Generalized local Fatou theorems and area integrals

Mair, B. A.; Philipp, Stan; Singman, David

doi:10.1090/S0002-9947-1990-0974520-3

Generalized local Fatou theorems and area integrals
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by B. A. Mair, Stan Philipp and David Singman

Trans. Amer. Math. Soc. 321 (1990), 401-413

DOI: https://doi.org/10.1090/S0002-9947-1990-0974520-3

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Abstract:

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.

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