Gaussian bounds for propagators perturbed by potentials

Vitali Liskevich

School of Mathematics University of Bristol Bristol BS8 1TW, UK

        V.Liskevich@bristol.ac.uk

Hendrik Vogt

Fachrichtung Mathematik Technische Universität Dresden 01062 Dresden, Germany

                                         vogt@math.tu-dresden.de

Jürgen Voigt

Fachrichtung Mathematik Technische Universität Dresden 01062 Dresden, Germany

                         voigt@math.tu-dresden.de

Abstract

We develop the perturbation theory for propagators, with the objective to prove Gaussian bounds. Let U be a strongly continuous propagator, i.e., a family of operators describing the solutions of a non-autonomous evolution equation, on an L_{p}-space, and assume that U is positive and satisfies Gaussian upper and lower bounds. Let V be a (time-dependent) potential satisfying certain Miyadera conditions with respect to U. We show that then the perturbed propagator enjoys Gaussian upper and lower bounds as well. To prepare the necessary tools, we extend the perturbation theory of strongly continuous propagators and the theory of absorption propagators.

2000 MSC: 47D06, 35K15, 47A55, 47B65 Keywords: propagator, evolution family, evolution semigroup, perturbation theory, Miyadera perturbation, non-autonomous Kato class, Gaussian bound

Introduction

The validity of Gaussian bounds for fundamental solutions of second-order parabolic equations with non-smooth coefficients goes back to the paper of Aronson [4]. Aronson studied the parabolic equation with measurable coefficients that depend on space and time variables. Besides a uniformly elliptic second order part and a first order part, a multiplication term (potential) was allowed that was supposed to satisfy a certain integrability property. Starting from the publication of [4] the topic has a rich history, and numerous extensions and generalisations have been obtained; cf. [11, 13, 7, 8, 36, 32, 20, 6] for (subsets of) ℝ^{N} and [18, 27, 9, 37, 14] for manifolds.

For the case of Schrödinger operators −Δ+V, in [1] the Kato class of potentials was shown to be the proper class for the L_{1}-perturbation theory of the corresponding C_{0}-semigroups, and in [33] it was shown that the fundamental solution of the perturbed heat equation still satisfies upper and lower Gaussian estimates. However, it was only in the last years that parabolic equations with more general time-dependent potentials were studied and the nonautonomous Kato class as the proper extension of the Kato class was introduced; cf. [43], [44], [30], [15], [16].

Essentially following [43] we say that a potential V ∈ L_{1,loc}((0, ∞) × ℝ^{N}) is in the nonautonomous 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−y|^{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 [30] 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. [30]).

In [44], Zhang studied the fundamental solution u_{V} of the parabolic equation

u_{t} = ∇ · (a(t, x)∇u) − V (t, x)u.  (0.1)

He proved that if a is uniformly elliptic and V_{κ} := V (·/κ, ·) ∈ NK for a suitable κ > 0 then there are c_{1}, c_{2} > 1 such that u_{V} satisfies the two-sided Gaussian bounds

\frac{1}{c_{1}} (t − s)^{−N/2}e^{−c_{2} \frac{|x−y|^{2}}{(t−s)}} ⩽ u_{V} (x, t; y, s) ⩽ c_{1}(t − s)^{−N/2}e^{− \frac{|x−y|^{2}}{c_{2}(t−s)}}  (0.2)

for s < t ⩽ s + 1. (Analogous estimates were shown for parabolic equations with sub-elliptic principal part.) 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. The bounds (0.2) are called global if they hold for all t > s.

As is observed in [30], for the well-posedness of the Cauchy problem for (0.1) in L_{1}(ℝ^{n}) only the condition N^{+}(V_{κ}) < κ/M is needed, where κ, M are such that u_{0}(x, t; y, s) ⩽ Mk_{κ(t−s)}(x − y). This is derived by means of the non-autonomous Miyadera perturbation theorem [25, Thm. 3.4]. In fact, the above condition implies that V is a small Miyadera perturbation of the unperturbed propagator corresponding to the Cauchy problem for (0.1) with V = 0. However, a condition controlling N^{+}(V_{κ}) (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_{κ}). The latter condition is responsible for the well-posedness of the adjoint Cauchy problem

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

in L_{1}(ℝ^{n}) (see Section 3.1).

It was noted in [44] that the Gaussian bounds (0.2) for the fundamental solution of (0.1) remain valid if V_{κ} ∈ N̂K and the quantities N^{±}(V_{κ}) are sufficiently small. The results of the present paper imply that the upper bound in (0.2) holds if the negative part V^{−} of V satisfies N^{±}(V_{κ}^{−}) < κ/M (with κ, M as in the previous paragraph); cf. Theorem 3.10. It follows from the general approach that we present in this paper that no restrictions on the positive part V^{+} of V are needed. The lower bound in (0.2) is established under the assumption N^{±}(V_{κ}^{+}) < ∞ and the condition (on V^{−}) that the Cauchy problem for (0.1) 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 [44]. The global lower bound under these conditions is new. The corresponding result in the autonomous case is due to Semenov; see [31, (6.6)].

The aim of this paper is to develop a perturbation method that enables one to derive the above results in the general framework of positive propagators on L_{p}(μ)-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 (additive time-dependent) perturbation we consider 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 [25] for a more extensive discussion.

This approach using the Duhamel formula works rather directly in the case of bounded perturbations. In order to include more general perturbations we apply the Miyadera perturbation theorem for C_{0}-semigroups. The connection between the theory of C_{0}-semigroups and (non-autonomous) propagators is established by the use of evolution semigroups. This method was first introduced by Howland ([17]) and further developed in [12], [22]; see also [29] and references therein. For the special case of positivity preserving propagators on L_{p}-spaces and perturbations 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 ([41], [42], [2], [21]). So far, in the non-autonomous case only perturbations by positive potentials have been studied; see [24]. In the present paper we develop the theory for sign-changing potentials and provide the link to the evolution semigroup approach; cf. subsection 2.1.

We further develop the perturbation theory for propagators U acting simultaneously on all L_{p} spaces, 1 ⩽ p ⩽ ∞. In this case, using Stein interpolation we show that it suffices to require the potential V to be a small Miyadera perturbation of U, with bound γ, in order to obtain that U_{V} extends to a strongly continuous propagator on L_{p}, 1 ⩽ p < 1/γ (see Remark 3.5(b)). This generalises one of the main results in [15], where the above was obtained for the heat equation with time-dependent potential from the non-autonomous Kato class NK, which corresponds to an infinitesimally small Miyadera perturbation.

In order to obtain a Gaussian upper bound for the perturbed propagator U_{V} one first needs to establish an L_{1}–L_{∞} estimate, which is a diagonal upper bound for the integral kernel. Assuming that the unperturbed propagator U on L_{1} satisfies an L_{1}–L_{∞} estimate, and V is a small Miyadera perturbation of U, with bound γ, we obtain an L_{1}–L_{κ} estimate (with 1 < κ < 1/γ) for the perturbed propagator U_{V}; cf. Proposition 3.3. Under the additional assumption that V satisfies a backward Miyadera condition (see (3.3) below), we also obtain an L_{κ′}–L_{∞} estimate. From these two estimates together we deduce the desired L_{1}–L_{∞} estimate by an iteration argument. The Gaussian upper bound for the perturbed propagator then follows from the upper bound for the unperturbed propagator and the diagonal bound for the perturbed one. We achieve this by means of an interpolation inequality for propagators that is well-known for the case of Schrödinger semigroups (see Proposition 3.8). We provide a purely analytical argument for the proof of this inequality, based on complex interpolation, thus avoiding previously used probabilistic tools or the Trotter product formula (for the latter see [31, (6.2)], [3, Thm. 2.4]).

By the same interpolation inequality we obtain a Gaussian lower bound for the perturbed propagator under the assumption that the unperturbed propagator satisfies a diagonal upper bound and a Gaussian lower bound.

The paper is organised as follows. In Section 1 we recall the notions and further develop the theory of propagators (or evolution families) on Banach spaces and the corresponding evolution semigroups. We provide a short introduction to bounded perturbations of propagators. The main part of Section 1, i.e. subsection 1.3, consists in a review of the Miyadera perturbation theorem for C_{0}-semigroups and its application to evolution semigroups and propagators. The main result of Section 1, Theorem 1.16, is a slight improvement of [25, Theorem 3.4]. We provide a new and more structured proof of this result, singling out important equivalences between relations occurring in the propagator context and in the semigroup context; cf. Theorems 1.12 and 1.14. The final subsection 1.4 of Section 1 deals with positive perturbations of positive propagators on Banach lattices.

Section 2 deals with absorption propagators. In subsection 2.1 we study the general theory, in subsection 2.2 we apply the results of subsection 1.3 to potentials and show, in particular, that the two approaches are consistent.

Section 3 contains the main results of the paper, namely the stability of Gaussian upper and lower bounds under perturbation by potentials. In subsection 3.1 we show the diagonal upper bound, in subsection 3.2 the Gaussian upper bound, and in subsection 3.3 the Gaussian lower bound.

1 Propagators, evolution semigroups, and perturbation theory

1.1 Propagators and evolution semigroups

Let J ⊆ ℝ be an interval, D_{J} := {(t, s) ∈ J × J; t ⩾ s}. 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

(P1) U(t, t) = I, U(t, r) = U(t, s) U(s, r) for all t ⩾ s ⩾ r in J.

If additionally

(P2) U: D_{J} → L(X) is strongly continuous

then U = (U(t, s))_{(t,s)∈D_{J}} is called a strongly continuous (s.c.) propagator.

The next lemma is useful in applications for verifying the strong continuity of propagators. For t ∈ ℝ we use the notation J_{⩾t} := {s ∈ J; s ⩾ t}, and analogously J_{>t}, J_{⩽t}, J_{<t}.

1.1 Lemma. ([16, 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}. 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}, x ∈ X. Let t_{n} → t_{0}, s_{n} → s_{0}. Without restriction 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 s.c. propagators are locally bounded. The following example shows that separate strong continuity of a propagator in not sufficient to obtain local boundedness (as was claimed in [5, Def. 2.32]). In particular it shows that condition (i) of Lemma 1.1 is needed for the conclusion.

1.2 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, φ_{n} = \frac{1}{n} on [b_{n}, 1]. Define U by

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

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

Then U is a propagator on ℓ_{2} with parameter interval [0, 1]. Further, [0, t] ∋ s ↦ U(t, s) is strongly continuous for all t ∈ [0, 1], and [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.

1.3 Remark. Note that, in the previous example, the unboundedness of U occurs close to the diagonal. In fact, it is easy to see that a separately strongly continuous propagator U with parameter interval J is locally bounded on {(t, s) ∈ D_{J}; t − s ⩾ δ}, for all δ > 0.

