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 $x\in L^r_{\infty}(I_{2\pi})$.