)||_{1→1}^{1/k′}
⩽ (c_{k}e^{r_{k}(t−s)})^{1/k}(Me^{ω_{γ}(t−s)})^{1/k′} =: φ(t − s),

and in the same way

||h_{k(t−s)}^{−1}U_{V/β}(t, s)h_{k(t−s)}^{1−2/k}||_{k→∞} ⩽ φ(t − s).

Another application of Corollary 2.37 yields

||h_{k(t−s)}^{−1+2/q}U_{V/β}(t, s)h_{k(t−s)}^{1−2/p}||_{p→q} ⩽ φ(t − s)  (3.22)

for all 1 ⩽ p < q ⩽ ∞ with \frac{1}{p} − \frac{1}{q} = \frac{1}{k}.

Let now (t, s) ∈ D′ J, and let 1 = p_{0} < p_{1} < ⋯ < p_{k} = ∞ and s = t_{0} < t_{1} < ⋯ < t_{k} = t satisfy \frac{1}{p_{j−1}} − \frac{1}{p_{j}} = \frac{1}{k} and t_{j} − t_{j−1} = \frac{t−s}{k} for j = 1, …, k. Then from (3.22) we infer that

||h_{t−s}^{−1}U_{V/β}(t, s)h_{t−s}^{−1}||_{1→∞} ⩽ ∏_{j=1}^{k} ||h_{t−s}^{−1+2/p_{j}}U_{V/β}(t_{j}, t_{j−1})h_{t−s}^{1−2/p_{j−1}} ||_{p_{j−1}→p_{j}}
⩽ φ(\frac{t−s}{k})^{k} = c_{k}M^{k−1}e^{ω_{1}(t−s)},

where ω_{1} = \frac{r_{k}}{k} + \frac{ω_{γ}}{k′}, so p_{t,s}^{V/β}(x, y) ⩽ c_{k}M^{k−1}h_{t−s}(x)h_{t−s}(y)e^{ω_{1}(t−s)} for a.e. x, y ∈ Ω. By Corollary 2.39 and the upper bound (3.19) for p_{t,s} we conclude that

p_{t,s}^{V}(x, y) ⩽ p_{t,s}^{V/β}(x, y)^{β}p_{t,s}(x, y)^{1−β} ⩽ ch_{t−s}(x)h_{t−s}(y)e^{ω(t−s)−(1−β)ψ_{t,s}(x,y)}

for all a.e. x, y ∈ Ω, with c = c_{k}^{β}M^{β(k−1)} and ω = βω_{1} +(1−β)A. This implies the assertion since β = k′γ and hence βω_{1} = k′γ \frac{r_{k}}{k} + γω_{γ}.

3.24 Remark. (a) The basic interpolation and iteration idea behind the L_{1}–L_{∞}-estimate in the above proof already appears in [Voi86; proof of Prop. 6.3], in the context of Schrödinger semigroups on ℝ^{n}. In this case one does not need the weight h_{t−s} in the estimates. It seems that for kernel estimates on manifolds, this type of estimate is hardly ever used. The argument in [Stu93; proof of Prop. 4.7] is similar, but not quite as straightforward.

(b) As explained in the previous two sections, theorems on upper bounds can also be used to prove stability of lower bounds. Assume, e.g., that the propagator U is L_{1}- and L_{∞}-bounded, that (3.20) holds with r_{k} = 0 for all k ∈ ℕ, and that V ⩾ 0 is a forward and backward Miyadera perturbation of U with constants (∞, γ), for some γ ⩾ 0. Then U_{−V/β} is L_{1}- and L_{∞}-bounded for β > γ, so p_{t,s}^{−V/β}(x, y) ⩽ ch_{t−s}(x)h_{t−s}(y) by Theorem 3.23 (applied with A = 0 and ψ_{t,s} = 0). From the estimate p_{t,s}^{V} ⩾ (p_{t,s}^{−V/β})^{−β}(p_{t,s})^{1+β} it thus follows that the lower bound

p_{t,s}(x, y) ⩾ c_{1}h_{t−s}(x)h_{t−s}(y)e^{−ψ_{t,s}(x,y)}

is, up to a factor 1 + β in front of ψ_{t,s}, stable under perturbation by V.

