Arch. Math. 82 (2004) 415–431 0003–889X/04/050415–17 DOI 10.1007/s00013-004-0585-2 © Birkh¨ auser Verlag, Basel, 2004 Archiv der Mathematik Maximal regularity for evolution equations in weighted L p -spaces By Jan Pr ¨ uss and Gieri Simonett Abstract. Let X be a Banach space and let A be a closed linear operator on X. It is shown that the abstract Cauchy problem ˙ u(t) + Au(t) = f(t), t > 0, u(0) = 0, enjoys maximal regularity in weighted L p -spaces with weights ω(t) = t p(1−µ) , where 1/p < µ, if and only if it has the property of maximal L p -regularity. Moreover, it is also shown that the derivation operator D = d/dt admits an H ∞ -calculus in weighted L p -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 ˙ u(t) + Au(t) = f (t ), t> 0, u(0) = 0, ( 1.1) where f ∈ L 1,loc (R + ; X). In the following we say that the Cauchy problem ( 1.1) has the property of maximal L p -regularity if for each function f ∈ L p (R + ; X) there exists a unique solution u ∈ W 1 p (R + ; X) ∩ L p (R + ; D(A)). We define MR p (X) to be the class of all operators A that admit maximal L p -regularity for ( 1.1) in X. Let us recall some well-known facts about this class. If A ∈ MR p (X) for some p ∈ (1, ∞), then A ∈ MR q (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]. If A ∈ MR p (X) for some p ∈ (1, ∞), then A generates an exponentially stable analytic C 0 -semigroup in X. A proof of this fact is contained in Hieber and Pr ¨ uss [6], see also Pr ¨ uss [7, Section 10]. In this note we consider the question of maximal regularity for the weighted L p -spaces L p,µ (R + ; X) :={f : R + → X : t 1−µ f ∈ L p (R + ; X)}. We say that A has maximal L p,µ -regularity if for each f ∈ L p,µ (R + ; X) there is a unique function u ∈ L p,µ (R + ; X) such that ˙ u, Au ∈ L p,µ (R + ; X), and such that u solves ( 1.1). Mathematics Subject Classification (2000): 35K90, 47A60, 35K55.