Regularity properties of sectorial operators: extrapolation, counterexamples and generic classes

Abstract

We study the interplay of different regularity properties of sectorial operators. The three main results deal with maximal regularity and the holomorphic functional calculus of sectorial operators.

In the first main result we construct the first explicit examples of generators of bounded strongly continuous analytic semigroups on UMD-spaces without maximal regularity. In the second result we give a negative answer to the extrapolation problem for maximal regularity. In the third result we give a complete characterization of the class of sectorial operators on reflexive L^p-spaces that have a bounded holomorphic functional calculus.