In the context of complete Riemannian manifolds with non-negative Ricci curvature, the above is a generalisation to the time-dependent situation of the fact that the Li-Yau estimate is stable under perturbation by a positive Green bounded (time-independent) potential. This stability theorem is due to [Zha00; Thm. C]; see also [Gri06; Thm. 10.5] for a different proof and [Tak07; Thm. 2] for a generalisation to perturbation by measures.

(c) Assume that one has good control of the exponential growth bound of U_{V/β} in L_{2}(μ), for some β ∈ (γ, 1). Moreover, suppose that h_{t} ⩽ h_{1} a.e. for all t > 1. Then one can proceed as follows to obtain an alternative bound on p_{t,s} for large t − s. As in the above proof one shows that there exists C > 0 such that

||U_{V/β}(s + 1, s)h_{1}^{−1}||_{1→2} ⩽ C and ||h_{1}^{−1}U_{V/β}(s + 1, s)||_{2→∞} ⩽ C

for all s ∈ J satisfying s + 1 ∈ J. Therefore, if M ⩾ 1 and ω_{2,β} ∈ ℝ are such that ||U_{V/β}(t, s)||_{2→2} ⩽ Me^{ω_{2,β}(t−s)} for all (t, s) ∈ D_{J}, then

||h_{1}^{−1}U_{V/β}(t, s)h_{1}^{−1}||_{1→∞} ⩽ C^{2}Me^{ω_{2,β}(t−s−2)}

and hence

p_{t,s}^{V}(x, y) ⩽ p_{t,s}^{V/β}(x, y)^{β}p_{t,s}(x, y)^{1−β} ⩽ (C^{2}Me^{−2ω_{2,β}})^{β}h_{1}(x)h_{1}(y)e^{ω(t−s)−(1−β)ψ_{t,s}(x,y)}

for all (t, s) ∈ D_{J} with t − s ⩾ 2 and a.e. x, y ∈ Ω, with ω = βω_{2,β} + (1 − β)A, where we have used that h_{t−s} ⩽ h_{1} a.e.

In the remainder of the section we discuss the case of classical Gaussian bounds in more detail, i.e., we assume that ψ_{t,s}(x, y) = a\frac{d(x,y)^{2}}{t−s} with a > 0 and a measurable metric d on Ω. Moreover, we assume that (Ω, d) is separable. Then one can use the following version of Davies’ trick to refine the estimate of Theorem 3.23. We will use the weight functions ρ_{z} defined by ρ_{z}(x) := e^{−d(x,z)}, for z ∈ Ω.

3.25 Lemma. Let A, α, β ∈ ℝ and a > 0, and let (t, s) ∈ D′ J. Then the estimate

p_{t,s}(x, y) ⩽ h_{t−s}(x)^{α}h_{t−s}(y)^{β} exp(A(t − s) − a\frac{d(x, y)^{2}}{t − s} )

holds for a.e. x, y ∈ Ω if and only if

||h_{t−s}^{−α}ρ_{z}^{λ}U(t, s)ρ_{z}^{−λ}h_{t−s}^{−β}||_{1→∞} ⩽ exp(A(t − s) + \frac{λ^{2}}{4a}(t − s))  (3.23)

for all z ∈ Ω and all λ ⩾ 0.

Proof. By the Dunford-Pettis theorem, (3.23) holds if and only if

p_{t,s}(x, y) ⩽ h_{t−s}(x)^{α}h_{t−s}(y)^{β} exp(A(t − s) + \frac{λ^{2}}{4a}(t − s) + λd(x, z) − λd(y, z)) (3.24)

for a.e. x, y ∈ Ω. Observe that

inf\{\frac{λ^{2}}{4a}(t − s) + λd(x, z) − λd(y, z); z ∈ Ω, λ ⩾ 0\} = −a\frac{d(x, y)^{2}}{t − s} ,

the infimum being attained for z = x and λ = \frac{2ad(x,y)}{t−s}. This implies the assertion since the right-hand side of (3.24) is continuous in z and λ, and both Ω and [0, ∞) are separable.

