Maximal regularity of solutions for the tempered fractional Cauchy problem

Abstract

Let $X$ be a Banach space. Given a closed linear operator $A$ defined on $X$ we show that, in vector-valued H\"older spaces $C^{\alpha}(\R,X)\, \, (0<\alpha<1)$, maximal regularity for the abstract Cauchy problem can be characterized solely in terms of a spectral property of the operator $A$, when we equip the Cauchy problem with the tempered fractional derivative. In particular, we show that generators of bounded analytic semigroups admit maximal regularity.