On Paley-Wiener and Hardy theorems for N A groups

摘要

Let N be a H-type group and let S=NA be an one dimensional solvable extension of N. For the Helgason Fourier transform on S we prove the following analogue of Hardy's theorem. Let (f) over cap (lambda, Y, Z) stand for the Helgason Fourier transform of f and let h(alpha) denote the heat kernel associated to the Laplace-Beltrami operator. Suppose a function f on S satisfies the conditions f(x) less than or equal to ch(alpha)(x) and [GRAPHICS] for all xis an element ofS, lambdais an element of R where gamma > k-1/2, k being the dimension of the centre of N. Then f=0 or f=ch(alpha) depending on whether alpha