On combinations of the circle shifts and some one-dimensional integral operators

S. B. Klimentov

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 1, pp. 30-40

Résumé

The diffeomorphism $\zeta=\zeta(e^{is})$ of the unit circle and the operator $\Psi \varphi(t) = \frac{1}{\pi i} \int\nolimits_{\Gamma} \left[\frac{\zeta'(\tau)}{\zeta(\tau)-\zeta(t)} - \frac{1}{\tau-t} \right] \varphi(\tau)d \tau$ are under consideration. The main results can be stated as follows: If $\zeta(t) \in C^{1,\alpha}(\Gamma)$, $0\alpha\leqslant 1$, $\varphi(t) \in C^{0,\beta}(\Gamma)$, $0\beta \leqslant 1$, $\mu=\alpha+\beta\leqslant 2$, then $\Psi \varphi (t) \in C^{\mu}(\Gamma)$ for $\mu 1$. Moreover, the following inequality holds: \begin{equation*} \|\Psi \varphi (t)\|_{C^{\mu}(\Gamma)} \leqslant {\rm const} \|\varphi(t)\|_{C^{0,\beta}(\Gamma)}, \end{equation*} where the constant depends on $\|\zeta\|_{C^{1,\alpha}(\Gamma)}$ only. If $\mu=1$, then $ \Psi \varphi (t) \in C^{\mu -\varepsilon}(\Gamma)$ for all $0\varepsilon\mu$ and the similar inequality holds. If $\mu>1$, then $ \Psi \varphi (t) \in C^{1,\mu -1}(\Gamma)$, and \begin{equation*} \|\Psi \varphi (t)\|_{C^{1,\mu-1}(\Gamma)} \leqslant {\rm const} \|\varphi(t)\|_{C^{0,\beta}(\Gamma)}, \end{equation*} where the constant depends on $\|\zeta\|_{C^{1,\alpha}(\Gamma)}$ only. If $\zeta(t) \in C^{1,\alpha}(\Gamma)$, $0\alpha\leqslant 1$, $\varphi(t) \in C^{1,\beta}(\Gamma)$, $0\beta \leqslant 1$, then $ \Psi \varphi (t) \in C^{1,\alpha}(\Gamma)$, and \begin{equation*} \|\Psi \varphi (t)\|_{C^{1,\alpha}(\Gamma)} \leqslant \mathrm{const}\, \|\varphi(t)\|_{C^{0,1}(\Gamma)} \leqslant \mathrm{const}\, \|\varphi(t)\|_{C^{1,\beta}(\Gamma)}, \end{equation*} where the constant depends on $\|\zeta\|_{C^{1,\alpha}(\Gamma)}$ only. The index $\alpha$ in the left-hand side of the last inequality can not be improved. The appropriate example is given.