Revista Matemática Iberoamericana

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Volume 28, Issue 1, 2012, pp. 141–156
DOI: 10.4171/RMI/671

Published online: 2012-01-20

Invertibility of convolution operators on homogeneous groups groups

Paweł Głowacki[1]

(1) Uniwersytet Wrocławski, Wroclaw, Poland

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 $$ |D^{\alpha}a(\xi)|\le C_{\alpha}(1+|\xi|)^{m-|\alpha|}. $$ 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.

Keywords: Fourier transform, multipliers, symbol classes, convolution operators, homogeneous groups, maximal estimates, parametrices

Głowacki Paweł: Invertibility of convolution operators on homogeneous groups groups. Rev. Mat. Iberoamericana 28 (2012), 141-156. doi: 10.4171/RMI/671