P.Yu. Glazyrina, N.S. Payuchenko. On Kolmogorov’s inequality for the first and second derivatives on the axis and on the period ... P. 84-95

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 April 4, 2022

Revised May 2, 2022

Accepted May 4, 2022

Funding Agency: The reported study was funded by RFBR, project number 20-31-90124.

Polina Yurevna Glazyrina, Cand. Sci. (Phys.-Math.), Docent, Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: polina.glazyrina@urfu.ru

Nikita Slavich Payuchenko, Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: aueiyo@gmail.com

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Cite this article as: P.Yu. Glazyrina, N.S. Payuchenko. On Kolmogorov’s inequality for the first and second derivatives on the axis and on the period. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 2, pp. 84–95.