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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 $L\sb p$-boundedness of convolution singular integral operators on curves $\gamma$ of the form $\gamma= x+ ig(x)$, $x\in\bbfR$, where $g$ is a Lipschitz function satisfying $\Vert g'\Vert\sb \infty\le N$. Let $\text{tan}\sp{-1}(N)< \mu<\varepsilon/2$, $1<p<\infty$, $\sp 0 S\sb \mu= {}\sp 0 S\sb{\mu\sp +}\cup {}\sp 0 S\sb{\mu\sp -}$, where $\sp 0 S\sb{\mu\sp \pm}$ denotes the open section consisting of the complex numbers $z$ satisfying $\vert\text{arg}(\pm z)\vert<\mu$. Suppose $\Phi$ and $\Phi\sb 1$ are holomorphic functions on $\sp 0 S\sb \mu$ and $\sp 0 S\sb{\mu\sp +}$, $z\Phi(z)$ and $\Phi\sb 1(z)$ are bounded in $z$, $\Phi\sb 1'(z)= \Phi(z)+ \Phi(-z)$ for all $z\in \sp 0 S\sb{\mu\sp +}$. Then there exists a bounded linear operator $B$ on $L\sb p(\gamma)$ defined for all $u\in L\sb p(\gamma)$ and almost all $z\in\gamma$ by $$Bu(z)=\lim\sb{\varepsilon\to 0\sp +}\left(\int\sb{\vert z- \zeta\vert\ge\varepsilon} \Phi(z-\zeta)u(\zeta)d\zeta+ \Phi\sb 1(\varepsilon{\bold t}(z))\right),$$ where ${\bold t}(z)$ is the unit tangent vector to $\gamma$ at $z$. For the entire collection see [Zbl 0762.00003].

42B20Singular and oscillatory integrals, several variables
45E10Integral equations of the convolution type
47B35Toeplitz operators, Hankel operators, Wiener-Hopf operators
47G10Integral operators