Inequalities for derivatives of functions in the spaces <i>L_p</i>

Abstract

The following sharp inequality for local norms of functions $x \in L^{r}_{\infty,\infty}(\textbf{R})$ is proved: $$\frac1{b-a}\int\limits_a^b|x'(t)|^qdt \leq \frac1{\pi}\int\limits_0^{\pi}|\varphi_{r-1}(t)|^q dt \left(\frac{||x||_{L_{\infty}(\textbf{R})}}{||\varphi_r||_{\infty}}\right)^{\frac{r-1}rq}||x^{(r)}||^q_{\infty}r,\quad r \in \textbf{N},$$ where $\varphi_r$ is the perfect Euler spline, takes place on intervals $[a, b]$ of monotonicity of the function $x$ for $q \geq 1$ or for any $q > 0$ in the cases of $r = 2$ and $r = 3.$ As a corollary, well-known A. A. Ligun's inequality for functions $x \in L^{r}_{\infty}$ of the form $$||x^{(k)}||_q \leq \frac{||\varphi_{r-k}||_q}{||\varphi_r||_{\infty}^{1-k/r}} ||x||^{1-k/r}_{\infty}||x^{(r)}||^{k/r}_{\infty},\quad k,r \in \textbf{N},\quad k < r, \quad 1 \leq q < \infty,$$ is proved for $q \in [0,1)$ in the cases of $r = 2$ and $r = 3.$