Convolution on homogeneous groups

Let G be a homogeneous group with homogeneous dimension Q, and let So denote the space of Schwartz functions on G with all moments vanishing. Let be the usual Euclidean Fourier transform. For j∈R, we let be the space of J, smooth away from 0, satisfying |α^∂J(ξ)|?Cβ|ξ|^j−|β|, where both |ξ| and |β| are taken in the homogeneous sense. We characterize , and show that as elements of . If j₁,j₂,j₁+j₂>−Q, one can replace So, by S, S^′ in this result. A key ingredient of our proof is a lemma from the fundamental wavelet paper from 1985 by Frazier and Jawerth [4]. We believe that, in turn, our result will be useful in the theory of wavelets on homogeneous groups.