For the following definition we refer to [15, Def. 3]. A backward propagator on X is a function W: D_{J}^{∗} → L(X), where D_{J}^{∗} := {(s, t) ∈ J × J; s ⩽ t}, satisfying

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

1.4 Remarks. (a) W is a backward propagator if and only if W̃: D_{−J} → L(X), defined by W̃(t, s) := W (−t, −s) ((t, s) ∈ D_{−J}) is a propagator.

(b) Let U be a propagator on X. Define U_{∗}: D_{J}^{∗} → L(X^{∗}) by U_{∗}(s, t) := U(t, s)^{∗}. Then U_{∗} is a backward propagator on X^{∗}. This shows that taking the adjoint of a propagator is connected with time reversal.

Let U be a s.c. propagator on X. If U is exponentially bounded, i.e., there exist M ⩾ 1, ω ∈ ℝ such that

||U(t, s)|| ⩽ Me^{ω(t−s)}  ((t, s) ∈ D_{J}),  (1.1)

then with the propagator U one can associate an evolution semigroup T = (T (t))_{t⩾0} on L_{p}(J; X) by

T (t)f(s) := 11_{J}(s − t) U(s, s − t)f(s − t).

From (1.1) one obtains ||T (t)|| ⩽ Me^{ωt} for all t ⩾ 0. In the present paper we use evolution semigroups on L_{1}(J; X) only.

We refer to [17], [12], [22], [29] for the concept of evolution semigroups, properties and historical remarks.

1.5 Remarks. (a) It is important to note that the correspondence between s.c. propagators and evolution semigroups is one-to-one: If U and Ũ are s.c. propagators giving rise to the same evolution semigroup T then U = Ũ. Indeed, let x ∈ X. For φ ∈ C_{c}^{1}(ℝ), f := φx (∈ L_{1}(J; X)), t ⩾ 0 we have

11_{J}(s − t)φ(s − t) Ũ(s, s − t)x = T (t)f(s) = 11_{J}(s − t)φ(s − t) U(s, s − t)x

for a.e. s ∈ J. Thus Ũ(t, s)x = U(t, s)x for a.e. (t, s) ∈ D_{J} and hence, due to the strong continuity, for all (t, s) ∈ D_{J}. (See also [22, Thm. 4.9].)

(b) For many purposes, only the time local behaviour of propagators is of interest. In these cases, (exponential) boundedness can be assumed without loss of generality, by replacing the interval J by a compact subinterval.

1.2 Bounded perturbations of propagators

In this subsection we describe the perturbation of propagators by bounded operator functions. As above, let U be a s.c. propagator on X with parameter interval J ⊆ ℝ. Let B: J → L(X) be strongly measurable and bounded.

The following considerations show that there exists a unique s.c. propagator U_{B} with parameter interval J 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,  (1.2)

for all (t, s) ∈ J, x ∈ X. The standard contraction principle argument shows uniqueness. The solution by iteration leads to the terms

U_{0}(t, s) := U(t, s), U_{j}(t, s) := ∫_{s}^{t} U_{j−1}(t, τ)B(τ)U(τ, s) dτ (j = 1, …)

(strong integrals). Then it is standard to show that

U_{B}(t, s) := ∑_{j=0}^{∞} U_{j}(t, s)  (1.3)

defines a s.c. propagator U_{B} satisfying (1.2), where the series converges in operator norm, uniformly absolutely on compact subsets of D_{J}. (With M := sup_{s⩽σ⩽τ⩽t} ||U(τ, σ)||, C := sup_{τ∈[s,t]} ||B(τ)|| we obtain

||U_{j}(t, s)|| ⩽ \frac{(t − s)^{j}}{j!} M^{j+1}C^{j};  (1.4)

the propagator property is a consequence of the formula

U_{j}(t, r) = ∑_{k=0}^{j} U_{j−k}(t, s) U_{k}(s, r) (t ⩾ s ⩾ r in J, j ∈ ℕ),

which is proved by induction.) We note that the terms U_{j}(t, s) also satisfy

\begin{aligned}U_{j}(t, s) & = ∫_{s⩽τ_{1}⩽...⩽τ_{j}⩽t} U(t, τ_{j})B(τ_{j})U(τ_{j}, τ_{j−1}) … B(τ_{1})U(τ_{1}, s) d(τ_{1}, …, τ_{j}) \\ & = ∫_{s}^{t} U(t, τ)B(τ)U_{j−1}(τ, s) dτ. \end{aligned}  (1.5)

From this representation one immediately concludes 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  (1.6)

for all (t, s) ∈ J, x ∈ X.

The existence and uniqueness of U_{B} was shown in [24, Thm. 2.1] by evolution semigroup methods. The method presented above is used in the proof of the following result and in the proof of Proposition 3.6.

1.6 Proposition. Let U and B be as above. Then z ↦ U_{zB}(t, s) is analytic, for all (t, s) ∈ D_{J}.

Proof. This is immediate from the series expansion

U_{zB}(t, s) = ∑_{j=0}^{∞} z^{j}U_{j}(t, s).

1.7 Remark. The following two properties of bounded perturbations can be obtained from uniqueness in Duhamel’s formula (or from evolution semigroup considerations).

(a) If c ∈ 𝕂 and B is the constant function B(τ) = cI, then U_{B}(t, s) = e^{c(t−s)}U(t, s) ((t, s) ∈ D_{J}).

(b) For a second function B_{1} (with the same properties as B) we have

(U_{B})_{B_{1}} = U_{B+B_{1}} (= (U_{B_{1}})_{B}).

This shows that formula (1.6) can also be derived from formula (1.2): consider U as obtained as a perturbation of U_{B} by −B.

The following convergence result for perturbed propagators strengthens [24, Prop. 2.3(b)]. It will be needed in the proof of Theorem 2.7.

1.8 Proposition. Let U, B be as above, and let (B_{n})_{n∈ℕ} be a sequence of bounded, strongly measurable functions B_{n}: J → L(X) with c := sup_{t∈J, n∈ℕ} ||B_{n}(t)|| < ∞, B_{n}(t) → B(t) strongly for a.e. t ∈ J.

Then U_{B_{n}}(t, s) → U_{B}(t, s) strongly as n → ∞, uniformly for (t, s) in compact subsets of D_{J}.

Proof. By Remark 1.7(b) we have U_{B_{n}} = (U_{B})_{B_{n}−B}, so without restriction we can assume B = 0. Let (t_{0}, s_{0}) ∈ D_{J}, x ∈ X. Then

C := {U(t, s)x; s_{0} ⩽ s ⩽ t ⩽ t_{0}}

is a compact subset of X. This implies that B_{n}(t)y → B(t)y = 0 (n → ∞) uniformly for y ∈ C, for a.e. t ∈ J. Therefore,

∫_{s_{0}}^{t_{0}} sup_{y∈C} ||B_{n}(τ)y|| dτ → 0 (n → ∞),

by the dominated convergence theorem. (Note that there exists a countable dense subset C_{0} of C, so

τ ⟼ sup_{y∈C} ||B_{n}(τ)y|| = sup_{y∈C_{0}} ||B_{n}(τ)y||

is measurable.) We obtain that

||U_{B_{n}}(t, s)x − U(t, s)x|| ⩽ ∫_{s}^{t} ||U_{B_{n}}(t, τ)B_{n}(τ)U(τ, s)x|| dτ

⩽ sup_{τ∈[s,t]} ||U_{B_{n}}(t, τ)|| ∫_{s_{0}}^{t_{0}} sup_{y∈C} ||B_{n}(τ)y|| dτ.

With M := sup_{s_{0}⩽s⩽t⩽t_{0}} ||U(t, s)|| we have ||U_{B_{n}}(t, τ)|| ⩽ Me^{Mc(t_{0}−s_{0})} by (1.4), so the assertion follows.

1.3 Unbounded perturbations and the Miyadera perturbation theorem

In this subsection we discuss a non-autonomous version of the Miyadera perturbation theorem and related issues. The main subject is the relation between s.c. propagators and the corresponding evolution semigroups.

For the convenience of the reader, we start with a formulation of the Miyadera perturbation theorem for semigroups (see [40, 41, 25]).

Let T be a C_{0}-semigroup on a Banach space E, with generator G. Let B be a G-bounded operator. Then B is called a Miyadera perturbation of T (with constants (α, γ)) if there exist α ∈ (0, ∞], γ ⩾ 0 such that

∫_{0}^{α} ||BT (t)f|| dt ⩽ γ||f||  (1.7)

for all f ∈ D(G). The operator B is called a small Miyadera perturbation if α, γ can be chosen such that γ < 1, and infinitesimally small if the infimum of all possible γ is zero.

For a Miyadera perturbation B of T it is easy to see that for all t > 0 there exists c > 0 such that

∫_{0}^{t} ||BT (τ)f|| dτ ⩽ c||f||  (1.8)

for all f ∈ D(G) (cf. [40, p. 168]).

The G-boundedness of B can be replaced by a slightly weaker hypothesis. The fact that B has a Miyadera bounded extension under this weaker hypothesis is in fact part of the Miyadera perturbation theorem as proved in [40]. We single out this part in the following lemma.

1.9 Lemma. Let E, T, G be as above, and let D ⊆ D(G) be a dense subspace of E with T (t)D ⊆ D (t ⩾ 0). Assume that B is a linear operator with D(B) = D such that

(i) [0, ∞) ∋ t ↦ BT (t)f ∈ E is continuous for all f ∈ D; (ii) there are constants α ∈ (0, ∞], γ ⩾ 0 such that (1.7) holds for all f ∈ D.

Then B has a unique G-bounded extension B̂ with D(B̂) = D(G). Inequality (1.7) holds for all f ∈ D(G), with B replaced by B̂, i.e., B̂ is a Miyadera perturbation of T. If B is closable then B̂ ⊆ B.

(Note that the hypotheses imply that D is a core for G.)

1.10 Theorem. Let E, T, G be as above, and let B be a small Miyadera perturbation of T. Then the following assertions hold:

(a) G + B generates a C_{0}-semigroup T_{B}.

(b) For f ∈ D(G) and t ⩾ 0 we have

T_{B}(t)f = T (t)f + ∫_{0}^{t} T_{B}(t − τ)BT (τ)f dτ,  (1.9)
T_{B}(t)f = T (t)f + ∫_{0}^{t} T (t − τ)BT_{B}(τ)f dτ.  (1.10)

Moreover, T_{B} is the only C_{0}-semigroup satisfying (1.9).

We refer to [40, Thm. 1 and Rem. 2(a)] and [41, Thm. 1.1] for the proof of Lemma 1.9 and Theorem 1.10. (In fact, T_{B} in Theorem 1.10 is constructed as a solution of the Duhamel formula (1.9); then formula (1.10) is a consequence of part (a).)

