Maximal regularity for evolution equations in weighted Lp - spaces

LetX be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem u̇(t)+ Au(t) = f (t), t > 0, u(0) = 0, enjoys maximal regularity in weighted Lp-spaces with weights ω(t) = tp(1−μ), where 1/p < μ, if and only if it has the property of maximal Lp-regularity. Moreover, it is also shown that the derivation operator D = d/dt admits anH∞-calculus in weighted Lp-spaces. Introduction. Let X be a Banach space and let A be a closed linear operator on X with domain D(A). We consider the abstract Cauchy problem u̇(t)+ Au(t) = f (t), t > 0, u(0) = 0, ( 1.1) where f ∈ L1,loc(R+;X). In the following we say that the Cauchy problem ( 1.1) has the property of maximal Lp-regularity if for each function f ∈ Lp(R+;X) there exists a unique solution u ∈ W 1 p(R+;X)∩Lp(R+;D(A)). We defineMRp(X) to be the class of all operators A that admit maximal Lp-regularity for ( 1.1) in X. Let us recall some well-known facts about this class. If A ∈ MRp(X) for some p ∈ (1,∞), thenA ∈ MRq(X) for all q ∈ (1,∞). This was first observed by Sobolevskii [9], and was then rediscovered several times, e.g. by Cannarsa and Vespri [3]. IfA ∈ MRp(X) for some p ∈ (1,∞), thenA generates an exponentially stable analytic C0-semigroup inX. A proof of this fact is contained in Hieber and Prüss [6], see also Prüss [7, Section 10]. In this note we consider the question of maximal regularity for the weighted Lp-spaces Lp,μ(R +;X) := {f : R+ → X : t1−μf ∈ Lp(R+;X)}. We say that A has maximal Lp,μ-regularity if for each f ∈ Lp,μ(R+;X) there is a unique function u ∈ Lp,μ(R+;X) such that u̇, Au ∈ Lp,μ(R+;X), and such that u solves ( 1.1). Mathematics Subject Classification (2000): 35K90, 47A60, 35K55. 416 Jan Prüss and Gieri Simonett arch. math. We will show that A ∈ MRp(X) implies that A also has maximal Lp,μ-regularity, that is A ∈ MRp,μ(X), provided μ > 1/p. Moreover, we show that the reverse conclusion is also true. The restriction on μ comes from several facts. The first one is the embedding Lp,μ(R +;X) ↪→ L1,loc(R+;X), which is valid for μ > 1/p. The second one is due to Hardy’s inequality which reads ∞ ∫