Generalized Local Fatou Theorems and Area Integrals

Let $X$ be a space of homogeneous type and $W$ a subset of $X \times (0,\infty )$. 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.