For the remainder of this subsection we fix the following assumptions. Let X be a Banach space, U an exponentially bounded s.c. propagator on X with parameter interval J, T the corresponding evolution semigroup on E := L_{1}(J; X), and G the generator of T . Further, let X_{s} ⊆ X (s ∈ J) be dense subspaces such that U(t, s)X_{s} ⊆ X_{t} for all (t, s) ∈ D_{J}. Let

D := lin{J ∋ s ↦ φ(s)U(s, r)x; r ∈ J, x ∈ X_{r}, φ ∈ C_{c}^{1}(ℝ) with spt φ ⊆ [r, ∞)} ⊆ E.

We emphasize that the space D depends on the choice of the family (X_{s})_{s∈J}. Let (B(t))_{t∈J} be a family of linear operators in X. We define a linear operator B in E by

D(B) := {f ∈ E; f(s) ∈ D(B(s)) for a.e. s ∈ J, Bf := B(·)f(·) ∈ E}.

1.11 Remarks. (a) Due to the strong continuity of U, the space D is dense in C_{c}( ^{∘} J; X) and hence in E. (Use a suitable partition of unity; cf. [28, proof of Prop. 1.13]). Let r ∈ J, x ∈ X_{r}, φ ∈ C_{c}^{1}(ℝ) with spt φ ⊆ [r, ∞), so that f_{0} := φ(·)U(·, r)x ∈ D. Then for all t ⩾ 0 we have

T (t)f_{0} = 11_{J}(· − t)U(·, · − t)φ(· − t)U(· − t, r)x = φ(· − t)U(·, r)x ∈ D,

so T (t)D ⊆ D and D ⊆ D(G). (As a consequence, D is a core for G.)

(b) Assume that D ⊆ D(B). Then Bf_{0} = φ(·)B(·)U(·, r)x ∈ E, with f_{0} as in (a). We infer that B(·)U(·, r)x ∈ L_{1,loc}(J_{>r}) for all r ∈ J, x ∈ X_{r}.

Moreover, BT (t)f_{0} = φ(· − t)B(·)U(·, r)x for all t ⩾ 0. Thus, for all f ∈ D the function [0, ∞) ∋ t ↦ BT (t)f ∈ E is continuous (even differentiable).

1.12 Theorem. Let α ∈ (0, ∞], γ ⩾ 0. Then the following are equivalent.

(i) For all s ∈ J, x ∈ X_{s} the function J_{⩾s} ∋ t ↦ B(t)U(t, s)x is defined almost everywhere and measurable, and

∫_{s}^{s+α} 11_{J}(t)||B(t)U(t, s)x|| dt ⩽ γ||x||.  (1.11)

If this condition is satisfied, we will say that (B(t))_{t∈J} is a Miyadera perturbation of U ( with constants (α, γ)), and that γ is a Miyadera bound.

(ii) D ⊆ D(B) and, for all f ∈ D,

∫_{0}^{α} ||BT (t)f||_{E} dt ⩽ γ||f||_{E}.

(Under this condition, the operator B̂|_{D} from Lemma 1.9 is a Miyadera perturbation of T ; recall from Remark 1.11(b) that t ↦ BT (t)f is continuous.)

Proof. ‘(i)⇒(ii)’ is shown in [25, proof of Thm. 3.4(a)].

‘(ii)⇒(i)’. Let s ∈ J, x ∈ X_{s}. For ε ∈ (0, α) let 0 ⩽ φ_{ε} ∈ C_{c}^{1}(ℝ) such that spt φ_{ε} ⊆ [s, s + ε], ∫ φ_{ε} = 1, and let f_{ε} := φ_{ε}(·)U(·, s)x (∈ D).

From Remark 1.11(b) we obtain that the function

J × [0, α] ∋ (t, τ) ↦ BT (τ)f_{ε}(t) = φ_{ε}(t − τ)B(t)U(t, s)x

is measurable, so

∫_{J} ∫_{0}^{α} φ_{ε}(t − τ) dτ ||B(t)U(t, s)x|| dt = ∫_{0}^{α} ∫_{J} ||φ_{ε}(t − τ)B(t)U(t, s)x|| dt dτ
= ∫_{0}^{α} ||BT (τ)f_{ε}||_{E} dτ ⩽ γ||f_{ε}||_{E}.

The assumptions on φ_{ε} imply that ∫_{0}^{α} φ_{ε}(t − τ) dτ = 1 for t ∈ [s + ε, s + α]. Therefore,

∫_{J} 11_{[s+ε,s+α]}(t)||B(t)U(t, s)x|| dt ⩽ γ ∫_{J} φ_{ε}(t)||U(t, s)x|| dt.

Letting ε → 0 we obtain (1.11).

1.13 Remark. It is interesting to note that B̂|_{D} is a Miyadera perturbation of T if and only if B̂|_{D} is G-bounded. Indeed, necessity is clear, so assume that B̂|_{D} is G-bounded. Then there are β, c ⩾ 0 such that ||Bf||_{1} ⩽ β||Gf||_{1} + c||f||_{1} for all f ∈ D. Let α > 0. For s ∈ J, x ∈ X_{s}, φ ∈ C_{c}^{1}(ℝ), 0 ⩽ φ ⩽ 11_{[s,s+α]}, ||φ′||_{1} ⩽ 2, f := φ(·)U(·, s)x we have Gf = −φ′(·)U(·, s)x by Remark 1.11(a) and hence

∫_{J} φ(t)||B(t)U(t, s)x|| dt = ||Bf||_{1} ⩽ β ∫_{J} |φ′(t)|||U(t, s)x|| dt + c ∫_{J} φ(t)||U(t, s)x|| dt

⩽ (2β + cα) sup_{s⩽t⩽s+α} ||U(t, s)x||.

Now choose (φ_{n}) ⊆ C_{c}^{1}(ℝ) such that 0 ⩽ φ_{n} ↑ 11_{[s,s+α]}, ||φ′_{n}||_{1} ⩽ 2. Then we obtain (1.11) with γ = (2β + cα) sup_{0⩽t−s⩽α} ||U(t, s)||.

For t ∈ J, t > inf J we define

X_{t−} := ⋃_{s∈J_{<t}} U(t, s)X_{s}.

Note that X_{t−} ⊆ X_{t}, U(t, s)X_{s−} ⊆ X_{t−}, and that X_{t−} is a dense subspace of X, for all (t, s) ∈ D_{J}, s > inf J.

1.14 Theorem. Let W be an exponentially bounded s.c. propagator on X, S the corresponding evolution semigroup on E. Let U, T , B be as above and assume that D ⊆ D(B). Then the following are equivalent.

(i) For all (t, s) ∈ D_{J}, s > inf J, x ∈ X_{s−}:

W (t, s)x = U(t, s)x + ∫_{s}^{t} W (t, τ)B(τ)U(τ, s)x dτ.  (1.12)

(ii) For all t ⩾ 0, f ∈ D:

S(t)f = T (t)f + ∫_{0}^{t} S(t − τ)BT (τ)f dτ.

(The assumption D ⊆ D(B) implies that the integrals in (i) and (ii) are defined; see Remark 1.11(b).)

If (ii) holds and, in addition, B(·)U(·, s)x ∈ L_{1,loc}(J_{⩾s}) for all s ∈ J, x ∈ X_{s}, then (1.12) holds for all (t, s) ∈ D_{J}, x ∈ X_{s}. This additional assumption is satisfied if (B(t))_{t⩾0} is a Miyadera perturbation of U.

Proof. We first show that (ii) is equivalent to (ii’) For all t ⩾ 0, r ∈ J, x ∈ X_{r}, and a.e. s > r with s + t ∈ J:

W (s + t, s)U(s, r)x = U(s + t, r)x + ∫_{s}^{s+t} W (s + t, τ)B(τ)U(τ, r)x dτ.  (1.13)

Let t ⩾ 0, r ∈ J, x ∈ X_{r}. By Remark 1.11(b), 11_{J_{>r}}(·)B(·)U(·, r)x is measurable. Since W is strongly continuous we obtain that

v: D_{J} → X, v(s, τ) := 11_{J_{>r}}(τ)W (s, τ)B(τ)U(τ, r)x

is measurable.

Let now φ ∈ C_{c}^{1}(ℝ) with spt φ ⊆ [r, ∞), f := φ(·)U(·, r)x (∈ D). Let 0 ⩽ τ ⩽ t. Then BT (τ)f = φ(· − τ)B(·)U(·, r)x, so

S(t − τ)BT (τ)f = 11_{J}(· − t + τ)W (·, · − t + τ)φ(· − t)B(· − t + τ)U(· − t + τ, r)x
= φ(· − t)v(·, · − t + τ) =: u(τ, ·),

with u: [0, t] × J → X measurable, u(τ, ·) ∈ E for all τ ∈ [0, t]. Moreover,

S(t)f = 11_{J}(· − t)W (·, · − t)φ(· − t)U(· − t, r)x = φ(· − t)W (·, · − t)U(· − t, r)x.

By the above computations, (ii) is equivalent to

φ(· − t)W (·, · − t)U(· − t, r)x = φ(· − t)U(·, r)x + ∫_{0}^{t} u(τ, ·) dτ  (1.14)

for all t ⩾ 0, r ∈ J, x ∈ X_{r}, φ ∈ C_{c}^{1}(ℝ) with spt φ ⊆ [r, ∞). Since u is measurable,

(∫_{0}^{t} u(τ, ·) dτ)(s) = ∫_{0}^{t} u(τ, s) dτ = φ(s − t) ∫_{s−t}^{s} v(s, τ) dτ

for a.e. s ∈ J. We conclude that (1.14) holds for all φ ∈ C_{c}^{1}(ℝ) with spt φ ⊆ [r, ∞) if and only if

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

for a.e. s ∈ J with s − t > r, i.e., if (1.13) holds for a.e. s > r with s + t ∈ J.

We now prove the equivalence of (i) and (ii’). Obviously, (i) implies (ii’), so assume that (ii’) holds. It is easy to see that the terms in (1.13) depend continuously on s, so (1.13) holds for all s > r with s + t ∈ J. For r ∈ J, y ∈ X_{r}, (t, s) ∈ D_{J} with s > r we thus obtain

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

i.e., (i) holds.

Finally, assume that (ii) holds, and that B(·)U(·, s)x ∈ L_{1,loc}(J_{⩾s}) for all s ∈ J, x ∈ X_{s}. If r ∈ J, x ∈ X_{r}, t ⩾ 0 with r + t ∈ ^{∘} J then, again by continuity in s, (1.13) holds for s = r. This proves (1.12) for all (t, s) ∈ D_{J} with t ∈ ^{∘} J, x ∈ X_{s}. To conclude the proof, observe that the terms in (1.12) are continuous in t.

