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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 $\{z\zeta^{-1} :z, \zeta\in\Gamma$, $z\ne\zeta\}$, 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.

42B20Singular and oscillatory integrals, several variables
42B10Fourier type transforms, several variables
30D55H (sup p)-classes (MSC2000)
42B30$H^p$-spaces (Fourier analysis)
Full Text: EuDML