Estimates for (\overline∂-𝜇∂)⁻¹ and Calderón’s theorem on the Cauchy integral

Semmes, Stephen W.

doi:10.1090/S0002-9947-1988-0927688-X

Estimates for $(\overline \partial -\mu \partial )^ {-1}$ and Calderón’s theorem on the Cauchy integral
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by Stephen W. Semmes

Trans. Amer. Math. Soc. 306 (1988), 191-232

DOI: https://doi.org/10.1090/S0002-9947-1988-0927688-X

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Abstract:

One can view the Cauchy integral operator as giving the solution to a certain $\overline \partial$ problem. If one has a quasiconformal mapping on the plane that takes the real line to the curve, then this $\bar \partial$ problem on the curve can be pulled back to a $\bar \partial - \mu \partial$ problem on the line. In the case of Lipschitz graphs (or chordarc curves) with small constant, we show how a judicial choice of q.c. mapping and suitable estimates for $\bar \partial - \mu \partial$ gives a new approach to the boundedness of the Cauchy integral. This approach has the advantage that it is better suited to related problems concerning ${H^\infty }$ than the usual singular integral methods. Also, these estimates for the Beltrami equation have application to quasiconformal and conformal mappings, taken up in a companion paper.

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