In the following lemma we need the hypothesis that J is left half open. This assumption will be avoided further on.

1.15 Lemma. Assume that J is left open and right closed, that B|_{D} has a G-bounded extension B̂|_{D}, and that G +B̂|_{D} generates a C_{0}-semigroup T_{B} on E. Then T_{B} is an evolution semigroup.

Proof. The assertion follows from [25, Thm. 2.4, (a)⇔(c)] (see also [22, Thm. 4.12] for the case of bounded J). For details we refer the reader to [25, middle of p. 352]; note that, by the closed graph theorem, condition (R) ([25, p. 345]) is equivalent to the requirement that D(G) is a dense subset of C_{0}(J).

The essentials of the following result were shown in [25]. In fact, equation (1.15) was shown there only for x ∈ X_{s−}, and J was assumed to be right closed.

We give a new proof of this result, where in Theorems 1.12 and 1.14 we have singled out important equivalences between relations occuring in the propagator context and in the semigroup context.

1.16 Theorem. ([25, Thm. 3.4, Cor. 3.5]) Assume that (B(t))_{t∈J} is a small Miyadera perturbation of U, i.e., property (i) of Theorem 1.12 holds for some α > 0, 0 ⩽ γ < 1.

(a) Then there is a unique exponentially bounded s.c. propagator U_{B} on X satisfying

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U_{B}(t, τ)B(τ)U(τ, s)x dτ  (1.15)

for all (t, s) ∈ D_{J} and x ∈ X_{s}.

(b) Assume in addition that B(t) is closed for a.e. t ∈ J. Then property (i) of Theorem 1.12 holds with X_{t} = X (t ∈ J); in particular, (1.15) holds for all x ∈ X, (t, s) ∈ D_{J}.

Moreover, for x ∈ X, s ∈ J, the function B(·)U_{B}(·, s)x is defined almost everywhere and locally integrable on J_{⩾s}, and

U_{B}(t, s)x = U(t, s)x + ∫_{s}^{t} U(t, τ)B(τ)U_{B}(τ, s)x dτ  (1.16)

for all x ∈ X, (t, s) ∈ D_{J}.

The evolution semigroup associated with U_{B} has generator G + B.

Proof. (a) We first show that any s.c. propagator U_{B} satisfying (1.15) is exponentially bounded. Let c := sup{||U(t, s)||; (t, s) ∈ D_{J}, t − s ⩽ α}. Let (t, s) ∈ D_{J} with t − s ⩽ α, M := sup_{s⩽τ⩽t} ||U_{B}(t, τ)||. Then, for all s ⩽ σ ⩽ t, x ∈ X_{σ},

||U_{B}(t, σ)x|| ⩽ ||U(t, σ)x|| + M ∫_{σ}^{t} ||B(τ)U(τ, σ)x|| dτ ⩽ c||x|| + Mγ||x||,

so ||U_{B}(t, σ)|| ⩽ c + Mγ for all s ⩽ σ ⩽ t. Thus M ⩽ \frac{c}{1−γ}, and we conclude that U_{B} is exponentially bounded.

The uniqueness assertion is obtained by an analogous argument.

By Remark 1.11(a) and Theorem 1.12, the conditions of Lemma 1.9 are satisfied for T , G, D, B|_{D} as above. Hence, B|_{D} has a G-bounded extension B̂|_{D}, which is a small Miyadera perturbation of T . Thus, by Theorem 1.10, G + B̂|_{D} generates a C_{0}-semigroup T_{B} on E satisfying

T_{B}(t)f = T (t)f + ∫_{0}^{t} T_{B}(t − τ)BT (τ)f dτ  (1.17)

for all t ⩾ 0, f ∈ D.

For the case that J is left open and right closed, the assertion now follows from Lemma 1.15 and Theorem 1.14, (ii)⇒(i).

Next we study the case that J is right closed but a := inf J ∈ J, i.e., J is not left open. Let J̃:= J \ {a}. By the above we obtain a unique exponentially bounded s.c. propagator U_{B} with parameter interval J̃ satisfying (1.15) for all (t, s) ∈ DJ̃ and x ∈ X_{s}. We now extend this propagator to a propagator with parameter interval J, defining operators U_{B}(s, a) ∈ L(X) by

U_{B}(s, a)x := U(s, a)x + ∫_{a}^{s} U_{B}(s, τ)B(τ)U(τ, a)x dτ  (s ∈ J, x ∈ X_{a}).  (1.18)

Applying U_{B}(t, s) to both sides of (1.18) we obtain that

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

Now the propagator property U_{B}(t, s)U_{B}(s, a) = U_{B}(t, a) easily follows from (1.15). By the dominated convergence theorem, U_{B}(s, a)x → x strongly as s → a, for all x ∈ X_{a}. By Lemma 1.1 we conclude that U_{B} is strongly continuous on D_{J}.

Finally, assume that J is not right closed. Then there exists a sequence (J_{n}) of right closed subintervals of J with ⋃ J_{n} = J. By the above, for each n we obtain a unique perturbed propagator satisfying (1.15) for all (t, s) ∈ D_{J_{n}} and x ∈ X_{s}. The uniqueness implies that two such perturbed propagators coincide on their common domain.

(b) Observe that B is closed due to our additional assumption. By Lemma 1.9 we obtain that B ( ⊇ B̂|_{D}) is a small Miyadera perturbation of T (in particular, G + B = G + B̂|_{D} is the generator of T_{B}). This means that the inequality in Theorem 1.12(ii) is satisfied for all f ∈ D(G). Let now D be as above, but with X_{t} = X (t ∈ J). Then D ⊆ D(G), so we obtain that property (i) of Theorem 1.12 holds with X_{t} = X. Thus, by part (a), (1.15) holds for all x ∈ X, (t, s) ∈ D_{J}.

Let T̃_{B} be the evolution semigroup associated with U_{B}. By Theorem 1.14 we infer from the above that (1.17) also holds with T_{B} replaced by T̃_{B}, so the uniqueness assertion of Theorem 1.10 implies T̃_{B} = T_{B}. We conclude that G + B is the generator of T̃_{B}.

By [23, Lemma 1.1], B is also a (not necessarily small) Miyadera perturbation of T_{B}, i.e., there exist α_{1} > 0, c ⩾ 0 such that

∫_{0}^{α_{1}} ||BT_{B}(τ)f||_{E} dτ ⩽ c||f||_{E}

for all f ∈ D(G). Moreover, by Theorem 1.10(b) we obtain

T_{B}(t)f = T (t)f + ∫_{0}^{t} T (t − τ)BT_{B}(τ)f dτ

for all t ⩾ 0, f ∈ D(G). The remaining assertions of (b) now follow from Theorems 1.12 and 1.14, applied with D (⊆ D(G)) constructed from U_{B} instead of U, and X_{t} = X.

1.17 Remark. (a) If in Theorem 1.16 the operators B(t) are not closable then the perturbed propagator U_{B} may depend on the choice of the spaces X_{s}.

(b) We note that one can also prove Theorem 1.16 without using evolution semigroups; cf. [39].

We conclude this subsection with a result that is used to establish strong continuity in cases when both Duhamel formulas hold. In this result we assume X_{s} to be a dense subspace of X, for all s ∈ J, but we do not require U(t, s)X_{s} ⊆ X_{t}.

1.18 Proposition. Let W be a locally bounded propagator on X. Assume that, for all (t, s) ∈ D_{J}, x ∈ X_{s}, the functions B(·)W (·, s)x, W (t, ·)B(·)U(·, s)x are integrable on [s, t], and

W (t, s)x = U(t, s)x + ∫_{s}^{t} U(t, τ)B(τ)W (τ, s)x dτ,  (1.19)
W (t, s)x = U(t, s)x + ∫_{s}^{t} W (t, τ)B(τ)U(τ, s)x dτ.  (1.20)

Then W is strongly continuous.

Proof. This is an immediate consequence of Lemma 1.1 and the following lemma.

1.19 Lemma. Let W be a locally bounded propagator on X.

(a) Assume that, for all (t, s) ∈ D_{J}, x ∈ X_{s}, the function B(·)W (·, s)x is integrable on [s, t], and (1.19) holds. Then W (·, s) is strongly continuous on J_{⩾s} for all s ∈ J.

(b) Assume that, for all (t, s) ∈ D_{J}, x ∈ X_{s}, the function W (t, ·)B(·)U(·, s)x is integrable on [s, t], and (1.20) holds. Then W (t, ·) is strongly continuous on J_{⩽t} for all t ∈ J.

Proof. (a) Let s ∈ J, x ∈ X_{s}. Since

W (t, s)x = U(t, s)x + ∫_{J} 11_{[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 dominated convergence. This shows the assertion since X_{s} is dense in X and W is locally bounded.

(b) Fix t ∈ J, s_{0} ∈ J_{⩽t}. Let r ∈ J_{<s_{0}} (r = s_{0} in case s_{0} = inf J), x ∈ X_{r}, x_{s} := U(s, r)x for s ∈ [r, t]. By (1.20) 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 (1.20) again, we obtain

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 (1.20).) Thus, by dominated convergence,

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 obtain

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.

1.4 Domination for propagators and evolution semigroups

In this subsection we assume that X is a Banach lattice. Let U, Ũ be exponentially bounded s.c. propagators on X, and let T , T̃ be the evolution semigroups on L_{1}(J; X), associated with U, Ũ, respectively.

1.20 Lemma. (a) T̃(t) ⩾ 0 for all t ⩾ 0 if and only if Ũ(t, s) ⩾ 0 for all (t, s) ∈ D_{J}.

(b) Let T̃(t) ⩾ 0 for all t ⩾ 0. Then |T (t)f| ⩽ T̃(t)|f| for all t ⩾ 0, f ∈ L_{1}(J; X) if and only if |U(t, s)x| ⩽ Ũ(t, s)|x| for all (t, s) ∈ D_{J}, x ∈ X.

Proof. (a) The “if” part is obvious. Suppose that T̃(t) ⩾ 0 for all t ⩾ 0. Let 0 ⩽ x ∈ X, t ⩾ 0. Then 11_{J}(s − t)Ũ(s, s − t)x ⩾ 0 for a.e. s ∈ J. Therefore (t, s) ↦ Ũ(t, s)x is positive a.e. and continuous, hence Ũ(t, s)x ⩾ 0 for all (t, s) ∈ D_{J}.

(b) is proved in the same way as (a).

