The Cauchy integral, Calderón commutators, and conjugations of singular integrals in 𝑅ⁿ

Murray, Margaret A. M.

doi:10.1090/S0002-9947-1985-0784001-6

The Cauchy integral, Calderón commutators, and conjugations of singular integrals in $\textbf {R}^ n$
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by Margaret A. M. Murray

Trans. Amer. Math. Soc. 289 (1985), 497-518

DOI: https://doi.org/10.1090/S0002-9947-1985-0784001-6

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Abstract:

We consider the Cauchy integral and Hilbert transform for Lipschitz domains in the Clifford algebra based on ${R^n}$. The Hilbert transform is shown to be the generating function for the Calderón commutators in ${R^n}$. We make use of an intrinsic characterization of these commutators to obtain ${L^2}$ estimates. These estimates are used to show the analyticity of the Hilbert transform and of the conjugation of singular integral operators by bi-Lipschitz changes of variable in ${R^n}$.

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