It is important to get a grip on the constants ω and β in the kernel bound (3.21). Remark 3.24(c) only gives some information about the constant ω. We now show that a suitable weighted L_{2}-estimate for U_{V} leads to good control of both ω and β. We formulate this as a result on the unperturbed propagator U since it does not involve perturbation theory.

In fact, Gaussian estimates have the following “self-improving” property that is particularly useful in the time-independent symmetric situation. In [Ouh06; Thm. 1], a similar result is proved in the context of Schrödinger semigroups on manifolds, and it is already noted there that the method also works in greater generality. Thus, the main feature of our result is not the more general context but the simpler and more transparent proof.

3.26 Proposition. Assume that t ↦ h_{t}(x) is decreasing for all x ∈ Ω and that the kernels p_{t,s} of the operators U(t, s) satisfy

p_{t,s}(x, y) ⩽ (h_{t−s}(x)^{2} ∧ h_{t−s}(y)^{2}) exp(A(t − s) − a\frac{d(x, y)^{2}}{t − s} )  (3.25)

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω, with constants A ∈ ℝ and a > 0. Moreover, assume that there exist M ⩾ 1 and ω, ω_{2} ∈ ℝ such that ||U(t, s)||_{p→p} ⩽ Me^{ω(t−s)} and

||ρ_{z}^{λ}U(t, s)ρ_{z}^{−λ}||_{2→2} ⩽ e^{(ω_{2}+λ^{2})(t−s)}  (3.26)

for all (t, s) ∈ D_{J}, p = 1, ∞, z ∈ Ω and λ ⩾ 0. Then

p_{t,s}(x, y) ⩽ eMh_{ε(t−s)}(x)h_{ε(t−s)}(y) exp(ω_{2}(t − s) − \frac{d(x, y)^{2}}{4(t − s)})

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω, where

ε = \frac{1}{2} ∧ ((ω + A − 2ω_{2})(t − s) + (\frac{1}{a} − 2)\frac{d(x, y)^{2}}{4(t − s)})^{−1}.

(We assume without loss of generality that ω + A ⩾ 2ω_{2} and a < \frac{1}{2}.)

Proof. Let z ∈ Ω and λ ⩾ 0. For (t, s) ∈ D′ J we obtain by (3.25) and Lemma 3.25 that

||ρ_{z}^{2λ}U(t, s)ρ_{z}^{−2λ}h_{t−s}^{−2}||_{1→∞} ⩽ exp(A(t − s) + \frac{(2λ)^{2}}{4a} (t − s))

and hence, by Corollary 2.37,

||ρ_{z}^{λ}U(t, s)ρ_{z}^{−λ}h_{t−s}^{−1}||_{1→2} ⩽ ||U(t, s)||_{1→1}^{1/2}||ρ_{z}^{2λ}U(t, s)ρ_{z}^{−2λ}h_{t−s}^{−2}||_{1→∞}^{1/2}
⩽ M^{1/2} exp(\frac{ω+A}{2}(t − s) + \frac{λ^{2}}{2a}(t − s)).

An analogous estimate holds for ||h_{t−s}^{−1}ρ_{z}^{λ}U(t, s)ρ_{z}^{−λ}||_{2→∞}. Let now (t, s) ∈ D′ J and ε ⩽ \frac{1}{2}, and let s_{1}, t_{1} ∈ (s, t) be such that s_{1} − s = t − t_{1} = ε(t − s). Then the above, together with (3.26), yields

||h_{ε(t−s)}^{−1}ρ_{z}^{λ}U(t, s)ρ_{z}^{−λ}h_{ε(t−s)}^{−1}||_{1→∞}
⩽ ||h_{t−t_{1}}^{−1}ρ_{z}^{λ}U(t, t_{1})ρ_{z}^{−λ}||_{2→∞}||ρ_{z}^{λ}U(t_{1}, s_{1})ρ_{z}^{−λ}||_{2→2}||ρ_{z}^{λ}U(s_{1}, s)ρ_{z}^{−λ}h_{s_{1}−s}^{−1}||_{1→2}
⩽ M exp((ω + A)ε(t − s) + \frac{λ^{2}}{a} ε(t − s) + (ω_{2} + λ^{2})(1 − 2ε)(t − s))
⩽ M exp((ω + A − 2ω_{2} + (\frac{1}{a} − 2)λ^{2})ε(t − s) + (ω_{2} + λ^{2})(t − s)).

