## 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