J Austral Math Soc Ser A 45 pp117--126, 1988.

Regularity of Mean-Values

Christopher Meaney

Abstract

Let X be the d-dimensional sphere or a compact, simply 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 Î X and a varies over a torus A in the group of isometries of X. For each of these cases there is an interval p₀ < p £ 2, where the p₀ depends on the geometry of X, such that if f is in L^p(X) then there is a set of 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.

1980 AMS Subject Classification: 42C10, 43A15

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Authors

Christopher Meaney

Department of Mathematics, Australian National University, Canberra, ACT 2601, Australia.