Invertibility of convolution operators on homogeneous groups

Details Invertibility of convolution operators on homogeneous groups Invertibility of convolution operators on homogeneous groups

2010-07-08

arxiv.org/abs/1007.1429v2

Mathematics

Functional Analysis

17 pages, see also http://www.math.uni.wroc.pl/~glowacki

Scientific paper

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

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