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

We present a result of $L^{p}$ continuity of singular integrals of Calder\'{o}n-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.