Generalized local Fatou theorems and area integrals

Abstract

ABSTRACT, Let X be a space of homogeneous type and W a subset of X x (0,00). Then, under minimal conditions on W, we obtain a relationship between two modes of convergence at the boundary X for functions defined on W. This result gives new local Fatou theorems of the Carleson-type for solutions of Laplace, parabolic and Laplace-Beltrami equations as immediate consequences of the classical results. Lusin area integral characterizations for the existence of limits within these more general approach regions are also obtained.