On the boundedness of the Cauchy singular operator (Q2774468)

The authors consider the Cauchy singular integral operator NEWLINE\[NEWLINE S_\Gamma f(t):=\frac 1{\pi i}\lim_{\varepsilon\to 0} \int_{\Gamma_{\tau,\varepsilon}} \frac{f(\tau) d\tau}{\tau-t},\quad \Gamma_{\tau,\varepsilon}:=\{t\in\Gamma: \|t -\tau\|\geq\varepsilon\},\quad t,\tau\in\Gamma\tag{1}NEWLINE\]NEWLINE on a contour \(\Gamma\) representing an infinite family of concentric circumferences NEWLINE\[NEWLINE \Gamma:=\left[\bigcup_{n=1}^\infty\Gamma_n\right] \cup\{0\},\quad \Gamma_n:=\{z\in\mathbb{C}: |z|=r_n\},\quad \sum_{n=1}^\infty r_n<+\infty, NEWLINE\]NEWLINE where \(r_1>r_2>\cdots\) is a strongly decaying sequence. The criteria for the continuity property \(S_\Gamma: \mathbb{L}_p(\Gamma)\to\mathbb{L}_p(\Gamma)\) of operator (1) in the Lebesgue space for \(1<p<\infty\) is found and reads: NEWLINE\[NEWLINE\int_\Gamma[\chi(t)]^\sigma dt<+\infty,\quad \chi(t):=\sup_{r>0}\frac{\ell_\Gamma(t,r)}r, \quad t\in\Gamma\tag{2}NEWLINE\]NEWLINE Here \(\ell_\Gamma(t,r)\) is the length of the part of the contour \(\Gamma\) inside the circle of the radius \(r\) centered at \(t\in\Gamma\). The condition NEWLINE\[NEWLINE\ell_\Gamma(t,r)\leq C r^{\nu+\varepsilon},\quad \nu=\frac{pq}{pq+p-q}\tag{3}NEWLINE\]NEWLINE with any \(\varepsilon>0\) and a constant \(C\) independent of \(t\) and \(r\), is sufficient for the criterion (2) to hold (with \(\varepsilon=0\) the condition (3) is only necessary for (2) to hold).