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Keywords:
square root of the Poisson kernel; approach regions; almost everywhere convergence; maximal functions; Orlicz spaces
square root of the Poisson kernel; approach regions; almost everywhere convergence; maximal functions; Orlicz spaces
Summary:
If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of $L^{p}$ and weak $L^{p}$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces $L^{\Phi }$ having the property $L^{\infty }\subset L^{\Phi }\subset L^{p}$, $1\le p<\infty $. The second contains spaces $L^{\Phi }$ that resemble $L^{p}$ spaces.
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