Behavior of singular integral with Hilbert kernel at weak continuity point of density

Abstract

We consider the singular integral with the Hilbert kernel $$ I(\gamma_0)=\int\limits^{2\pi}_{0} \varphi(\gamma)\cot\frac{\gamma-\gamma_0}{2} \,d\gamma, $$ whose density $\varphi(\gamma)$ is a continuous in $[0, 2\pi]$ function, $\gamma_0~\in~[0, 2\pi]$, $\varphi(0)=\varphi(2\pi)$, and the integral is treated in the sense of its principal value. We assume that in the vicinity of a fixed point $\gamma = c$, $c\in(c^{-},c^{+})\subset[0, 2\pi]$, $c^{+}-c^{-}<1$, the density $\varphi(\gamma)$ satisfies the representation $ \varphi(\gamma)=\frac{\Phi(\gamma)}{\left(-\ln \sin^2 \frac{\gamma-c}{2}\right)^{\beta}},\, \gamma \in (c^{-},c^{+}), $ where $\Phi(\gamma)$ is a given continuous in $[c^{-},c]$, $[c,c^{+}]$ function with not necessarily coinciding one-sided limits $\Phi(c-0)$ and $\Phi(c+0)$, $\beta$ is a given number, and $\beta>1$. We suppose that the representations $\Phi(\gamma)-\Phi(c\pm0) = \frac{\chi(\gamma)}{\left( -\ln \sin^2 \frac{\gamma-c}{2}\right)^{\delta}}, $ $ \chi'(\gamma)=\frac{\nu(\gamma)}{\left(-\ln \sin^2 \frac{\gamma-c}{2}\right)\tan\frac{\gamma-c}{2}}, $ hold, where $\delta>0$ is a given number, $\chi(\gamma)$, $\nu(\gamma)$ are given functions continuous in each of the intervals $[c^{-},c]$, $[c,c^{+}]$, $\nu(c\pm0)=0$, $\Phi(c+0)$ is taken as $\gamma > c$, $\Phi(c-0)$ is taken as $\gamma < c$. We prove that under the above conditions the representation \begin{align*} I(\gamma_0)&-I(c)= \frac{\Phi(c-0)-\Phi(c+0)}{(\beta-1)\left(-\ln\sin^2\frac{\gamma_0-c}{2}\right)^{\beta-1}} \\ &- \frac{U(c+0)-U(c-0)}{\tilde{\beta}(\tilde{\beta}-1) \left(-\ln\sin^2\frac{\gamma_0-c}{2}\right)^{\tilde{\beta}-1}}+ o\left(\frac{1}{\left(-\ln\sin^2\frac{\gamma_0-c}{2}\right)^{\tilde{\beta}-1}}\right) +O\left(\frac{1}{\left(-\ln\sin^2\frac{\gamma_0-c}{2}\right)^{\beta}}\right), \end{align*} holds as $\gamma_0\to c$. Here $\tilde{\beta}=\beta+\delta$, $\beta>1$, $\delta>0$, $U(c+0)-U(c-0)=\tilde{\beta}\big(\chi(c+0)-\chi(c-0)\big)$. We also consider the case $\beta=1$. A distinguishing feature of the paper is that while studying the behavior of the considered singular integral in the vicinity of the weak continuity point of its density, we need the Hölder condition no for the density neither for a component of the density. This feature allowed us to extend the range of possible applications of our results.