Generalized local Fatou theorems and area integrals

Abstract

Let X X be a space of homogeneous type and W W a subset of X × ( 0 , ∞ ) X \times (0,\infty ) . Then, under minimal conditions on W W , we obtain a relationship between two modes of convergence at the boundary X X for functions defined on W 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.