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Fourier analysis on starlike Lipschitz surfaces. (English) Zbl 0991.42009

The author considers singular integrals of a type generalizing the Cauchy integral on Lipschitz surfaces. Specifically, if \(\Sigma\) is a starlike Lipschitz surface in \(R \times R^n\), the author considers singular integrals with kernel \(K(x-y)\), where \(K\) is a Clifford-valued function defined on a “sector” which is cylindrically equivariant with respect to rotations of the \(R^n\) variable (the same way that the Cauchy kernel \(K(x-y) = C (x-y) / |x-y|^{n+1}\) is). Such kernels \(K\) can (formally at least) be given by multipliers \(b(k)\), where \(k\) indexes the monomial functions \(P^{(-k)} = c_n \Delta^{(n-1)/2}( x^{-k})\) and \(k\) ranges over the integers. The main result of the author’s paper is that if \(b\) can be extended to \(k\) in a certain complex sector, then \(K\) obeys good Calderón-Zygmund type estimates on a certain heart-shaped region, and the corresponding operator is then bounded on \(L^2\) of starlike Lipschitz surfaces. As a consequence the author creates a bounded holomorphic functional calculus for operators of Dirac type on these surfaces.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B15 Multipliers for harmonic analysis in several variables
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