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Approach regions for the square root of the Poisson kernel and bounded functions. (English) Zbl 0903.31002

Summary: If the Poisson kernel of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of \(L^p\) boundary functions converge almost everywhere at the boundary, along approach regions wider than the ordinary nontangential cones. The sharp approach region, defined by means of a monotone function, increases with \(p\). We make this picture complete by determining along which approach regions one has almost everywhere convergence for \(L^\infty\) boundary functions.

MSC:

31A20 Boundary behavior (theorems of Fatou type, etc.) of harmonic functions in two dimensions
42B25 Maximal functions, Littlewood-Paley theory
43A85 Harmonic analysis on homogeneous spaces
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References:

[1] Sjögren, Pacific J. Math. 131 pp 361– (1988) · Zbl 0601.31001
[2] DOI: 10.1006/aima.1996.0035 · Zbl 0878.46020
[3] Rönning, A convergence result for square roots of the Poisson kernel in the bidisk (1993)
[4] Sjögren, Théorie du Potentiel, Proceedings Orsay 1983 1096 pp 141– (1984)
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