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Convolution singular integral operators on Lipschitz curves. (English) Zbl 0791.42012

Cheng, M.-T. (ed.) et al., Harmonic analysis. Proceedings of the special program at the Nankai Institute of Mathematics, Tianjin, PR China, March- July, 1988. Berlin etc.: Springer-Verlag. Lect. Notes Math. 1494, 142-162 (1991).
The article intends to present prerequisites from harmonic analysis and operator theory needed to establish the \(L_ p\)-boundedness of convolution singular integral operators on curves \(\gamma\) of the form \(\gamma= x+ ig(x)\), \(x\in\mathbb{R}\), where \(g\) is a Lipschitz function satisfying \(\| g'\|_ \infty\leq N\). Let \(\text{tan}^{-1}(N)< \mu<\varepsilon/2\), \(1<p<\infty\), \(^ 0 S_ \mu= {}^ 0 S_{\mu^ +}\cup {}^ 0 S_{\mu^ -}\), where \(^ 0 S_{\mu^ \pm}\) denotes the open section consisting of the complex numbers \(z\) satisfying \(|\text{arg}(\pm z)|<\mu\). Suppose \(\Phi\) and \(\Phi_ 1\) are holomorphic functions on \(^ 0 S_ \mu\) and \(^ 0 S_{\mu^ +}\), \(z\Phi(z)\) and \(\Phi_ 1(z)\) are bounded in \(z\), \(\Phi_ 1'(z)= \Phi(z)+ \Phi(-z)\) for all \(z\in ^ 0 S_{\mu^ +}\). Then there exists a bounded linear operator \(B\) on \(L_ p(\gamma)\) defined for all \(u\in L_ p(\gamma)\) and almost all \(z\in\gamma\) by \[ Bu(z)=\lim_{\varepsilon\to 0^ +}\left(\int_{| z- \zeta|\geq\varepsilon} \Phi(z-\zeta)u(\zeta)d\zeta+ \Phi_ 1(\varepsilon{\mathbf t}(z))\right), \] where \({\mathbf t}(z)\) is the unit tangent vector to \(\gamma\) at \(z\).
For the entire collection see [Zbl 0762.00003].

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47G10 Integral operators
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