Choosing ε = ε_{λ} := \frac{1}{2} ∧ (ω + A − 2ω_{2} + (\frac{1}{a} − 2)λ^{2})^{−1}(t − s)^{−1}, we infer that

||h_{ε(t−s)}^{−1}ρ_{z}^{λ}U(t, s)ρ_{z}^{−λ}h_{ε(t−s)}^{−1}||_{1→∞} ⩽ M exp(1 + (ω_{2} + λ^{2})(t − s)).

Now observe that λ ↦ h_{ε_{λ}}(x) is increasing for all x ∈ Ω since t ↦ h_{t}(x) is decreasing. We thus obtain the assertion as in Lemma 3.25, letting ℚ_{⩾0} ∋ λ ↑ \frac{d(x,y)}{2(t−s)}.

3.27 Remark. (a) The factor (h_{t−s}(x)^{2} ∧ h_{t−s}(y)^{2}) in assumption (3.25) is chosen instead of the (larger) factor h_{t−s}(x)h_{t−s}(y) in order to simplify the result. If the kernel estimate (3.19) is satisfied with ψ_{t,s}(x, y) = a\frac{d(x,y)^{2}}{t−s}, and similarly as in (3.20) one has p_{t,s}(x, y) ⩽ h_{t−s}(x)^{1+q}h_{t−s}(y)^{1−q}e^{r_{3}(t−s)} for a.e. x, y ∈ Ω and q = ±2, then (3.25) holds with Ã = \frac{1}{2}(A + r_{3}) and ã = \frac{a}{2} in place of A and a.

There is no factor in front of λ^{2} in the weighted L_{2}-estimate (3.26) since we assume that the metric d is defined in such a way that (3.26) holds. In the context of C_{0}-semigroups generated by elliptic operators this is satisfied by the Davies metric.

Note that the above proof does not require strong continuity (or strong measurability) of U. Moreover, the result is also valid for propagators that are not positive: Then one has to use Stein interpolation instead of Corollary 2.37 in the proof, and one obtains an estimate for |p_{t,s}(x, y)|.

(b) Assume that there exist C, n > 0 such that h_{εt} ⩽ Cε^{−n/4}h_{t} a.e. for all t > 0 and all 0 < ε < 1. Then Proposition 3.26 implies that

p_{t,s}(x, y) ⩽ eMC^{2}h_{t−s}(x)h_{t−s}(y)g(t − s, d(x, y))

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω, where

g(t, r) = [2 + (ω + A − 2ω_{2})t + (\frac{1}{a} − 2)\frac{r^{2}}{4t} ]^{n/2} exp(ω_{2}t − \frac{r^{2}}{4t}).

(c) The assumption on the h_{t} in part (b) holds, e.g., if h_{t}(x) = cμ(B(x, \sqrt{t}))^{−1/2} and μ satisfies the volume growth condition μ(B(x, R)) ⩽ C^{2}(\frac{R}{r})^{n}μ(B(x, r)) for all R > r > 0. It follows from [Sik04; Thm. 4] that in this situation the exponent \frac{n}{2} in the estimate of part (b) can be replaced with \frac{n−1}{2} if, in addition, ω_{2} = 0 and the propagator U comes from a selfadjoint C_{0}-semigroup.

Even if the propagator U comes from a selfadjoint C_{0}-semigroup: The perturbed situation is usually a non-symmetric one since the potential is time-dependent. Suppose, e.g., that V is an infinitesimally small Miyadera perturbation of U, with backward Miyadera constant close to 1. Then the kernel estimate from Theorem 3.23 is rather poor since 1 − β is close to 0. If one has no good control of ||ρ_{z}^{λ}U_{V} (t, s)ρ_{z}^{−λ}||_{2→2}, then instead of Proposition 3.26 one can use the following theorem to obtain a better estimate. The result may seem rather technical; in Remark 3.29 below we explain how it can be applied in the context of Riemannian manifolds with Ricci curvature bounded below. For this application it is important that the constants in the estimate are explicit.

