The Fatou Property for General Approximate Identities on Metric Measure Spaces

G. A. Karagulyan ; I. N. Katkovskaya ; V. G. Krotov

Matematičeskie zametki, Tome 110 (2021) no. 2, pp. 204-220.

Résumé

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.

Détail

Keywords: metric measure space, approximate identity, Fatou property
Mots-clés : Poisson integral.

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