Invertibility of convolution operators on homogeneous groups

2010/07/08 by Pawel Glowacki, Glowacki, Pawel
Mathematics · #22E30 (Primary) #47G30 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:22E30 #msc:47G30

paper · pdf · doi:10.48550/arxiv.1007.1429

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

arxiv created 2010/09/16 · arxiv updated 2010/09/17

Abstract

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=A is a smooth function on the dual \Ges and satisfies the estimates |D^αa(ξ)|≤ C_α(1+|ξ|)^m-|α|. Let A∈ S⁰(\Ge). Then the operator f↦ f⋆\widetildeA(x) is bounded on L²(\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 S⁰(\Ge) as well. As a corollary we generalize Melin's theorem on the parametrix construction for Rockland operators.

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