# 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 $S_{m}(g)$ on a homogeneous Lie algebra $g$ if its Abelian Fourier transform $a=A^$ is a smooth function on the dual $g_{⋆}$ and satisfies the estimates

$∣D_{α}a(ξ)∣≤C_{α}(1+∣ξ∣)_{m−∣α∣}.$

Let $A∈S_{0}(g)$. Then the operator $f↦f⋆A~(x)$ is bounded on $L_{2}(g)$. Suppose that the operator is invertible and denote by $B$ the convolution kernel of its inverse. We show that $B$ belongs to the class $S_{0}(g)$ 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