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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

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Full Text: DOI
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