Reduction of the Kolmogorov inequality for a non negative part of the second derivative on the real line to the inequality for convex functions on an interval

N. S. Payuchenko

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 18 (2021) no. 2, pp. 1625-1638

Résumé

In this paper we delve into connection between sharp constants in the inequalities $$\|y'\|_{L_q(\mathbb{R})}\le K_+ \sqrt{\|y\|_{L_r(\mathbb{R})}\|y''_+\|_{L_p(\mathbb{R})} },$$ $$\|u'\|_{L_q(0,1)}\le \overline{K} \sqrt{\|u\|_{L_r(0,1)} \|u''\|_{L_p(0,1)}},$$ where the second one is considered for convex functions $u(x)$, $x\in[0,1]$ with an absolutely continuous derivative that vanishes at the point $x=0$. We prove that $K_+=\overline{K}$ under conditions $1 \le q,r,p\infty$ and $1/r+1/p=2/q$.

Détail

Keywords: Kolmogorov inequality, inequalities between norms of function and its derivatives, non-negative part of the second derivative
Mots-clés : exact constant.