3.28 Theorem. Assume that for all θ ∈ (0, 1] there exist c_{θ}, a_{θ} > 0 and A_{θ} ∈ ℝ such that

p_{t,s}(x, y) ∨ p_{t,s}(y, x) ⩽ c_{θ}h_{(t−s)/θ}(x)^{1/θ}h_{(t−s)/θ}(y)^{2−1/θ} exp(A_{θ}(t − s) − a_{θ} \frac{d(x, y)^{2}}{t − s} )

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω. Let γ_{1}, γ_{∞} > 0 satisfy σ := γ_{1} + γ_{∞} ⩽ 1, and let V be a potential such that V/γ_{1} is weakly U-admissible. Assume that there exist M_{p} ⩾ 1 and ω_{p} ∈ ℝ such that

||U_{(V/γ_{p})∨(−n)}(t, s)||_{p→p} ⩽ M_{p}e^{ω_{p}(t−s)} ((t, s) ∈ D_{J}, n ∈ ℕ, p = 1, ∞).

Then

p_{t,s}^{V}(x, y) ⩽ ch_{t−s}(x)h_{t−s}(y) exp(ω(t − s) − a\frac{d(x, y)^{2}}{t − s} )

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω, where c = (M_{1}^{θ}M_{∞}^{1−θ}c_{θ}^{θ}c_{1−θ}^{1−θ})^{σ}c_{1}^{1−σ} (with θ := \frac{γ_{1}}{σ}), ω = σθ(1−θ)(ω_{1} +ω_{∞})+σθ^{2}A_{θ} +σ(1−θ)^{2}A_{1−θ} +(1−σ)A_{1} and a = σ \frac{a_{θ}a_{1−θ}}{a_{θ} + a_{1−θ}} +(1−σ)a_{1}.

Note that the slightly sophisticated formulation of the condition on the L_{p}-norm bound of U_{V/γ_{p}} is only needed for the case p = ∞ and γ_{∞} < γ_{1}: Then the potential V/γ_{∞} need not be weakly U-admissible. In the case p = 1 or γ_{1} ⩽ γ_{∞} one can simply require ||U_{V/γ_{p}}(t, s)||_{p→p} ⩽ M_{p}e^{ω_{p}(t−s)} for all (t, s) ∈ D_{J}. The estimate assumed for the p_{t,s} can be viewed as a combined version of the bounds (3.19) and (3.20).

Proof of Theorem 3.28. Without loss of generality assume that V is bounded from below, and observe that 1 − θ = \frac{γ_{∞}}{σ}. Let z ∈ Ω and λ ⩾ 0. Then Lemma 3.25 yields

N_{θ} := ||h_{(t−s)/θ}^{−1/θ}ρ_{z}^{λ/θ}U(t, s)ρ_{z}^{−λ/θ}h_{(t−s)/θ}^{1/θ−2}||_{1→∞} ⩽ c_{θ} exp(A_{θ}(t − s) + \frac{(λ/θ)^{2}}{4a_{θ}} (t − s))

and hence, due to Theorem 2.38 and Corollary 2.37,

||h_{(t−s)/θ}^{−1}ρ_{z}^{λ}U_{V/σ}(t, s)ρ_{z}^{−λ}h_{(t−s)/θ}^{1−2θ}||_{1/θ→∞} ⩽ ||U_{V/γ_{∞}}(t, s)||_{∞→∞}^{1−θ}N_{θ}^{θ}
⩽ c_{∞,θ} exp(ω_{∞,θ}(t − s) + \frac{λ^{2}}{4θa_{θ}} (t − s))

for all (t, s) ∈ D′ J, where c_{∞,θ} := M_{∞}^{1−θ}c_{θ}^{θ} and ω_{∞,θ} := (1 − θ)ω_{∞} + θA_{θ}. In the same way,

