We say that a tempered distribution belongs to the class on a homogeneous Lie algebra if its Abelian Fourier transform is a smooth function on the dual and satisfies the estimates
Let . Then the operator is bounded on . Suppose that the operator is invertible and denote by the convolution kernel of its inverse. We show that belongs to the class 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