Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications
In the context of the spaces of homogeneous type, given a family of operators that look like approximations of the identity, new sharp maximal functions are considered. We prove a good-$\lambda$ inequality for Muckenhoupt weights, which leads to an analog of the Fefferman–Stein estimate for the classical sharp maximal function. As a consequence, we establish weighted norm estimates for certain singular integrals, defined on irregular domains, with Hörmander conditions replaced by some estimates which do not involve the regularity of the kernel. We apply these results to prove the boundedness of holomorphic functional calculi on Lebesgue spaces with Muckenhoupt weights. In particular, some applications are given to second order elliptic operators with different boundary conditions.