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Estimates for $ (\overline\partial-\mu\partial)\sp {-1}$ and Calderón's theorem on the Cauchy integral


Author: Stephen W. Semmes
Journal: Trans. Amer. Math. Soc. 306 (1988), 191-232
MSC: Primary 30E20; Secondary 30C60, 42B20
DOI: https://doi.org/10.1090/S0002-9947-1988-0927688-X
MathSciNet review: 927688
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0927688-X
Keywords: $ \overline \partial $, Cauchy integral, quasiconformal mapping, BMO, Carleson measures
Article copyright: © Copyright 1988 American Mathematical Society

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