The Cauchy integral, Calderón commutators, and conjugations of singular integrals in 𝑅ⁿ

Murray, Margaret A. M.

doi:10.1090/S0002-9947-1985-0784001-6

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
DOI: https://doi.org/10.1090/S0002-9947-1985-0784001-6
MathSciNet review: 784001
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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 ${R^n}$ .

[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 466568, https://doi.org/10.1073/pnas.74.4.1324
[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, https://doi.org/10.2307/2007065
[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, https://doi.org/10.1007/BFb0069149
[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 89334, https://doi.org/10.1002/cpa.3160100204
[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.