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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 variableargument of Caffarelli and Peral and adapt this theorem to account fornon-local weak reverse H\"older estimates. These non-local weak reverseH\"older estimates appear for example in the investigation of non-localelliptic integrodifferential operators. This originates from the fact that hereonly a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, andSire. As an application, we prove resolvent estimates and maximal regularityproperties in $\mathrm{L}^p$-spaces of non-local elliptic integrodifferentialoperators.

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