Sharp Remez type inequalities of various metrics with non-symmetric restrictions on functions

Abstract

UDC 517.5

For any $p\in (0, \infty],$ $\omega > 0,$ $\beta \in (0, 2 \omega)$, and arbitrary measurable set $B \subset I_d := [0, d],$ $\mu B \le \beta,$ we obtain the sharp inequality of Remez type
$$
\|x_{\pm}\|_\infty \le
\frac{\|(\varphi+c)_{\pm}\|_\infty}{\|\varphi+c\|_{L_p(I_{2\omega}
\setminus B^c_y)}} \left\|x \right\|_{L_{p} \left(I_d \setminus B
\right)}
$$
on the set $S_{\varphi}(\omega)$ of $d$-periodic functions $x$ having zeros with given the sine-shaped $2\omega$-periodic comparison function $\varphi$, where $c\in [-\|\varphi\|_\infty, \|\varphi\|_\infty]$ satisfies the condition
$$
\|x_{+}\|_\infty \cdot
\|x_{-}\|^{-1}_\infty = \|(\varphi+c)_{+}\|_\infty \cdot
\|(\varphi+c)_{-}\|^{-1}_\infty ,
$$
$B^c_y:=\{t\in [0, 2\omega]:|\varphi(t)+c| > y \}$ and $y$ is such that $\mu B^c_y = \beta$.

In particular, we obtain such type inequalities on the Sobolev sets of periodic functions and on the spaces of trigonometric polynomials and splines with given quotient $\|x_{+}\|_\infty / \|x_-\|_\infty$.