Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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



The Cauchy integral, Calderón commutators, and conjugations of singular integrals in $ {\bf R}\sp n$

Author: Margaret A. M. Murray
Journal: Trans. Amer. Math. Soc. 289 (1985), 497-518
MSC: Primary 42B20; Secondary 47B38, 47G05
MathSciNet review: 784001
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the Cauchy integral and Hilbert transform for Lipschitz domains in the Clifford algebra based on $ {R^n}$. The Hilbert transform is shown to be the generating function for the Calderón commutators in $ {R^n}$. We make use of an intrinsic characterization of these commutators to obtain $ {L^2}$ estimates. These estimates are used to show the analyticity of the Hilbert transform and of the conjugation of singular integral operators by bi-Lipschitz changes of variable in $ {R^n}$.

References [Enhancements On Off] (What's this?)

  • [1] F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
  • [2] A.-P. Calderón, Cauchy integrals on Lipschitz curves and related operators, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 4, 1324–1327. MR 0466568
  • [3] A.-P. Calderón, Commutators, singular integrals on Lipschitz curves and applications, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 85–96. MR 562599
  • [4] R. R. Coifman, A. McIntosh, and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur 𝐿² pour les courbes lipschitziennes, Ann. of Math. (2) 116 (1982), no. 2, 361–387 (French). MR 672839,
  • [5] R. R. Coifman, Y. Meyer, and E. M. Stein, Un nouvel éspace fonctionnel adapté à l’étude des opérateurs définis par des intégrales singulières, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 1–15 (French). MR 729344,
  • [6] -, Some new functions spaces and their applications to harmonic analysis, J. Funct. Anal. (to appear).
  • [7] Gerald B. Folland, Introduction to partial differential equations, Princeton University Press, Princeton, N.J., 1976. Preliminary informal notes of university courses and seminars in mathematics; Mathematical Notes. MR 0599578
  • [8] Robert P. Gilbert and James L. Buchanan, First order elliptic systems, Mathematics in Science and Engineering, vol. 163, Academic Press, Inc., Orlando, FL, 1983. A function theoretic approach. MR 715259
  • [9] C. B. Morrey Jr. and L. Nirenberg, On the analyticity of the solutions of linear elliptic systems of partial differential equations, Comm. Pure Appl. Math. 10 (1957), 271–290. MR 0089334,
  • [10] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [11] Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
  • [12] G. C. Verchota, Layer potentials and boundary value problems for Laplace's equation in Lipschitz domains, Ph. D. Thesis, University of Minnesota, 1982.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B20, 47B38, 47G05

Retrieve articles in all journals with MSC: 42B20, 47B38, 47G05

Additional Information

Keywords: Commutators with singular integrals in $ {R^n}$, Cauchy integrals in $ {R^n}$, conjugation of singular integrals by changes of variable
Article copyright: © Copyright 1985 American Mathematical Society