Poisson integrals of regular functions. (English) Zbl 0613.31005
Let \(C_ p^{\alpha}\) be the Banach space of \(L^ p\)-functions, introduced by A. P. Calderon and R. Scott [Stud. Math. 62, 75-92 (1978; Zbl 0399.46031)] and extensively discussed by R. A. Devore and R. C. Sharpley [Mem. Am. Math. Soc. 293 (1984; Zbl 0529.42005)]. The author defines a proper closed subspace \(F_ p^{\alpha}\subset C_ p^{\alpha}\) by an asymptotic condition and studies this space in some detail; in particular \(F_ p^{\alpha}\) contains the spaces of Bessel potentials of \(L^ p\)-functions, \(1<p<\infty\), and of functions in the local Hardy space \(h^ 1\). Tangential convergence in the special case of Bessel potentials is established by A. Nagel, W. Rudin and J. H. Shapiro [Ann. Math., II. Ser. 116, 331-360 (1982; Zbl 0531.31007)]. Here tangential convergence of Poisson integrals a.e. through appropriate approach regions, depending on \(\alpha\) and p, is proved for \(f\in F_ p^{\alpha}\). Furthermore it is shown, that the corresponding tangential maximal functions are of strong p-type, \(p\geq 1\).
Reviewer: L.Neckermann
MSC:
| 31B25 | Boundary behavior of harmonic functions in higher dimensions |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
Bessel potentials; \(L^ p\)-functions; local Hardy space; Tangential convergence; Poisson integrals; tangential maximal functionsReferences:
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