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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Estimates for $(\overline \partial -\mu \partial )^ {-1}$ 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
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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Keywords: <!– MATH $\overline \partial$ –> <IMG WIDTH="18" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img9.gif" ALT="$\overline \partial$">, Cauchy integral, quasiconformal mapping, BMO, Carleson measures
Article copyright: © Copyright 1988 American Mathematical Society