On Combinations of the Circle Shifts and Some One-Dimensional Integral Operators

Klimentov S. B.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 1.

Abstract:
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 \(\zeta(t) \in C^{1,\alpha}(\Gamma)\), \(0<\alpha\leq 1\), \(\varphi(t) \in C^{0,\beta}(\Gamma)\), \(0<\beta \leq 1\), \(\mu=\alpha+\beta\leq 2\), then \(\Psi \varphi (t) \in C^{\mu}(\Gamma)\) for \(\mu < 1\). Moreover, the following inequality holds: \( \|\Psi \varphi (t)\|_{C^{\mu}(\Gamma)} \leq {\rm const} \|\varphi(t)\|_{C^{0,\beta}(\Gamma)}, \) 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 \( \|\Psi \varphi (t)\|_{C^{1,\mu-1}(\Gamma)} \leq {\rm const} \|\varphi(t)\|_{C^{0,\beta}(\Gamma)}, \) where the constant depends on \(\|\zeta\|_{C^{1,\alpha}(\Gamma)}\) only. If \(\zeta(t) \in C^{1,\alpha}(\Gamma)\), \(0<\alpha\leq 1\), \(\varphi(t) \in C^{1,\beta}(\Gamma)\), \(0<\beta \leq 1\), then \( \Psi \varphi (t) \in C^{1,\alpha}(\Gamma)\), and \( \|\Psi \varphi (t)\|_{C^{1,\alpha}(\Gamma)} \leq \operatorname{const} \|\varphi(t)\|_{C^{0,1}(\Gamma)} \leq \operatorname{const} \|\varphi(t)\|_{C^{1,\beta}(\Gamma)}, \) 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.

Keywords: shift, singular integral operator.

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