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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
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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  • [BA] A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta. Math. 96 (1956), 125-142. MR 0086869 (19:258c)
  • [BDS] F. Brackx, R. Delanghe, and F. Sommer, Clifford analysis, Pitman Press, 1982.
  • [Ca] A. P. Calderón, Cauchy integrals on Lipschitz graphs and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 1324-1327. MR 0466568 (57:6445)
  • [Cr] L. Carleson, On $ {H^\infty }$ in multiply connected domains, Conference on Harmonic Analysis in honor of Antoni Zygmund, Vol. 2, Wadsworth, Belmont, Calif., 1983, pp. 349-372. MR 730079 (85g:30058)
  • [CM1] R. Coifman and Y. Meyer, Au-dela des opérateurs pseudo-différentiels, Astérisque 57 (1978). MR 518170 (81b:47061)
  • [CM2] -, Le théorème de Calderón par les méthods de variable reele, C. R. Acad. Sci. Paris Sér. A 289 (1979), 425-428.
  • [CM3] -, Une généralisation du théorème de Calderón sur l'integral de Cauchy, Fourier Analysis (Proc. Conf., El Escorial, Spain, 1979) (M. de Guzman and I. Peral, eds.), 1980. MR 582248 (81f:42001)
  • [CMM] R. Coifman, A. McIntosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur $ {L^2}$ pour les courbes Lipschitziennes, Ann. of Math. (2) 116 (1982), 361-387. MR 672839 (84m:42027)
  • [CMS] R. Coifman, Y. Meyer, and E. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304-335. MR 791851 (86i:46029)
  • [Dh] B. E. J. Dahlberg, On the absolute continuity of elliptic measures, preprint. MR 859772 (88i:35061)
  • [Dv] G. David, Opérateurs intégraux singuliers sur certaines courbes du plan complex, Ann. Sci. Ecole Norm. Sup. 17 (1984), 157-189. MR 744071 (85k:42026)
  • [FJK] E. Fabes, D. Jerison, and C. Kenig, Necessary and sufficient conditions for absolute continuity of elliptic harmonic measure, Ann. of Math. (2) 119 (1984), 121-141. MR 736563 (85h:35069)
  • [G] J. B. Garnett, Bounded analytic functions, Academic Press, 1981. MR 628971 (83g:30037)
  • [GJ] J. B. Garnett and P. W. Jones, The Corona theorem for Denjoy domains, Acta Math. 155 (1985), 29-40. MR 793236 (87e:30044)
  • [Je] J. L. Journé, Calderón-Zygmund operators, pseudo-differential operators, and the Cauchy integral of Calderón, Lecture Notes in Math., vol. 994, Springer-Verlag, Berlin and New York, 1983.
  • [Js1] P. W. Jones, $ {L^\infty }$ estimates for the $ \overline \partial $ problem in a half-plane, Acta Math. 150 (1983), 137-152. MR 697611 (84g:35135)
  • [Js2] -, Homeomorphisms of the line that preserve BMO, Ark. Mat. 21 (1983), 229-231. MR 727346 (86a:42028)
  • [Js3] -, On $ {L^\infty }$ solutions to $ \overline \partial $ in domains with thick boundary (to appear).
  • [JK] D. Jerison and C. Kenig, Hardy spaces, $ {A_\infty }$, and singular integrals on chord-arc domains, Math. Scand. 50 (1982), 221-247. MR 672926 (84k:30037)
  • [M] D. Marshall, Removable sets for bounded analytic functions, Linear and Complex Analysis Problem Book, Lecture Notes in Math., no. 1043, Springer-Verlag, Berlin and New York, 1984, pp. 485-490.
  • [P1] Chr. Pommerenke, Univalent functions, Vanhoeck and Ruprecht, Gottingen, 1975. MR 0507768 (58:22526)
  • [P2] -, Boundary behaviour of conformal mappings, Aspects of Contemporary Complex Analysis (D. Brannen and J. Clunie, eds.), Academic Press, 1980. MR 623475 (82i:30008)
  • [S] E. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
  • [Se1] S. Semmes, The Cauchy integral, chord-arc curves, and quasiconformal mappings, Proc. Bieberbach Conf. (Purdue University, 1985) (A. Baernstein, P. Duren, A. Marden, and D. Drasin, eds.), Math. Surveys, no. 21, Amer. Math. Soc, Providence, R.I., 1986. MR 875240 (88a:30088)
  • [Se2] -, Quasiconformal mappings and chord-arc curves, Trans. Amer. Math. Soc. 302 (1988), 233-263. MR 927689 (89j:30029)
  • [V1] N. Varopoulos, BMO functions and the $ \overline \partial $ equation, Pacific J. Math. 71 (1977), 221-273. MR 0508035 (58:22639a)
  • [V2] -, A remark on BMO and bounded harmonic functions, Pacific J. Math. 714 (1977), 257-259.
  • [T1] P. Tukia, The planar Shonflies theorem for Lipschitz maps, Ann. Acad. Math. Sci. Fenn. Ser. A I Math. 5 (1980), 49-72. MR 595177 (82e:57003)
  • [T2] -, Extension of quasisymmetric and Lipschitz embeddings of the real line into the plane, Math. Sci. Fenn. Ser. AI Math. 6 (1981), 89-94. MR 639966 (83d:30022)

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Keywords: $ \overline \partial $, Cauchy integral, quasiconformal mapping, BMO, Carleson measures
Article copyright: © Copyright 1988 American Mathematical Society

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