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 Hölder 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.