Invertibility of convolution operators on homogeneous groups groups

Abstract

We say that a tempered distribution $A$ belongs to the class $S^m(\mathfrak{g})$ on a homogeneous Lie algebra $\mathfrak{g}$ if its Abelian Fourier transform $a=\hat{A}$ is a smooth function on the dual ${\mathfrak{g}}^{\star }$ and satisfies the estimates

Let $A\in S^0(\mathfrak{g})$. Then the operator $f\mapsto f\star \tilde{A}(x)$ is bounded on $L^2(\mathfrak{g})$. Suppose that the operator is invertible and denote by $B$ the convolution kernel of its inverse. We show that $B$ belongs to the class $S^0(\mathfrak{g})$ as well. As a corollary we generalize Melin’s theorem on the parametrix construction for Rockland operators.