Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

Abstract

For arbitrary $[\alpha, \beta] \subset \textbf{R}$ and $p > 0$, we solve the extremal problem $$\int_{\alpha}^{\beta}|x^{(k)}(t)|^q dt \rightarrow \sup, \quad q \geq p, \quad k = 0, \quad \text{or} \quad q \geq 1, \quad k \geq 1,$$ on the set of functions $S^k_{\varphi}$ such that$\varphi ^{(i)}$ is the comparison function for $x^{(i)},\; i = 0, 1, . . . , k$, and (in the case $k = 0$) $L(x)_p \leq L(\varphi)_p$, where $$L(x)_p := \sup \left\{\left(\int^b_a|x(t)|^p dt \right)^{1/p}\; :\; a, b \in \textbf{R},\; |x(t)| > 0,\; t \in (a, b) \right\}$$ In particular, we solve this extremal problem for Sobolev classes and for bounded sets of the spaces of trigonometric polynomials and splines.