Inversion in an algebra of singular integral operators of generalized homogeneity
David Weiland
42B20singular integrals
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.
Indiana University Mathematics Journal
2008
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10.1512/iumj.2008.57.3430
10.1512/iumj.2008.57.3430
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Indiana Univ. Math. J. 57 (2008) 1235 - 1260
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