||h_{(t−s)/(1−θ)}^{1−2(1−θ)}ρ_{z}^{λ}U_{V/σ}(t, s)ρ_{z}^{−λ}h_{(t−s)/(1−θ)}^{−1}||_{1→1/θ} ⩽ c_{1,θ} exp(ω_{1,θ}(t − s) + \frac{λ^{2}}{4(1−θ)a_{1−θ}} (t − s)), with c_{1,θ} := M_{1}^{θ}c_{1−θ}^{1−θ} and ω_{1,θ} := θω_{1} + (1 − θ)A_{1−θ}. Let now (t, s) ∈ D′ J, and let r ∈ (s, t) be such that t − r = θ(t − s). Then r − s = (1 − θ)(t − s), so by the above we obtain that

||h_{t−s}^{−1}ρ_{z}^{λ}U_{V/σ}(t, s)ρ_{z}^{−λ}h_{t−s}^{−1}||_{1→∞}
⩽ ||h_{t−s}^{−1}ρ_{z}^{λ}U_{V/σ}(t, r)ρ_{z}^{−λ}h_{t−s}^{1−2θ}||_{1/θ→∞}||h_{t−s}^{2θ−1}ρ_{z}^{λ}U_{V/σ}(r, s)ρ_{z}^{−λ}h_{t−s}^{−1}||_{1→1/θ}
⩽ c_{1,θ}c_{∞,θ} exp(ω_{0}(t − s) + \frac{λ^{2}}{4a_{0}} (t − s)),

with ω_{0} := θω_{∞,θ} + (1 − θ)ω_{1,θ} and a_{0} > 0 defined by \frac{1}{a_{0}} = \frac{1}{a_{θ}} + \frac{1}{a_{1−θ}}, so a_{0} = \frac{a_{θ}a_{1−θ}}{a_{θ}+a_{1−θ}}. Therefore,

p_{t,s}^{V/σ}(x, y) ⩽ c_{1,θ}c_{∞,θ}h_{t−s}(x)h_{t−s}(y) exp(ω_{0}(t − s) − a_{0} \frac{d(x,y)^{2}}{t−s} )

for a.e. x, y ∈ Ω by Lemma 3.25. Using the estimate p_{t,s}^{V} ⩽ (p_{t,s}^{V/σ})^{σ}(p_{t,s})^{1−σ}, we conclude that the asserted kernel bound holds with c = (c_{1,θ}c_{∞,θ})^{σ}c_{1}^{1−σ},

ω = σω_{0} + (1 − σ)A_{1} = σ(θ(1 − θ)(ω_{1} + ω_{∞}) + θ^{2}A_{θ} + (1 − θ)^{2}A_{1−θ}) + (1 − σ)A_{1}

and a = σa_{0} + (1 − σ)a_{1}.

We conclude by demonstrating how Theorem 3.28 can be applied to stability of heat kernel estimates on Riemannian manifolds.

3.29 Remark. (a) Assume that Ω is a complete n-dimensional Riemannian manifold with non-negative Ricci curvature. Then by Bishop’s comparison principle there exists C_{0} ⩾ 1 such that

\frac{μ(B(x, R))}{μ(B(x, r))} ⩽ C_{0}(\frac{R}{r})^{n} (x ∈ Ω, R ⩾ r > 0),

and it follows that

\frac{μ(B(x, r))}{μ(B(y, r))} ⩽ \frac{μ(B(y, r + d(x, y)))}{μ(B(y, r))} ⩽ C_{0}(1 + \frac{d(x, y)}{r})^{n} (x, y ∈ Ω, r > 0).

Thus, if h_{t}(x) = cμ(B(x, \sqrt{t}))^{−1/2} then there exists C ⩾ 1 such that

\frac{h_{t}(x)}{h_{t}(y)} ⩽ C(1 + \frac{d(x, y)^{2}}{4t} )^{n/4} (x, y ∈ Ω, t > 0),  (3.27)

and h_{θt} ⩽ C_{0}^{1/2}θ^{−n_{0}/4}h_{t} pointwise for all t > 0 and all θ ∈ (0, 1], where n_{0} = n. In the following we will only use these two properties of the functions h_{t}, without requiring n_{0} = n. Note that (3.27) holds with n = 0 if the functions h_{t} are constant, as in the setting of Section 3.1. This is the reason why we have introduced the quantity n_{0} in the second property of the h_{t}.

