## 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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