${\mathrm{L}}^p$-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 which builds on a real variable argument of Caffarelli and Peral 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. As an application, we prove resolvent estimates and maximal regularity properties in ${\mathrm{L}}^p$-spaces of non-local elliptic integrodifferential operators.

17 pages