Invertibility of convolution operators on homogeneous groups groups

摘要

We say that a tempered distribution A belongs to the class S~m(g) on a homogeneous Lie algebra g if its Abelian Fourier transform a=? is a smooth function on the dual g* and satisfies the estimates |D ~αa(ξ)|≤C _α(1+|ξ|) ~(m-|α|). Let A ?S~0(g). Then the operator f → f * ?(χ) is bounded on L~2(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(g) as well. As a corollary we generalize Melin's theorem on the parametrix construction for Rockland operators.