A criterion for the boundedness of singular integrals on hypersurfaces
Author:
Stephen W. Semmes
Journal:
Trans. Amer. Math. Soc. 311 (1989), 501-513
MSC:
Primary 42B20; Secondary 42B25
DOI:
https://doi.org/10.1090/S0002-9947-1989-0948198-0
MathSciNet review:
948198
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: This paper gives geometric conditions on a hypersurface in ${{\mathbf {R}}^n}$ so that certain singular integrals on that hypersurface define bounded operators on ${L^2}$. These singular integrals include the Cauchy integral operator in the sense of Clifford analysis and in particular the double layer potential. For curves in the plane, this condition is more general than the chord-arc condition but less general than the Ahlfors-David condition. The main tool is the $T(b)$ theorem [DJS].
-
F. Brackx, R. Delanghe, and F. Sommer, Clifford analysis, Pitman, 1982.
- R. R. Coifman, G. David, and Y. Meyer, La solution des conjecture de Calderón, Adv. in Math. 48 (1983), no. 2, 144–148 (French). MR 700980, DOI https://doi.org/10.1016/0001-8708%2883%2990084-1
- Guy David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 157–189 (French). MR 744071 ---, Opérateurs d’intégrale singulière sur les surfaces régulières, Ann. Sci. École Norm. Sup. (to appear).
- G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56 (French). MR 850408, DOI https://doi.org/10.4171/RMI/17 ---, Calerón-Zygmund operators, para-accretive functions, and interpolation, preprint.
- Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI https://doi.org/10.1016/0001-8708%2882%2990054-8
- 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 University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR 0304972
- Akihito Uchiyama, A constructive proof of the Fefferman-Stein decomposition of BMO $({\bf R}^{n})$, Acta Math. 148 (1982), 215–241. MR 666111, DOI https://doi.org/10.1007/BF02392729 J. Väisälä, Quasimöbius invariance of uniform holes, preprint.
Retrieve articles in Transactions of the American Mathematical Society with MSC: 42B20, 42B25
Retrieve articles in all journals with MSC: 42B20, 42B25
Additional Information
Article copyright:
© Copyright 1989
American Mathematical Society