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On Kolmogorov's inequality for the first and second derivatives on the axis and on the period
P. Yu. Glazyrina, N. S. Payuchenko Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
Abstract:
We study the inequality $\|y'\|_{L_q(G)}\le K(r,p, G) \|y\|_{L_r(G)}^{1/2}\|y'' \|_{L_p(G)}^{1/2}$ on the real line $G=\mathbb{R}$ and on the period $\mathbb{T}$ for $q\in [1,\infty)$, $r\in (0, \infty]$, $p\in[1, \infty ]$, and $1/r+1/p=2/q$. We prove that the exact constant $K(r,p,\mathbb{R})$ is equal to the exact constant $K_1$ in the inequality $\|u'\|_{L_q[0,1]}\le K_1 \|u\|_{ L_r[0,1]}^{1/2} \|u''\|_{L_p[0,1]}^{1/2}$ over the set of convex functions $u(x)$, $x\in [0,1]$, having an absolutely continuous derivative and satisfying the condition $u'(0)=u(1)=0$. As a consequence of this statement, the equality $K(r,p,\mathbb{R})=K(r,p,\mathbb{T})$ established in 2003 by V. F. Babenko, V. A. Kofanov, and S. A. Pichugov for $r\ge 1$, is extended to $r\ge 1/2$. In addition, we give a new proof of the equality $K(r,1,\mathbb{R})=(r+1)^{1/(2(r+1))}$ for $p=1$, $r\in [1,\infty)$, and $q=2r/(r+1)$, which was established by V. V. Arestov and V. I. Berdyshev in 1975.
Keywords:
Kolmogorov's inequality, inequalities for norms of functions and their derivatives, exact constants, real axis, period.
Received: 04.04.2022 Revised: 02.05.2022 Accepted: 04.05.2022
Citation:
P. Yu. Glazyrina, N. S. Payuchenko, “On Kolmogorov's inequality for the first and second derivatives on the axis and on the period”, Trudy Inst. Mat. i Mekh. UrO RAN, 28, no. 2, 2022, 84–95
Linking options:
https://www.mathnet.ru/eng/timm1906 https://www.mathnet.ru/eng/timm/v28/i2/p84
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