On Kolmogorov's inequality for the first and second derivatives on the axis and on the period

P. Yu. Glazyrina   ; N. S. Payuchenko

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 84-95

Résumé

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.

Détail

Keywords: Kolmogorov's inequality, inequalities for norms of functions and their derivatives, real axis, period.
Mots-clés : exact constants

Bibliographie

[1] Arestov V.V., “Priblizhenie neogranichennykh operatorov ogranichennymi i rodstvennye ekstremalnye zadachi”, Uspekhi mat. nauk, 51:6 (1996), 89–124 | DOI | MR | Zbl

[2] Arestov V.V., Berdyshev V.I., “Neravenstva dlya differentsiruemykh funktsii”, Metody resheniya uslovno-korrektnykh zadach, no. 17, In-t matematiki i mekhaniki UNTs AN SSSR, Sverdlovsk, 1975, 108–138

[3] Babenko V.F., Kofanov V.A., Pichugov S.A., “Sravnenie tochnykh konstant v neravenstvakh dlya proizvodnykh na veschestvennoi pryamoi i okruzhnosti”, Ukr. mat. zhurn., 55:5 (2003), 579–589 | MR | Zbl

[4] Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A., Neravenstva dlya proizvodnykh i ikh prilozheniya, Naukova dumka, Kiev, 2003, 590 pp.

[5] Gabushin V.N., “Neravenstva mezhdu proizvodnymi v metrikakh $L_p$ pri $0$$p\leq \infty$”, Izv. AN SSSR. Ser. matematicheskaya, 40:4 (1976), 869–892 | MR

[6] Zernyshkina E.A., “Kolmogorov type inequality in $L_2$ on the real line with one-sided norm”, East J. Approx., 12:2 (2006), 127–150 | MR | Zbl

[7] Korneichuk N.P., Splainy v teorii priblizheniya, Nauka, M., 1984, 352 pp. | MR

[8] Landau E., “Einige Ungleichungen für zweimal differenzierbare Funktion”, Proc. London Math. Soc., 13 (1913), 43–49 | DOI | MR | Zbl

[9] Payuchenko N.S., “Reduktsiya neravenstva Kolmogorova dlya polozhitelnoi srezki vtoroi proizvodnoi na osi k neravenstvu dlya vypuklykh funktsii na otrezke”, Sib. elektron. mat. izv., 18:2 (2021), 1625–1638 | DOI | MR | Zbl

[10] Stein E.M., “Functions of exponential type”, Ann. Math., 65:3 (1957), 582–592 | DOI | MR | Zbl