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Regularity of mean-values

Part of: Abstract harmonic analysis Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Christopher Meaney
Christopher Meaney
Affiliation:
Department of MathematicsAustralian National UniversityCanberra, ACT 2601, Australia
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Abstract

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Let X be either the d-dimensional sphere or a compact, simply connected, simple, connected Lie group. We define a mean-value operator analogous to the spherical mean-value operator acting on integrable functions on Euclidean space. The value of this operator will be written as ℳ f (x, a), where x ∈ X and a varies over a torus A in the group of isometries of X. For each of these cases there is an interval pO < p ≦ 2, where the p0 depends on the geometry of X, such that if f is in Lp (X) then there is a set full measure in X and if x lies in this set, the function a ↦ℳ f(x, a) has some Hölder continuity on compact subsets of the regular elements of A.

Keywords

Symmetric spacespherical functionSobolev spacemean-value operatormaximal torus.

MSC classification

Secondary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 1 , August 1988 , pp. 117 - 126
Copyright
Copyright © Australian Mathematical Society 1988

References

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