Harmonic functions, Riesz potentials, and the Lipschitz spaces of Herz

Abstract

In the paper [5], T. M. Flett introduced a space of temperatures (solutions of the heat equation) on a half space which is isomorphic to the Lipschitz space -4(α; p9 q) of M. H. Taibleson ([16]). As a consequence, many results concerning Λ(α; p, q) were proved; in particular, he showed that the topological dual of Λ(VL p, 1), l:gp<oo, is isomorphic to -4( — α; p', oo). Some time later, R. Johnson ([10]), adopting Flett's idea, defined a space of temperatures which is isomorphic to the Lipschitz space Λ^q of C. S. Herz ([8]). His method leaned on a theory of Riesz potentials for temperatures. As an application of the theory developed, among other things, a characterization of temperatures whose are in Λ*tq (α<0) was given. In this paper, heavily influenced by [5] and [10], using an integral representation of the Riesz potential R*f(feL) in [14], we extend the definition of Riesz potential to a class of harmonic functions in a half space. Our first aim is to construct a space of harmonic functions in a half space which is isomorphic to AΛp%q. For this purpose, we show that boundary values of harmonic functions p-« ( § 5) satisfying