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Sobolev inequalities, the Poisson semigroup, and analysis on the sphere \(S^ n\). (English) Zbl 0766.46012

The principal result of this paper is to prove Stein’s conjecture about the action of the Poisson semigroup on \(L^ p(S^ n)\). Consider the smooth functions on the sphere \(S^ n\) with development in spherical harmonics \(f(\xi)=\sum^ \infty_{k=0}Y_ k(\xi)\), where \(Y_ k\) is a spherical harmonic of degree \(k\). The action of the Poisson semigroup is defined by \((P_ rf)(\xi)=\sum^ \infty_{k=0}r^ kY_ k(\xi)\) for \(-1\leq r\leq 1\).
The main result of the paper is the following theorem:
The Poisson semigroup defines a contraction mapping from \(L^ p(S^ n)\) to \(L^ q(S^ n)\) with \(1\leq p\leq q\leq\infty\) and \(r\) real if and only if \(| r|\leq[(p-1)/(q-1)]^{1/2}\). \(\| P_ rf\|_{L^ q(S^ n)}\leq\| f\|_{L^ p(S^ n)}\).
This theorem generalizes results from other papers: F. B. Weissler [J. Funct. Anal. 37, 218-234 (1980; Zbl 0463.46024)]; S. Janson [Ark. Mat. 21, 97-110 (1983; Zbl 0516.42022)]; C. E. Mueller and F. B. Weissler [J. Funct. Anal. 48, 252-283 (1982; Zbl 0506.46022)]; E. Nelson [J. Funct. Anal. 12, 211-227 (1973; Zbl 0273.60079)]; L. Gross [Am. J. Math. 97, 1061-1083(1975) (1976; Zbl 0318.46049)].
This note contains fundamental theorems, which are of interest for all mathematicians who investigate functional analysis and operator theory.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47D06 One-parameter semigroups and linear evolution equations
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