On support theorems

Abstract

We denote by ^ _ i the 7z—1 dimensional space consisting of elements ξ'. Let 2), S? and 0 M be the spaces of all C°°-functions with compact supports, all rapidly decreasing C°°-functions and all slowly increasing C°°-functions on Rn respectively. These spaces are provided with usual topologies of L. Schwartz \JΓ\. Let 2X and έf be the strong duals of 2) and Sf respectively and let O'c be the space of all rapidly decreasing distributions. We shall denote by Quiβn-x) the space 0M considered on Ξn-i. By the partial Fourier transform of T e Sf' we understand the Fourier transform of T with respect to the first n—1 variables which will be denoted by t(ξ\ t). For any A(ξ') e O M ( ^ _ I ) , we define the operator A(DX,) on y" as follows: The partial Fourier transform of A(DX,) T, Γ e / , is A(ξ') f(ξ\ t). In this paper we are concerned with the operator of the following form :