1.21 Proposition. Assume that U(t, s) ⩾ 0 for all (t, s) ∈ D_{J}. Let (B̃(t))_{t∈J} be a small Miyadera perturbation of U, and assume that the spaces X_{t} occuring in the definition (see Theorem 1.12(i)) are sublattices of X. Let B(t) be operators in X with D(B̃(t)) ⊆ D(B(t)) and

B̃(t)x ⩾ B(t)x ⩾ 0 (0 ⩽ x ∈ D(B̃(t)) ∩ X_{t−})

for all t ∈ J, t > inf J. Then (B(t))_{t∈J} is a small Miyadera perturbation of U, and

UB̃(t, s) ⩾ U_{B}(t, s) ⩾ 0 ((t, s) ∈ D_{J}).

Proof. Let s ∈ J. From the assumptions it is clear that (1.11) holds for all 0 ⩽ x ∈ X_{s} (with α, γ from the Miyadera condition for (B̃(t))_{t∈J}). Thus we can define an operator A_{s}: X_{s} → L_{1}((s, s + α) ∩ J; X) by

A_{s}x(t) := B(t)U(t, s)x.

The assumptions imply that A_{s} is positive. Therefore, |A_{s}x| ⩽ A_{s}|x| for all x ∈ X_{s}, and we obtain (1.11) for all x ∈ X_{s}, i.e., (B(t))_{t∈J} is a small Miyadera perturbation of U.

Let now B, B̃ be defined as in the previous subsection. By Theorem 1.12 we have D ⊆ D(B) ∩ D(B̃), with D ⊆ E = L_{1}(J; X) constructed from the spaces X_{t}.

By the definition of D, f(t) ∈ X_{t−} for all f ∈ D, t ∈ J. For 0 ⩽ f ∈ D we thus obtain B̃f ⩾ Bf ⩾ 0. From the Dyson-Phillips expansions for T_{B} and TB̃ (cf. [26]) we conclude that TB̃(t)f ⩾ T_{B}(t)f ⩾ 0 for all t ⩾ 0, 0 ⩽ f ∈ D. As in Remark 1.11(a) we see that {f ∈ D; f ⩾ 0} is dense in {f ∈ E; f ⩾ 0}. Thus the assertion follows from Lemma 1.20(b).

2 Absorption propagators

In this section we study the perturbation of propagators on L_{p}(μ) by real-valued potentials.

2.1 Abstract theory of positive absorption propagators

Let (Ω, μ) be a measure space, 1 ⩽ p < ∞, U a positive s.c. 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).) Let V: J × Ω → ℝ be measurable. Let V^{+}, V^{−} denote the positive and negative parts of V, respectively, i.e. V^{+} = V ∨ 0, V^{−} = (−V)^{+}. We are going to apply the perturbation results of subsection 1.2 to bounded potentials and to extend the definition of a perturbed propagator to suitable unbounded real-valued potentials.

For real-valued V ∈ L_{∞}(J ×Ω) we define a strongly measurable function B: J → L(L_{p}(μ)) by

B(τ) := −V (τ, ·) (τ ∈ J).

(In order to make this a valid definition we choose a bounded representative of V with measurable V (τ, ·) for all τ ∈ J.) 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.

The next proposition, which is a complete analogue of the corresponding statement for semigroups [42, Prop. 1.3], is the corner stone to the whole approach.

2.1 Proposition. Let U, U_{1}, U_{2} be positive s.c. propagators on L_{p}(μ), V, V_{1}, V_{2} ∈ L_{∞}(J ×Ω) real-valued. Then

(a) V_{1} ⩽ V_{2} implies U_{V_{1}} ⩾ U_{V_{2}} ⩾ 0;

(b) V ⩾ 0, U_{1} ⩽ U_{2} implies 0 ⩽ (U_{2})_{V} − (U_{1})_{V} ⩽ U_{2} − U_{1}.

Proof. (a) is proved in [24, Lemma 3.1].

(b) Choose c ∈ ℝ with V − c ⩽ 0 a.e. Formula (1.3) implies

(U_{1})_{V−c} ⩽ (U_{2})_{V−c}.

By Remark 1.7 we have

(U_{1})_{V} (t, s) = e^{−c(t−s)}(U_{1})_{V−c}(t, s) ((t, s) ∈ D_{J}),

and analogously for U_{2}. This proves the first of the asserted inequalities. The second, in the form U_{1} − (U_{1})_{V} ⩽ U_{2} − (U_{2})_{V}, then follows from the Duhamel formula (1.2).

As in the theory of absorption semigroups ([41], [42]), we approximate V by means of cut-offs V^{(n)} := (V ∧ n) ∨ (−n) (n ∈ ℕ) and define

U_{V} (t, s) := s-lim_{n→∞} U_{V^{(n)}}(t, s) ((t, s) ∈ D_{J}),  (2.1)

if the limits exist.

Obviously, U_{V} is a propagator in this case, and the inequalities stated in Proposition 2.1 for bounded potentials V, V_{1}, V_{2} are valid whenever the corresponding limits in (2.1) exist.

In the cases V ⩾ 0 and V ⩽ 0, the sequence (U_{V^{(n)}}(t, s))_{n∈ℕ} is monotone due to Proposition 2.1. This leads to the following definition.

2.2 Definition. Let V: J × Ω → ℝ be measurable.

(a) V ⩾ 0 is called U-admissible if (2.1) defines a s.c. propagator.

(b) V ⩽ 0 is called U-admissible if the limit in (2.1) exists and defines a s.c. propagator.

(c) V is called U-admissible if V^{+} and −V^{−} are U-admissible.

The crucial fact concerning the notion of admissibility with respect to s.c. propagators is in analogy to the corresponding fact for semigroups [42, Thm. 2.6], though the proof requires an additional argument involving Dini’s theorem. An important feature of the result is that the potential V is written as a difference of two positive potentials in a more general way than V = V^{+} − V^{−}.

2.3 Proposition. Let V_{±} ⩾ 0, assume that −V_{−} is U-admissible, and denote V := V_{+} − V_{−}.

(a) Then

(U_{−V_{−}})_{V_{+}}(t, s) = s-lim_{n→∞} U_{V^{(n)}}(t, s) ( = U_{V} (t, s))

for all (t, s) ∈ D_{J}.

(b) Assume additionally that V_{+} is U-admissible. Then V_{+} is U_{−V_{−}}-admissible, V_{−} is U_{V_{+}}-admissible, and

U_{V} = (U_{−V_{−}})_{V_{+}} = (U_{V_{+}})_{−V_{−}};

in particular, U_{V} is strongly continuous.

Proof. The arguments of the proof are taken from [38, proof of Prop. 1.6], except for the last argument in which Dini’s theorem is used.

(a) Let f ∈ L_{p}(μ), f ⩾ 0, (t, s) ∈ D_{J}. For m, n ∈ ℕ we have, by Proposition 2.1,

0 ⩽ (U_{−V_{−}})_{V_{+}^{(n)}}(t, s)f − (U_{−V_{−}^{(m)}})_{V_{+}^{(n)}}(t, s)f ⩽ U_{−V_{−}}(t, s)f − U_{−V_{−}^{(m)}}(t, s)f.  (2.2)

In order to proceed, we note the inequalities

V_{+}^{(n)} − V_{−}^{(2n)} ⩽ (V_{+} − V_{−})^{(n)} ⩽ V_{+}^{(2n)} − V_{−}^{(n)},

which imply

U_{V_{+}^{(n)}−V_{−}^{(2n)}}(t, s)f ⩾ U_{V^{(n)}}(t, s)f ⩾ U_{V_{+}^{(2n)}−V_{−}^{(n)}}(t, s)f.  (2.3)

Recall that (U_{−V_{−}^{(m)}})_{V_{+}^{(n)}} = U_{V_{+}^{(n)}−V_{−}^{(m)}} by Remark 1.7(b). Thus, using (2.2) first with m = 2n, and then with n replaced by 2n and m = n, we obtain that both the left hand side and the right hand side of (2.3) tend to (U_{−V_{−}})_{V_{+}}(t, s)f, and hence also U_{V^{(n)}}(t, s)f → (U_{−V_{−}})_{V_{+}}(t, s)f as n → ∞.

(b) Let f and (t, s) be as in the proof of (a). For m, n ∈ ℕ we have, by Proposition 2.1,

0 ⩽ (U_{V_{+}^{(n)}})_{m−V_{−}^{(m)}}(t, s)f − (U_{V_{+}})_{m−V_{−}^{(m)}}(t, s)f ⩽ U_{V_{+}^{(n)}}(t, s)f − U_{V_{+}}(t, s)f.

Hence, by Remark 1.7,

(U_{−V_{−}^{(m)}})_{V_{+}^{(n)}}(t, s)f = e^{m(t−s)}(U_{V_{+}^{(n)}})_{m−V_{−}^{(m)}}(t, s)f
→ e^{m(t−s)}(U_{V_{+}})_{m−V_{−}^{(m)}}(t, s)f = (U_{V_{+}})_{−V_{−}^{(m)}}(t, s)f

as n → ∞. Therefore, letting n → ∞ in (2.2) we obtain

0 ⩽ (U_{−V_{−}})_{V_{+}}(t, s)f − (U_{V_{+}})_{−V_{−}^{(m)}}(t, s)f ⩽ U_{−V_{−}}(t, s)f − U_{−V_{−}^{(m)}}(t, s)f → 0 (m → ∞),

and due to Dini’s theorem the latter convergence is uniform with respect to (t, s) on compact subsets of D_{J}. We conclude that (U_{−V_{−}})_{V_{+}} = (U_{V_{+}})_{−V_{−}} and that this propagator is strongly continuous.

2.4 Remark. Let V be U-admissible and assume that U_{−V^{−}} is exponentially bounded. Let T_{V}, T_{V^{(n)}} denote the evolution semigroups on E := L_{1}(J; L_{p}(μ)) corresponding to U_{V}, U_{V^{(n)}}, respectively. Then T_{V^{(n)}}(t) → T_{V} (t) strongly for all t ⩾ 0. Indeed, by Proposition 2.3(a), U_{V} (t, s) = s-lim_{n→∞} U_{V^{(n)}}(t, s) exists for all (t, s) ∈ D_{J}, and for f ∈ E, n ∈ ℕ we have

|T_{V^{(n)}}(t)f| ⩽ 11_{J}(· − t)U_{−V^{−}}(·, · − t)|f(· − t)|.

Thus the asserted convergence follows from the dominated convergence theorem.

Let T be the evolution semigroup on E corresponding to U, G the generator of T . Recall from Theorem 1.16(b) that G − V^{(n)} is the generator of T_{V^{(n)}}. In the same way as in [41, Cor. 2.7] we now obtain that the generator G_{V} of T_{V} is an extension of G−V. As a consequence, if G − V is a generator then G_{V} = G − V.

