IUMJ

Title: Inversion in an algebra of singular integral operators of generalized homogeneity

Authors: David Weiland

Issue: Volume 57 (2008), Issue 3, 1235-1260

Abstract:

A result of M. Christ on inversion of translation-invariant homogeneous singular integral operators on a Lie group having kernels with Sobolev-type smoothness is shown to hold without the homogeneity assumption. Instead we impose uniform size and smoothness conditions on the dilates of the kernel over a fixed annulus. These are analogous to the H\"{o}rmander condition for multipliers. Using natural embeddings between Sobolev and Lipschitz spaces we show that this generalization gives an inversion result for singular integral operators satisfying the standard Calder\'on-Zygmund Lipschitz estimates. Applications to the algebra of operators bounded on a Banach space of molecules are also given.