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

José María Martell¹

¹ Departamento de Matemáticas, C-XV Universidad Autónoma de Madrid 28049 Madrid, Spain

Studia Mathematica, Tome 161 (2004) no. 2, pp. 113-145

Résumé

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.

Détail

DOI : 10.4064/sm161-2-2

Keywords: context spaces homogeneous type given family operators look approximations identity sharp maximal functions considered prove good lambda inequality muckenhoupt weights which leads analog fefferman stein estimate classical sharp maximal function consequence establish weighted norm estimates certain singular integrals defined irregular domains rmander conditions replaced estimates which involve regularity kernel apply these results prove boundedness holomorphic functional calculi lebesgue spaces muckenhoupt weights particular applications given second order elliptic operators different boundary conditions