Convolution on homogeneous groups

Let G be a homogeneous group with homogeneous dimension Q, and let S o denote the space of Schwartz functions on G with all moments vanishing. Let   ˆ be the usual Euclidean Fourier transform. For j ∈ R , we let D j ⊆ S ˆ o ′ be the space of J, smooth away from 0, satisfying | ∂ α J ( ξ ) | ⩽ C β | ξ | j − | β | , where both | ξ | and | β | are taken in the homogeneous sense. We characterize D ˇ j , and show that D ˇ j 1 * D ˇ j 2 ⊆ D ˇ j 1 + j 2 as elements of S o ′ . If j 1 , j 2 , j 1 + j 2 > − Q , one can replace S o , S o ′ 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.