Note that L_{1}(J; L_{1}(μ)) is isomorphic to L_{1}(J × Ω, λ^{1} ⊗ μ), where λ^{1} is one-dimensional Lebesgue measure. Thus, in the case p = 1, T_{V} is an absorption semigroup on L_{1}(J × Ω, λ^{1} ⊗ μ), in the sense of [41]. (In the case p > 1, T_{V} is an absorption semigroup on an appropriate Banach function space; cf. [21].)

The remainder of this subsection is devoted to giving some more information concerning approximation and convergence of absorption propagators. These results will not be needed in the remaining part of the present paper.

2.5 Lemma. Let V_{1} ⩽ V_{2} ⩽ 0, V_{1} U-admissible. Then V_{2} is U-admissible.

Proof. First note that s-lim_{n→∞} U_{V_{2}^{(n)}}(t, s) exists for all (t, s) ∈ D_{J} since the sequence is dominated by U_{V_{1}}(t, s). The inequality

V_{1} + V_{2}^{(n)} − V_{1}^{(n)} ⩽ V_{1} + V_{2} − V_{1} = V_{2}

implies U_{V_{1}+V_{2}^{(n)}−V_{1}^{(n)}} ⩾ U_{V_{2}}. By Proposition 2.3(a) and Proposition 2.1 we obtain

0 ⩽ U_{V_{2}} − U_{V_{2}^{(n)}} ⩽ U_{V_{1}+V_{2}^{(n)}−V_{1}^{(n)}} − U_{V_{2}^{(n)}} = (U_{V_{1}})_{V_{2}^{(n)}−V_{1}^{(n)}} − (U_{V_{1}^{(n)}})_{V_{2}^{(n)}−V_{1}^{(n)}} ⩽ U_{V_{1}} − U_{V_{1}^{(n)}}.

By Dini’s theorem, the right hand side tends to zero strongly, uniformly on compact subsets of D_{J}, and the estimate shows that the same holds for U_{V_{2}} − U_{V_{2}^{(n)}}.

2.6 Remark. One would expect an analogous result for positive potentials, namely that V_{1} ⩾ V_{2} ⩾ 0, V_{1} U-admissible implies that V_{2} is U-admissible. However, we can only show this under the additional assumption lim sup_{s→t−} ||U(t, s)|| ⩽ 1 (t ∈ J).

We now prove an analogue of the non-monotone convergence result for semigroups (cf. [42, Thm. 2.6], [19, Prop. 1.11]). This generalises [24, Lemma 3.1(c)].

2.7 Theorem. Let V_{±} ⩾ 0, −V_{−} U-admissible, V := V_{+} − V_{−}. Let (V_{n})_{n∈ℕ} be such that −V_{−} ⩽ V_{n} ⩽ V_{+} for all n ∈ ℕ, V_{n} → V a.e.

(a) Then U_{V_{n}}(t, s) → U_{V} (t, s) strongly as n → ∞, for all (t, s) ∈ D_{J}.

(b) If additionally V_{+} is U-admissible then the strong convergence in (a) is uniform for (t, s) in compact subsets of D_{J}.

(Note that −V_{n}^{−} is U-admissible by Lemma 2.5 since V_{n}^{−} ⩽ V_{−}, and that therefore U_{V_{n}}(t, s) = s-lim_{k→∞} U_{V_{n}^{(k)}}(t, s) exists for all (t, s) ∈ D_{J}, by Proposition 2.3(a).)

Proof. As a first preliminary step, we prove the theorem for the case V_{+} = 0.

Let 0 ⩽ f ∈ L_{p}(μ), (t_{0}, s_{0}) ∈ D_{J}, ε > 0. From Dini’s theorem we obtain m ∈ ℕ such that

sup_{s_{0}⩽s⩽t⩽t_{0}} ||U_{V} (t, s)f − U_{V^{(m)}}(t, s)f|| ⩽ ε.

Observe that V_{n}^{(m)}(t, ·) → V^{(m)}(t, ·) strongly as n → ∞, for all t ∈ J, where we consider V_{n}^{(m)}(t, ·), V^{(m)}(t, ·) as multiplication operators in L(L_{p}(μ)). Thus, applying Proposition 1.8 we obtain N ∈ ℕ such that

sup_{s_{0}⩽s⩽t⩽t_{0}} ||U_{V^{(m)}}(t, s)f − U_{V_{n}^{(m)}}(t, s)f|| ⩽ ε.

for all n ⩾ N. From U_{V} (t, s)f ⩾ U_{V_{n}}(t, s)f ⩾ U_{V_{n}^{(m)}}(t, s)f we now conclude that

sup_{s_{0}⩽s⩽t⩽t_{0}} ||U_{V} (t, s)f − U_{V_{n}}(t, s)f|| ⩽ sup_{s_{0}⩽s⩽t⩽t_{0}} ||U_{V} (t, s)f − U_{V_{n}^{(m)}}(t, s)f|| ⩽ 2ε

for all n ⩾ N. This proves part (b) in this special case.

As a second preliminary step, we prove the theorem for the case V_{−} = 0.

Part (a) is proved in [24, Lemma 3.1(c)] (cf. also [41, Prop. A1]). The proof of part (b) is the same as the proof given above, only that now U_{V} (t, s)f ⩽ U_{V_{n}}(t, s)f ⩽ U_{V_{n}^{(m)}}(t, s)f. (Incidentally, in order to prove part (a), one just has to leave away the sup’s.)

Now we give the proof for the general case.

(a) Following [42, proof of Thm. 2.6] we define

V_{n,+} := (V_{n} + V_{−}) ∧ V_{+}, V_{n,−} := V_{n,+} − V_{n} ( = (−V_{n} + V_{+}) ∧ V_{−}).

Then

0 ⩽ V_{n,±} ⩽ V_{±}, V_{n,±} → V_{±} a.e.

By Proposition 2.3(a) we have U_{V_{n}} = (U_{−V_{n,−}})_{V_{n,+}}, so Proposition 2.1 implies

0 ⩽ (U_{−V_{−}})_{V_{n,+}} − U_{V_{n}} ⩽ U_{−V_{−}} − U_{−V_{n,−}}.  (2.4)

From this inequality, the assertion of (a) follows since U_{−V_{n,−}} → U_{−V_{−}} and (U_{−V_{−}})_{V_{n,+}} → (U_{−V_{−}})_{V_{+}} = U_{V} as n → ∞, as a consequence of the special cases V_{+} = 0, V_{−} = 0 studied initially.

(b) Under the additional hypothesis, V_{+} is U_{−V_{−}}-admissible by Proposition 2.3(b). Thus, inequality (2.4) together with the preliminary steps shows the assertion.

2.2 Miyadera class for potentials

Let (Ω, μ) be a measure space, 1 ⩽ p < ∞. Let U be an exponentially bounded positive s.c. propagator on L_{p}(μ) with parameter interval J ⊆ ℝ. Let V: J × Ω → ℝ be measurable. Without restriction, V (t, ·) is measurable for all t ∈ J and hence defines a (closed) multiplication operator in L_{p}(μ). Thus, by Theorem 1.16(b), V is a Miyadera perturbation of U if and only if there exist α > 0, γ ⩾ 0 such that

∫_{s}^{s+α} 11_{J}(t)||V (t)U(t, s)f||_{p} dt ⩽ γ||f||_{p}  (2.5)

for all s ∈ J, f ∈ L_{p}(μ), where we use the notation V (t) = V (t, ·).

In the following two theorems we formulate conditions under which the potential V is U-admissible. The first of these results is a generalisation of the strong continuity assertion in [24, Thm. 3.3].

2.8 Theorem. Let V ⩾ 0. Assume that for s ∈ J there exists a dense sublattice X_{s} of L_{p}(μ) such that ||V (·)U(·, s)f||_{p} is locally integrable on J_{⩾s}, for all f ∈ X_{s}. (This assumption is satisfied with X_{s} = X if V is a Miydera perturbation of U.) Then V is U-admissible, and for all (t, s) ∈ D_{J}, f ∈ X_{s} we have

U_{V} (t, s)f = U(t, s)f − ∫_{s}^{t} U_{V} (t, τ)V (τ)U(τ, s)f dτ  (2.6)
= U(t, s)f − ∫_{s}^{t} U(t, τ)V (τ)U_{V} (τ, s)f dτ.  (2.7)

Proof. Let (t, s) ∈ D_{J}, f ∈ X_{s}. For τ ∈ [s, t] we have, since U_{V^{(n)}} ⩽ U,

|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|.

Thus, by dominated convergence 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,

and obtain (2.6), (2.7). Now from Proposition 1.18 we conclude that U_{V} is strongly continuous.

2.9 Theorem. Let V ⩽ 0 be a small Miyadera perturbation of U. Then V is U-admissible, and (2.6), (2.7) hold for all (t, s) ∈ D_{J}, f ∈ L_{p}(μ).

Proof. Let T be the evolution semigroup associated with U, with generator G. By Theorem 1.16 there exists a s.c. propagator Ũ_{V} on L_{p}(μ) such that the evolution semigroup T̃_{V} associated with Ũ_{V} has generator G − V. Also by Theorem 1.16(b), G − V^{(n)} is the generator of the evolution semigroup T_{V^{(n)}} associated with U_{V^{(n)}}. Since −V^{(n)} ⩽ −V we obtain U_{V^{(n)}} ⩽ Ũ_{V} by Proposition 1.21.

Let now (t, s) ∈ D_{J}, f ∈ L_{p}(μ). From Theorem 1.16 we know that |V |(·)U(·, s)f, |V |(·)Ũ_{V} (·, s)f are locally integrable on J_{⩾s}. Thus, using

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

we obtain (2.6), (2.7) and hence the strong continuity of U_{V} as in the proof of Theorem 2.8.

For the case of a small Miyadera perturbation V of U, a perturbed propagator can be defined either by Theorem 1.16 or by using absorption propagators. The following result shows that these two approaches lead to the same object.

2.10 Corollary. Let T be the evolution semigroup associated with U, G the generator of T . Let V be a Miyadera perturbation of U, with V^{−} Miyadera small. Then V is U-admissible, and G − V is the generator of the evolution semigroup T_{V} associated with U_{V}.

Proof. We start with the following simple observation. If Ũ is a positive s.c. propagator on L_{p}(μ) with Ũ ⩽ U then a potential that is a small Miyadera perturbation of U is also Miyadera small with respect to Ũ.

