Invertibility of convolution operators on homogeneous groups

We say that a tempered distribution $A$ belongs to the class $S^m(\text{Ge})$ on a homogeneous Lie algebra $\text{Ge}$ if its Abelian Fourier transform $a=\widehat{A}$ is a smooth function on the dual $\text{Ges}$ and satisfies the estimates $$\mathrm{\mid }{D}^{\alpha }a(\xi )\mathrm{\mid }\le C_{\alpha }(1+\mathrm{\mid }\xi \mathrm{\mid }{)}^{m-\mathrm{\mid }\alpha \mathrm{\mid }}\mathrm{.}$$ Let $A\in S^0(\text{Ge})$. Then the operator $f\mapsto f\star \tilde{A}(x)$ is bounded on $L^2(\text{Ge})$. Suppose that the operator is invertible and denote by $B$ the convolution kernel of its inverse. We show that $B$ belongs to the class $S^0(\text{Ge})$ as well. As a corollary we generalize Melin’s theorem on the parametrix construction for Rockland operators.