# zbMATH — the first resource for mathematics

A criterion for the boundedness of singular integrals on hypersurfaces. (English) Zbl 0675.42015
Let $${\mathcal S}$$ be a d-dimensional hypersurface in $${\mathbb{R}}^{d+1}$$. Let k(x) be a kernel which is odd, homogeneous of degree -d, and smooth away from the origin. The paper contains geometric conditions on $${\mathcal S}$$ which guarantee that the singular integral $$Tf(x)=pv\int_{{\mathcal S}}k(x- y)f(y)dy$$ is bounded on $$L^ 2({\mathcal S})$$, and hence all $$L^ p({\mathcal S})$$ spaces since $${\mathcal S}$$ is a space of homogeneous type. This result generalizes earlier work of David. The proof involves Clifford analysis.
Reviewer: D.S.Kurtz

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
##### Keywords:
singular integral; Clifford analysis
Full Text:
##### References:
 [1] F. Brackx, R. Delanghe, and F. Sommer, Clifford analysis, Pitman, 1982. · Zbl 0529.30001 [2] R. R. Coifman, G. David, and Y. Meyer, La solution des conjecture de Calderón, Adv. in Math. 48 (1983), no. 2, 144 – 148 (French). · Zbl 0518.42024 · doi:10.1016/0001-8708(83)90084-1 · doi.org [3] Guy David, Opérateurs intégraux singuliers sur certaines courbes du plan complexe, Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 1, 157 – 189 (French). · Zbl 0537.42016 [4] -, Opérateurs d’intégrale singulière sur les surfaces régulières, Ann. Sci. École Norm. Sup. (to appear). · Zbl 0655.42013 [5] 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). · Zbl 0604.42014 · doi:10.4171/RMI/17 · doi.org [6] -, Calerón-Zygmund operators, para-accretive functions, and interpolation, preprint. [7] Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71 – 79. , https://doi.org/10.1016/0001-8708(82)90054-8 David S. Jerison and Carlos E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), no. 1, 80 – 147. · Zbl 0514.31003 · doi:10.1016/0001-8708(82)90055-X · doi.org [8] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501 [9] 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. · Zbl 0232.42007 [10] Akihito Uchiyama, A constructive proof of the Fefferman-Stein decomposition of BMO (\?$$^{n}$$), Acta Math. 148 (1982), 215 – 241. · Zbl 0514.46018 · doi:10.1007/BF02392729 · doi.org [11] J. Väisälä, Quasimöbius invariance of uniform holes, preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.