# Invertibility of convolution operators on homogeneous groups groups

### Paweł Głowacki

Uniwersytet Wrocławski, Wroclaw, Poland

## 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.

## Cite this article

Paweł Głowacki, Invertibility of convolution operators on homogeneous groups groups. Rev. Mat. Iberoam. 28 (2012), no. 1, pp. 141–156

DOI 10.4171/RMI/671