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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Estimates for $(\overline \partial -\mu \partial )^ {-1}$ and Calderón’s theorem on the Cauchy integral
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by Stephen W. Semmes PDF
Trans. Amer. Math. Soc. 306 (1988), 191-232 Request permission

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
  • © Copyright 1988 American Mathematical Society
  • 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