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Boundedness of singular integral operators with holomorphic kernels on star-shaped closed Lipschitz curves. (English) Zbl 0860.42013

Summary: The aim of this paper is to study singular integrals \(T\) generated by holomorphic kernels \(\Phi\) defined on a natural neighbourhood of the set \(\{z\zeta^{-1} :z, \zeta\in\Gamma\), \(z\neq\zeta\}\), where \(\Gamma\) is a star-shaped Lipschitz curve, \(\Gamma= \{\exp(iz): z=x+ iA(x)\), \(A'\in L^\infty [-\pi,\pi]\), \(A(-\pi) = A(\pi)\}\). Under suitable conditions on \(F\) and \(z\), the operators are given by \[ TF(z) = \text{p.v. } \int_\Gamma \Phi (z\eta^{-1}) F(\eta) {d\eta\over \eta}. \] We identify a class of kernels of the stated type that give rise to bounded operators on \(L^2(\Gamma, |d\Gamma |)\). We establish also transference results relating the boundedness of kernels on closed Lipschitz curves to corresponding results on periodic, unbounded curves.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
30D55 \(H^p\)-classes (MSC2000)
42B30 \(H^p\)-spaces
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