A sharp Remez type inequalities for the functions with asymmetric restrictions on the oldest derivative

Abstract

For odd $$$r\in \mathbb{N}$$$; $$$\alpha, \beta >0$$$; $$$p\in [1, \infty]$$$; $$$\delta \in (0, 2 \pi)$$$, any $$$2\pi$$$-periodic function $$$x\in L^r_{\infty}(I_{2\pi})$$$, $$$I_{2\pi}:=[0, 2\pi]$$$, and arbitrary measurable set $$$B \subset I_{2\pi},$$$ $$$\mu B \leqslant \delta/\lambda,$$$ where $$$\lambda=$$$ $$$\left({\left\|\varphi_{r}^{\alpha, \beta}\right\|_{\infty} \left\| {\alpha^{-1}}{x_+^{(r)}} + {\beta^{-1}}{x_-^{(r)}}\right\|_\infty}{E^{-1}_0(x)_\infty}\right)^{1/r}$$$, we obtain sharp Remez type inequality $$E_0(x)_\infty \leqslant \frac{\|\varphi_r^{\alpha, \beta}\|_\infty}{E_0(\varphi_r^{\alpha, \beta})^{\gamma}_{L_p(I_{2\pi} \setminus B_\delta)}} \left\|x \right\|^{\gamma}_{{L_p} \left(I_{2\pi} \setminus B \right)}\left\| {\alpha^{-1}}{x_+^{(r)}} + {\beta^{-1}}{x_-^{(r)}}\right\|_\infty^{1-\gamma},$$ where $$$\gamma=\frac{r}{r+1/p},$$$ $$$\varphi_r^{\alpha, \beta}$$$ is non-symmetric ideal Euler spline of order $$$r$$$, $$$B_\delta:= \left[M- \delta_2, M+ \delta_1 \right]$$$, $$$M$$$ is the point of local maximum of spline $$$\varphi_r^{\alpha, \beta}$$$ and $$$\delta_1 > 0$$$, $$$\delta_2 > 0$$$ are such that $$$\varphi_r^{\alpha, \beta}(M+ \delta_1) = \varphi_r^{\alpha, \beta}(M- \delta_2), \;\; \delta_1 + \delta_2 = \delta .$$$
In particular, we prove the sharp inequality of Hörmander-Remez type for the norms of intermediate derivatives of the functions $$$x\in L^r_{\infty}(I_{2\pi})$$$.