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 $\textbf {R}^ n$
HTML articles powered by AMS MathViewer

by Margaret A. M. Murray PDF

Trans. Amer. Math. Soc. 289 (1985), 497-518 Request permission

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

F. Brackx, Richard Delanghe, and F. Sommen, Clifford analysis, Research Notes in Mathematics, vol. 76, Pitman (Advanced Publishing Program), Boston, MA, 1982. MR 697564
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, DOI 10.1073/pnas.74.4.1324
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
R. 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), no. 2, 361–387 (French). MR 672839, DOI 10.2307/2007065
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, DOI 10.1007/BFb0069149

Some new functions spaces and their applications to harmonic analysis

Gerald B. Folland, Introduction to partial differential equations, Mathematical Notes, Princeton University Press, Princeton, N.J., 1976. Preliminary informal notes of university courses and seminars in mathematics. MR 0599578, DOI 10.1515/9780691213033
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
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, DOI 10.1002/cpa.3160100204
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972

Layer potentials and boundary value problems for Laplace’s equation in Lipschitz domains