On Hardy's Theorem on SU(1, 1)*

Abstract

The classical Hardy theorem asserts that f and its Fourier transform $\ifmmode\expandafter\hat\else\expandafter\^\fi{f}$ can not both be very rapidly decreasing. This theorem was generalized on Lie groups and also for the Fourier-Jacobi transform. However, on SU(1, 1) there are infinitely many “good” functions in the sense that f and its spherical Fourier transform $ \ifmmode\expandafter\tilde\else\expandafter\~\fi{f}$ both have good decay. In this paper, we shall characterize such functions on SU(1, 1).

Keywords

Heat kernel / Jacobi transform / Plancherel formula / 22E30 / 43A80 / 43A90 / 33C45

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