Singular integrals in nonhomogeneous spaces: $L^2$ and $L^p$ continuity from Hölder estimates

Abstract

We present a result of $L^p$ continuity of singular integrals of Calderón-Zygmund type in the context of bounded nonhomogeneous spaces, well suited to be applied to problems of a priori estimates for partial differential equations. First, an easy and selfcontained proof of $L^2$ continuity is got by means of $C^{\alpha }$ continuity, thanks to an abstract theorem of Krein. Then $L^p$ continuity is derived adapting known results by Nazarov-Treil-Volberg about singular integrals in nonhomogeneous spaces.