Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss

Abstract

For $1 <p,q <+\infty$, we show the maximal $L^q$ regularity for the inhomogeneous evolution equation in $L^p(\Omega)$ associated with the infinitesimal generator of a semigroup whose kernel satisfies suitable upper bounds, with $\Omega$ a subset of a space of homogeneous type.