Harvey Greenwald

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

Harvey Greenwald — Wed, 02 Jan 2008 08:53:31 PST

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.

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

Harvey Greenwald — Wed, 02 Jan 2008 08:53:28 PST

In this paper Lipschitz spaces of distributions are defined and various inclusion relations are shown. Certain properties such as completeness, separability, and the density of the testing space for appropriate Lipschitz spaces are proved. The Littlewood-Paley function is defined and used to prove inclusion relationships between Lipschitz and Lebesgue spaces.

On the Theory of Homogeneous Lipschitz Spaces and Campanato Spaces

Harvey Greenwald — Wed, 02 Jan 2008 08:53:25 PST

In this paper the equivalence between the Campanato spaces and homogeneous Lipschitz spaces is shown through the use of elementary and constructive means. These Lipschitz spaces can be defined in terms of derivatives as well as differences.