The Fatou Property for General Approximate Identities on Metric Measure Spaces

Abstract approximate identities on metric measure spaces are considered in this paper. We find exact conditions on the geometry of domains for which the convergence of approximate identities occurs almost everywhere for functions from the spaces $L^p$, $p\ge 1$. The results are illustrated with examples of Poisson kernels and their powers in the unit ball in $\mathbb{R}^n$ or $\mathbb{C}^n$, and also of convolutions with dilatations on $\mathbb{R}^n$. In all these examples, the conditions found are exact.