Mathrm {L}^p-extrapolation of non-local operators: Maximal regularity of elliptic integrodifferential operators with measurable coefficients

Abstract

The aim of this article is to deepen the understanding of the derivation of mathrm {L}^p-estimates of non-local operators. We review the mathrm {L}^p-extrapolation theorem of Shen (2005) which builds on a real variable argument of Caffarelli and Peral (1998) and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear, for example, in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire (2015). As an application, we prove resolvent estimates and maximal regularity properties in mathrm {L}^p-spaces of non-local elliptic integrodifferential operators.