Let r ⩾ 0. Observe that 1 + r ⩽ \frac{1}{ε}(1 + εr) ⩽ \frac{1}{ε}e^{εr} for all ε ∈ (0, 1] and 1 + r ⩽ e^{εr} for all ε > 1, and hence (1 + r)^{ν} ⩽ (\frac{ν}{ε} ∨ 1)^{ν}e^{εr} for all ε, ν > 0. Thus, from (3.27) we infer that

\frac{h_{t}(x)}{h_{t}(y)} ⩽ C(\frac{n}{4ε} ∨ 1)^{n/4} exp(ε\frac{d(x, y)^{2}}{4t} ) (x, y ∈ Ω, t, ε > 0).  (3.28)

Now assume that the kernel bound (3.19) holds with ψ_{t,s}(x, y) = \frac{d(x,y)^{2}}{4(t−s)} , i.e.,

p_{t,s}(x, y) ⩽ h_{t−s}(x)h_{t−s}(y) exp(A(t − s) − \frac{d(x, y)^{2}}{4(t − s)})

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω. Let ε > 0. Then for θ ∈ (0, 1) we obtain, using (3.28), that

p_{t,s}(x, y) ∨ p_{t,s}(y, x) ⩽ C_{0}θ^{−n_{0}/2}h_{(t−s)/θ}(x)h_{(t−s)/θ}(y) exp(A(t − s) − \frac{d(x, y)^{2}}{4(t − s)})

⩽ c_{θ,ε}h_{(t−s)/θ}(x)^{1/θ}h_{(t−s)/θ}(y)^{2−1/θ} exp(A(t − s) − a_{θ,ε} \frac{d(x, y)^{2}}{t − s} ),

with c_{θ,ε} = C_{0}θ^{−n_{0}/2}(C(\frac{n}{4ε} ∨ 1)^{n/4})^{1/θ−1} and a_{θ,ε} = \frac{1}{4} − \frac{εθ}{4}(1/θ − 1) ⩾ \frac{1}{4}(1 − ε).

One easily sees that the function (0, 1) ∋ θ ↦ θ^{θ}(1 − θ)^{1−θ} attains it minimum at θ = \frac{1}{2}, so

c_{θ,ε}^{θ}c_{1−θ,ε}^{1−θ} = C_{0}C(θ^{θ}(1 − θ)^{1−θ})^{−n_{0}/2}(\frac{n}{4ε} ∨ 1)^{n/4} ⩽ C_{1}(1 + \frac{n}{4ε})^{n/4},

where C_{1} = C_{0}C2^{n_{0}/2}. Moreover, \frac{1}{a_{θ,ε}} + \frac{1}{a_{1−θ,ε}} ⩽ \frac{8}{1−ε}. Thus, if the potential V satisfies the assumptions of Theorem 3.28, then

p_{t,s}^{V}(x, y) ⩽ c(1 + \frac{n}{4ε})^{σn/4}h_{t−s}(x)h_{t−s}(y) exp(ω(t − s) − a_{ε} \frac{d(x, y)^{2}}{t − s} )  (3.29)

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω, where c = (M_{1}^{θ}M_{∞}^{1−θ}C_{1})^{σ} (with θ := \frac{γ_{1}}{σ}),

ω = σθ(1 − θ)(ω_{1} + ω_{∞}) + σ(2θ^{2} − 2θ + 1)A + (1 − σ)A
= σθ(1 − θ)(ω_{1} + ω_{∞}) + (1 − 2σθ(1 − θ))A

and a_{ε} = σ \frac{1−ε}{8} + (1 − σ)\frac{1}{4} = \frac{1−σ/2}{4} − σ \frac{ε}{8}. Since the right-hand side of the estimate (3.29) is continuous in ε, we can choose ε such that \frac{n}{ε} = \frac{d(x,y)^{2}}{t−s}, so σ \frac{ε}{8} \frac{d(x,y)^{2}}{t−s} = \frac{σn}{8}, and with β := \frac{σ}{2} = \frac{1}{2}(γ_{1} + γ_{∞}) we conclude that

p_{t,s}^{V}(x, y) ⩽ ce^{βn/4}h_{t−s}(x)h_{t−s}(y)(1 + \frac{d(x, y)^{2}}{4(t − s)})^{βn/2} exp(ω(t − s) − (1 − β)\frac{d(x, y)^{2}}{4(t − s)})

for all (t, s) ∈ D′ J and a.e. x, y ∈ Ω.

