### Abstract

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 f(x, a), where x ϵ 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 po 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 ↦ f(x, a) has some Hölder continuity on compact subsets of the regular elements of A. 1980 Mathematics subject classification (Amer. Math. Soc.): 42 C 10, 43 A 15. Keywords and phrases: symmetric space, spherical function, Sobolev space, mean-value operator. operator. maximal torus.

Original language | English |
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Pages (from-to) | 117-126 |

Number of pages | 10 |

Journal | Journal of the Australian Mathematical Society |

Volume | 45 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1988 |

Externally published | Yes |

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## Cite this

*Journal of the Australian Mathematical Society*,

*45*(1), 117-126. https://doi.org/10.1017/S1446788700032328