Expansions for spherical functions on noncompact symmetric spaces

Abstract

Section 0 Lot G be a connected noncompact semisimple Lie group with finite center and real rank one; fix a maximal compact subgroup K. Our concern in this paper is Fourier analysis on the Riemannian symmetric space G]K. We shall analyze the local and global behavior of spherical functions, the boundedness of multiplier operators, and the inversion of differential operators. The core of the paper, however, is an analysis of how the size of a function is controlled by the size of its Fourier transform. There is an extensive literature on such topics, centered about the Paley-Wiener and Plancherel theorems. Our work relies heavily on these earlier ideas and techniques, to which detailed reference will be made in the body of the paper. The problems we wish to solve, however, require estimates more precise and of a different nature than are necessary for the Plancherel or Paley-Wienor theorem. Thus the first two sections of this paper are devoted to the construction of various asymptotic expansions for spherical functions; in later sections we show how these expansions may be applied to the Fourier analysis of G/K.