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DOI: 10.23671/VNC.2017.1.5819

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

Klimentov S. B.
Vladikakazian 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.