Lipschitz Spaces on the Surface of Unit Sphere in Euclidean n-Space

Harvey Greenwald, California Polytechnic State University - San Luis Obispo

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Published by Pacific Journal of Mathematics, Volume 50, Issue 1, pp. 63-80. Publisher website: http://pjm.math.berkeley.edu/pjm/about/journal/cover.html

Abstract

This paper is concerned with defining Lipschitz spaces on Σn-1 the surface of the unit sphere in Rn. The importance of this example is that Σn-1 is not a group but a symmetric space. One begins with functions in Lp(Σn-1),1≤p≤∞. Σn-1 is a symmetric space and is related in a natural way to the rotation group SO(n). One can then use the group SO(n) to define first and second differences for functions in Lp(Σn-1). Such a function is the boundary value of its Poisson integral. This enables one to work with functions which are harmonic. Differences can then be replaced by derivatives.