The U-admissibility of V follows from Theorems 2.8 and 2.9. There exists n ∈ ℕ such that \frac{1}{n}V^{+} is Miyadera small with respect to U. By Theorems 1.12 and 1.10, G − \frac{1}{n}V^{+} is a generator, so by Remark 2.4, G − \frac{1}{n}V^{+} is the generator of the evolution semigroup associated with U_{\frac{1}{n} V^{+}}. Proposition 2.1(a) implies that U_{\frac{1}{n} V^{+}} ⩽ U. By the initial observation, \frac{1}{n}V^{+} is also Miyadera small with respect to U_{\frac{1}{n} V^{+}}, so we can repeat the argument and obtain that G − V^{+} is the generator of the evolution semigroup associated with U_{V^{+}}.

Again by the initial observation, −V^{−} is Miyadera small with respect to U_{V^{+}}. Repeating the above argument once more, we conclude that (G − V^{+}) + V^{−} = G − V is the generator of T_{V}.

3 Bounds for the integral kernels of propagators

In this section we assume that (Ω, μ) is a σ-finite measure space. For p, q ∈ [1, ∞] and a linear operator B: D(B) ⊆ L_{p}(μ) → L_{q}(μ), we denote its norm by ||B||_{p→q}.

Throughout this section we assume that

(A1) U is a positive s.c. 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 satisfying the bound in (A1); cf. Remark 1.7(a). Observe that 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 assumption

(A2) U(t, s) is bounded from L_{1} to L_{∞}, and there are constants K, ν > 0, 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 interest. In fact, if (A1) holds and ||U(t, s)||_{1→∞} ⩽ K(t − s)^{−ν} for all (t, s) ∈ D′_{J}, 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, f(t) := t^{−ν} ∨ 1 we thus obtain

(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 ν_{1} = 0. With the appropriate changes, our subsequent results are valid under assumption (A2’) with f as defined in (3.1), too, but we confine ourselves to assumption (A2) for the sake of simplicity.

(b) By the Dunford-Pettis theorem, assumptions (A1) and (A2) imply that the operator U(t, s) is an integral operator, with a positive kernel bounded by K(t − s)^{−ν}e^{A(t−s)}.

3.1 Diagonal upper bound

3.3 Proposition. Assume (A1) and (A2). Let V be a small Miyadera perturbation of U, i.e., condition (2.5) is satisfied with some α ∈ (0, ∞] and γ ∈ [0, 1). Then for 1 ⩽ κ < \frac{1}{γ} there exists c > 0 such that

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

where ω = \frac{1}{κ}(\frac{1}{α} log \frac{L}{1−γκ}) + (1 − \frac{1}{κ})A. In particular, the operators U_{V} (t, s) are integral operators.

Proof. We will prove (3.2) for bounded V, with a constant c depending on V only via the Miyadera bound γ. The case of unbounded V then is immediate from the definition (2.1) of U_{V}.

Let T be the evolution semigroup on L_{1}(J; L_{1}(μ)) corresponding to U, G the generator of T . By Theorem 1.12, κV is a small Miyadera perturbation of T with constants (α, γκ), so by [40, Thm. 1c)] we obtain

||T_{κV} (t)|| ⩽ \frac{L}{1−γκ}e^{(^{\frac{1}{α}} log ^{\frac{L}{1−γκ}} )t} (t ⩾ 0).

Recall from Theorem 1.16(b) that T_{κV} is the evolution semigroup corresponding to U_{κV}. The definition of evolution semigroups thus implies

||U_{κV} (t, s)|| ⩽ \frac{L}{1−γκ}e^{(^{\frac{1}{α}} log ^{\frac{L}{1−γκ}} )(t−s)} ((t, s) ∈ D_{J}).

Now we consider the family of propagators U_{κzV}, with z from the strip 0 ⩽ Re z ⩽ 1. The corresponding evolution semigroups T_{κzV} have generators G − κzV. Thus, by the Trotter product formula, |T_{κzV} (t))| ⩽ T_{κ(Re z)V} (t) for all t ⩾ 0. By Lemma 1.20(b) we obtain, for all (t, s) ∈ D′_{J}, that

|U_{κibV} (t, s)| ⩽ U(t, s), |U_{κ(1+ib)V} (t, s)| ⩽ U_{κV} (t, s) (b ∈ ℝ),

and that z ↦ U_{κzV} (t, s) is bounded. Moreover, z ↦ U_{κzV} (t, s) is analytic by Proposition 1.6. Thus from the Stein interpolation theorem [34, p. 69] we obtain

||U_{V} (t, s)||_{1→κ} ⩽ ||U_{κV} (t, s)||_{1→1}^{1/κ}||U(t, s)||_{1→∞}^{1−1/κ}

and hence (3.2).

By [10, p. 63, Thm. 5] we obtain from (3.2) that U_{V} (t, s) is an integral operator.

3.4 Remark. Let V be such that V^{−} is a small Miyadera perturbation of U. Then Proposition 3.3 can be applied to U_{−V^{−}}, and the inequality U_{V} ⩽ U_{−V^{−}} implies that (3.2) also holds under the above more general assumption about V.

Similar observations apply to most of the subsequent results.

In order to obtain L_{1}–L_{∞}-estimates, we will need two more assumptions about the propagator U. The first of these is boundedness in L_{∞},

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

3.5 Remarks. (a) Let assumptions (A1) and (A3) be satisfied. Then ||U(t, s)||_{p→p} ⩽ L for all 1 ⩽ p ⩽ ∞, (t, s) ∈ D_{J} by Riesz-Thorin interpolation. Thus, U extends to a consistent family of s.c. propagators U_{p} on L_{p}(μ), 1 ⩽ p < ∞ (for f ∈ L_{1} ∩ 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} ∩ L_{∞}(μ)).

(b) In addition, let V be a small Miyadera perturbation of U, with bound γ. Let 1 ⩽ p < \frac{1}{γ}. For (t, s) ∈ D_{J} we then obtain as in the proof of Proposition 3.3 that

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

Thus, U_{V} extends to a consistent family of exponentially bounded s.c. propagators U_{p,V} on L_{p}(μ), 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)} (1 ⩽ p ⩽ q < \frac{1}{γ}),

where C > 0, 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 [15, Thm. 2].

If (A3) holds then ||U(t, s)^{∗}f||_{1} ⩽ L||f||_{1} for all (t, s) ∈ D_{J}, f ∈ L_{1} ∩ L_{∞}(μ). We thus obtain a bounded backward propagator U_{∗} on L_{1}(μ) (we do not consider U_{∗} as a backward propagator on L_{∞}(μ) as in Remark 1.4(b)). The following assumption on U is needed for technical reasons. (Our results are in fact true without this assumption; cf. [39]).

(A4) U_{∗} is strongly continuous on L_{1}(μ).

The definition of a Miyadera perturbation for a s.c. backward propagator is reduced to the case of s.c. propagators by means of Remark 1.4(a). With the obvious modifications, Proposition 3.3 also holds for s.c. backward propagators. We note that V is a Miyadera perturbation of the s.c. backward propagator U_{∗} on L_{1}(μ) if there exist α ∈ (0, ∞], γ ⩾ 0 such that

∫_{s−α}^{s} 11_{J}(τ)||V (τ)U(s, τ)^{∗}f||_{1} dτ ⩽ γ||f||_{1}  (s ∈ J, f ∈ L_{1} ∩ L_{∞}(μ)).  (3.3)

In the case of the heat equation the above coincides with the backward Kato condition mentioned in the introduction.

3.6 Proposition. Assume (A1) to (A4). Let V be a small Miyadera perturbation of U and U_{∗}, with constants (α, γ), γ < 1 (valid for both U and U_{∗}). Then there exist c > 0, ω ∈ ℝ such that

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

More precisely, for θ ∈ (γ, 1) there exists c > 0 such that the above holds with ω = θ(\frac{1}{α} log \frac{L}{1−γ/θ} ) + (1 − θ)A.

Proof. (The following arguments are similar to those in [35, Thm. 5.1]. In [3, Thm. 3.3], a similar result is proved by an extrapolation argument.) Let θ ∈ (γ, 1). There exists k ∈ ℕ such that β := θk′ ∈ (γ, 1), where \frac{1}{k} + \frac{1}{k′} = 1. Then V_{1} := V/β is a small Miyadera perturbation of U and U_{∗}, with constants (α, γ_{1}), γ_{1} = \frac{γ}{β}. Thus, by Proposition 3.3 there exists c_{1} > 0 such that

||U_{V_{1}}(t, s)||_{1→k′} ⩽ c_{1}(t − s)^{− \frac{ν}{k}} e^{ω_{1}(t−s)} =: φ(t − s) and ||(U_{∗})_{V_{1}}(s, t)||_{1→k′} ⩽ φ(t − s)

for all (t, s) ∈ D′_{J}, with ω_{1} := \frac{1}{k′} (\frac{1}{α} log \frac{L}{1−γ/θ} ) + \frac{1}{k}A (note that γ_{1}k′ = γ/θ).

At the end of the proof we will show that (U_{∗})_{V_{1}}(s, t) = U_{V_{1}}(t, s)^{∗} on L_{1} ∩ L_{∞}(μ). By duality we then obtain

||U_{V_{1}}(t, s)||_{k→∞} ⩽ φ(t − s).

The Riesz-Thorin convexity theorem implies that

||U_{V_{1}}(t, s)||_{p→q} ⩽ φ(t − s) for 1 ⩽ p < q ⩽ ∞, \frac{1}{p} − \frac{1}{q} = \frac{1}{k}.

Now choosing p_{0} = 1 < p_{1} < ⋯ < p_{k} = ∞ such that \frac{1}{p_{j−1}} − \frac{1}{p_{j}} = \frac{1}{k} and t_{0} = s < t_{1} < ⋯ < t_{k} = t such that t_{j} − t_{j−1} = \frac{t−s}{k} (j = 1, …, k), we obtain

||U_{V_{1}}(t, s)||_{1→∞} ⩽ ∏_{j=1}^{k} ||U_{V_{1}}(t_{j}, t_{j−1})||_{p_{j−1}→p_{j}} ⩽ φ(\frac{t−s}{k})^{k} = c_{1}^{k}k^{ν}(t − s)^{−ν}e^{ω_{1}(t−s)}.

This implies, as in the proof of Proposition 3.3, that

||U_{V} (t, s)||_{1→∞} ⩽ ||U_{V_{1}}(t, s)||_{1→∞}^{β}||U(t, s)||_{1→∞}^{1−β} ⩽ c(t − s)^{−ν}e^{ω(t−s)},

with ω = βω_{1} + (1 − β)A = θ(\frac{1}{α} log \frac{L}{1−γ/θ} ) + (1 − β(1 − \frac{1}{k}))A, and the assertion follows.

