ℰ′ _{♮ ♮} (X) Spectral Analysis on Domains of Noncompact Symmetric Spaces of Arbitrary Rank

Abstract

In this chapter a new approach to problems of spectral analysis on symmetric spaces X=G/K is developed. The main advantage of this approach is that it enables us to prove a symmetric space analog of the local version of the Brown–Schreiber–Taylor theorem and investigate the problem of existence of a nontrivial solution for systems of convolution equations in more detail. For the case where rankX=1, some new results concerning the exponential representation problem are also presented. The remainder of the chapter consists of applications to the Zalcman two-radii problem on X, which deals with the characterization of a function via integral ball means. When the group G is complex, a definitive local version of the Zalcman two-radii theorem is proved.