Summarising, we have achieved the following improvement of the bound (3.21) on the perturbed kernel. An application of Theorem 3.23 leads to the condition β > γ_{1} ∨ γ_{∞} for the bound to be valid, whereas the above shows that the bound is also valid for β > \frac{1}{2}(γ_{1} + γ_{∞}), or even for β = \frac{1}{2}(γ_{1} + γ_{∞}) if one allows for the polynomial correction factor (1 + \frac{d(x,y)^{2}}{4(t−s)} )^{βn/4}. If V is an infinitesimally small Miyadera perturbation of U, and a backward Miyadera perturbation of U with bound γ < 1, then the condition β > γ is replaced with the condition β > \frac{γ}{2}. This is a considerable improvement since 1 − γ may be close to 0, but 1 − \frac{γ}{2} > \frac{1}{2}.

(b) Now assume, more generally, that Ω is a complete n-dimensional Riemannian manifold with lower bounded Ricci curvature. We indicate how the application of Theorem 3.28 presented in part (a) can be adapted to this situation; we point out that then the constants A_{θ} in the estimation will also depend on θ.

As in [Stu92b; proof of Lemma 2.2(a)] one obtains from Bishop’s comparison principle and the explicit formula for the volume of balls in hyperbolic space there exists κ > 0 (depending on the bound for the Ricci curvature) such that

\frac{μ(B(x, R))}{μ(B(x, r))} ⩽ \frac{f_{n}(κR)}{f_{n}(κr)} (x ∈ Ω, R ⩾ r > 0),

where f_{n}(r) := ∫_{0}^{r}(sinh \frac{t}{n−1})^{n−1}dt for all r > 0 (we assume that n ⩾ 2). One easily shows that there exists a constant c_{n} > 1 such that \frac{1}{c_{n}} g_{n} ⩽ f_{n} ⩽ c_{n}g_{n}, where g_{n}(r) := (r ∧ 1)^{n}e^{r}. From this one deduces that \frac{f_{n}(R)}{f_{n}(r)} ⩽ c_{n}^{2}(\frac{R}{r})^{n}e^{R−r} for all R ⩾ r > 0, and as in (a) it follows that

\frac{μ(B(x, r))}{μ(B(y, r))} ⩽ \frac{f_{n}(κr + κd(x, y))}{f_{n}(κr)} ⩽ c_{n}^{2}(1 + \frac{d(x, y)}{r})^{n}e^{κd(x,y)} (x, y ∈ Ω, r > 0) (3.30)

(which improves on the estimate provided in [Stu92b; Lemma 2.2(a)]). Using the inequality κd(x, y) ⩽ ε\frac{d(x,y)^{2}}{4(t−s)} + \frac{1}{ε}κ^{2}(t − s), one can then proceed as in part (a) to verify the assumptions of Theorem 3.28 with suitable constants c_{θ}, A_{θ} and a_{θ}.

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Versicherung

Hiermit versichere ich, daß ich die vorliegende Arbeit ohne unzulässige Hilfe Dritter und ohne Benutzung anderer als der angegebenen Hilfsmittel angefertigt habe; die aus fremden Quellen direkt oder indirekt übernommenen Gedanken sind als solche kenntlich gemacht.

Ich habe bisher kein Habilitationsgesuch an einer anderen Hochschule gestellt.

Ich erkenne die Habilitationsordnung der Fakultät Mathematik und Naturwissenschaften an der Technischen Universität Dresden vom 10. Januar 1996 an.

Dresden, 9. 6. 2010