Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications

Abstract

In the context of the spaces of homogeneous type, given a family of op- erators that look like approximations of the identity, new sharp maximal functions are considered. We prove a good- inequality for Muckenhoupt weights, which leads to an analog of the Feerman{Stein estimate for the classical sharp maximal function. As a consequence, we establish weighted norm estimates for certain singular integrals, dened on irregular domains, with Hormander conditions replaced by some estimates which do not involve the regularity of the kernel. We apply these results to prove the bounded- ness of holomorphic functional calculi on Lebesgue spaces with Muckenhoupt weights. In particular, some applications are given to second order elliptic operators with dieren t boundary conditions.