JournalsrmiVol. 28, No. 1pp. 141–156

Invertibility of convolution operators on homogeneous groups groups

  • Paweł Głowacki

    Uniwersytet Wrocławski, Wroclaw, Poland
Invertibility of convolution operators on homogeneous groups groups cover
Download PDF

Abstract

We say that a tempered distribution AA belongs to the class Sm(g)S^m(\mathfrak{g}) on a homogeneous Lie algebra g\mathfrak{g} if its Abelian Fourier transform a=A^a=\hat{A} is a smooth function on the dual g\mathfrak{g}^{\star} and satisfies the estimates

Dαa(ξ)Cα(1+ξ)mα.|D^{\alpha}a(\xi)|\le C_{\alpha}(1+|\xi|)^{m-|\alpha|}.

Let AS0(g)A\in S^0(\mathfrak{g}). Then the operator ffA~(x)f\mapsto f\star\tilde{A}(x) is bounded on L2(g)L^2(\mathfrak{g}). Suppose that the operator is invertible and denote by BB the convolution kernel of its inverse. We show that BB belongs to the class S0(g)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