Finally, we show that ⟨(U_{∗})_{V_{1}}(s, t)f, g⟩ = ⟨f, U_{V_{1}}(t, s)g⟩ for all f, g ∈ L_{1} ∩ L_{∞}(μ). Taking into account Remark 1.4(a), we obtain from (1.3) that

(U_{∗})_{V_{1}}(s, t) = ∑_{j=0}^{∞} U_{∗,j}(s, t),

with

U_{∗,0}(s, t) := U_{∗}(s, t), U_{∗,j}(s, t) := ∫_{s}^{t} U_{∗,j−1}(s, τ)( − V_{1}(τ))U_{∗}(τ, t) dτ (j = 1, …)

(strong integrals). From (1.5) we infer by induction that

⟨U_{∗,j}(s, t)f, g⟩ = ⟨f, U_{j}(t, s)g⟩ (j = 0, …)

where U_{j} are the constituents of the series for U_{V_{1}}. Summing the last identity over j we obtain the assertion.

3.2 Off-diagonal upper bound

More strongly than in (A2) we now assume that the operators U(t, s) are integral operators with kernels u_{0}(t, ·; s, ·) satisfying the Gaussian type upper bound

u_{0}(t, x; s, y) ⩽ K(t − s)^{−ν}e^{A(t−s)−ψ(t,x;s,y)}  ((t, s) ∈ D′_{J}, x, y ∈ Ω),  (3.4)

with constants K, ν > 0, A ∈ ℝ and a measurable function ψ ⩾ 0.

3.7 Remark. If Ω = ℝ^{N}, ν = \frac{N}{2} , A = 0 and ψ(t, x; s, y) = a\frac{d(x,y)^{2}}{t−s} for some a > 0 then (3.4) becomes the classical Gaussian bound. By the results of [4], this bound holds for the fundamental solution of (0.1) (with V = 0).

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 [9, 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)^{− \frac{N}{2}} has to be replaced by (t − s)^{− \frac{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 a bound of the type (A2) is valid (so our results do not apply to this case).

The next result is the main tool to obtain a off-diagonal upper bound for the kernel of the perturbed propagator. It will also be crucial for the lower bound.

3.8 Proposition. Assume (A1) and (A2). Let V_{0}, V_{1} be measurable, with V_{0}^{−}, V_{1}^{−} small Miyadera perturbations of U. Let θ ∈ (0, 1), V_{θ} := (1 − θ)V_{0} + θV_{1}. Then V_{θ}^{−} is a small Miyadera perturbation of U, and for all f ∈ L_{1}(μ), g ∈ L_{∞}(μ) we have

|⟨U_{V_{θ}}(t, s)f, g⟩| ⩽ ⟨U_{V_{0}}(t, s)|f|, |g|⟩^{1−θ} ⟨U_{V_{1}}(t, s)|f|, |g|⟩^{θ} ((t, s) ∈ D_{J}).

Moreover, U_{V_{θ}}(t, s) is an integral operator for (t, s) ∈ D′_{J}, and for the kernel we have

u_{V_{θ}} ⩽ u_{V_{0}}^{1−θ}u_{V_{1}}^{θ}.  (3.5)

Proof. The first assertion is clear. By the definition of absorption propagators it suffices to prove the second assertion in the case of bounded V_{0}, V_{1}. We now proceed as in [42, proof of Prop. 1.4]. Fix (t, s) ∈ D_{J}, f ∈ L_{1}(μ), g ∈ L_{∞}(μ). Define the function

F (z) := ⟨U_{(1−z)V_{0}+zV_{1}}(t, s)f, g⟩ = ⟨(U_{V_{0}})_{z(V_{1}−V_{0})}(t, s)f, g⟩

on the strip 0 ⩽ Re z ⩽ 1. As in the proof of Proposition 3.3, we obtain that F is bounded and analytic, and

|F (0 + ib)| ⩽ ⟨U_{V_{0}}(t, s)|f|, |g|⟩, |F (1 + ib)| ⩽ ⟨U_{V_{1}}(t, s)|f|, |g|⟩ (b ∈ ℝ).

By the Hadamard three lines lemma we conclude that

|⟨U_{V_{θ}}(t, s)f, g⟩| = |F (θ)| ⩽ ⟨U_{V_{0}}(t, s)|f|, |g|⟩^{1−θ} ⟨U_{V_{1}}(t, s)|f|, |g|⟩^{θ}.

From Proposition 3.3 we obtain that U_{V_{θ}}(t, s) is an integral operator. The last assertion now follows from Lemma 3.9 below.

3.9 Lemma. Let A, B, C be positive integral operators on L_{p}(μ). Let a, b, c be the corresponding integral kernels. Suppose that there exists θ ∈ (0, 1) such that for 0 ⩽ f, g ∈ L_{1} ∩ L_{∞}(μ) one has

⟨Af, g⟩ ⩽ ⟨Bf, g⟩^{θ}⟨Cf, g⟩^{1−θ}.

Then

a ⩽ b^{θ}c^{1−θ} a.e.

Proof. It follows from Young’s inequality that, for any δ > 0,

⟨Af, g⟩ ⩽ θδ⟨Bf, g⟩ + (1 − θ)δ^{− \frac{θ}{1−θ}} ⟨Cf, g⟩.

This implies

a ⩽ θδb + (1 − θ)δ^{− \frac{θ}{1−θ}} c a.e.

Optimizing the right hand side over rational δ’s we obtain the assertion.

3.10 Theorem. Let U be a propagator satisfying (A1) to (A4) and the Gaussian type upper bound (3.4). Let V be a Miyadera perturbation of both U and U_{∗}, V^{−} Miyadera small with constants (α, γ), γ < 1. Then the propagator U_{V} consists of integral operators U_{V} (t, s), and for γ < β < 1 there exist c > 0, ω ∈ ℝ such that the kernel u_{V} satisfies

u_{V} (t, x; s, y) ⩽ c(t − s)^{−ν}e^{ω(t−s)−(1−β)ψ(t,x;s,y)} ((t, s) ∈ D′_{J}, x, y ∈ Ω).

More precisely, for θ ∈ (γ, β) there exists c > 0 such that the above holds with ω = θ(\frac{1}{α} log \frac{L}{1−γ/θ} ) + (1 − θ)A.

Proof. Let V_{1} := V/β, θ_{1} := θ/β. By Proposition 3.6 there exists c_{1} > 0 such that

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

with ω_{1} = θ_{1}(\frac{1}{α} log \frac{L}{1−γ/θ} ) + (1 − θ_{1})A. From Proposition 3.8 we obtain u_{V} ⩽ u_{V_{1}}^{β}u_{0}^{1−β}, so the assertion follows with ω = βω_{1} + (1 − β)A.

3.11 Remark. If V^{−} is an infinitesimally small Miyadera perturbation of U and 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.

The validity of the upper bound (3.4) with A = 0, i.e., a bound global in time, is of special interest for questions of long time behaviour. By Theorem 3.10 the global upper bound (3.4) is stable under perturbation by the potential V if V^{−} is a Miyadera perturbation of U and U_{∗} with constants (∞, γ), γ < 1.

3.3 Lower bound

With the methods from the previous subsections we obtain the following result about the lower bound of the integral kernel.

3.12 Theorem. Let U be a propagator satisfying (A1) to (A4) and the Gaussian type lower bound

u_{0}(t, x; s, y) ⩾ K_{1}(t − s)^{−ν}e^{−A_{1}(t−s)−ψ(t,x;s,y)}  ((t, s) ∈ D′_{J}, x, y ∈ Ω),  (3.6)

with some K_{1}, ν_{1} > 0, A_{1} ∈ ℝ, and a measurable function ψ ⩾ 0. Let V be a Miyadera perturbation of U, with V^{−} Miyadera small. Assume that V^{+} is a Miyadera perturbation of U_{∗}, and let (α, γ) be Miyadera constants of V^{+} with respect to both U and U_{∗}. Then for β > γ there are constants c > 0, ω ∈ ℝ such that the kernel u_{V} of U_{V} satisfies

u_{V} (t, x; s, y) ⩾ c(t − s)^{− \frac{N}{2}} e^{−ω(t−s)−(1+β)ψ(t,x;s,y)} ((t, s) ∈ D′_{J}, x, y ∈ Ω).

More precisely, for θ ∈ (γ, β) there exists c > 0 such that the above holds with ω = θ(\frac{1}{α} log \frac{L}{1−γ/θ} ) + (β − θ)A + (1 + β)A_{1}.

Proof. Note that U_{V} ⩾ U_{V^{+}}. Thus, U_{V} has a kernel by Proposition 3.3, and it suffices to prove the lower estimate for V = V^{+}. By the assumption, −V/β is a small Miyadera perturbation of U and U_{∗}. Therefore, by Proposition 3.8,

u_{0} ⩽ u−V/β^{\frac{β}{1+β}}uV^{\frac{1}{1+β}} , so u_{V} ⩾ (u_{−V/β})^{−β}u_{0}^{1+β}.

By Proposition 3.6 there exists c_{1} > 0 such that ||U_{−V/β}(t, s)||_{1→∞} ⩽ c_{1}(t − s)^{−ν}e^{ω_{1}(t−s)} for all (t, s) ∈ D′_{J}, with ω_{1} as in the proof of Theorem 3.10. Thus the assertion follows with ω = βω_{1} + (1 + β)A_{1}.

3.13 Remark. Observations similar to those in Remark 3.11 concerning sharp and global upper bounds, respectively, also apply to the lower bounds. Assume that V^{−} is a small Miyadera perturbation of U. If V^{+} is an infinitesimally small Miyadera perturbation of U and U^{∗} we obtain sharp lower bounds for t − s ⩽ 1.

If A = 0 in the diagonal upper bound and A_{1} = 0 in (3.6) then the global lower bound is stable under perturbation by the potential V if V^{+} is a Miyadera perturbation of U and U_{∗} with constants (∞, γ), with any γ ⩾ 0. Up to our knowledge, the latter result is new even in the case of the heat equation with a time-dependent potential.

3.14 Remark. Assume that U and U^{∗} are L_{1}-norm preserving and that U satisfies classical Gaussian upper and lower bounds (cf. Remark 3.7). This is the case, e.g., if U corresponds to the Cauchy problem for (0.1) with V = 0. If V ⩽ 0 then the conditions of Theorem 3.10 are close to necessary. Indeed, if U_{V} satisfies a Gaussian upper bound then U_{V} and U_{V}^{∗} are exponentially bounded on L_{1}(ℝ^{N}). Then one can follow the argument in the proof of [41, Prop. 4.6] in order to show that V is a (not necessarily small) Miyadera perturbation of U and U^{∗}. If V ⩾ 0 then the conditions of Theorem 3.12 are in fact necessary. This follows by a similar argument.

Acknowledgement. The first named author would like to acknowledge the support of Alexander von Humboldt Foundation and to thank the Institut für Analysis at the Technische Universität Dresden for hospitality.

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