Abstract:
We consider inequalities of the form
‖f(k)‖Lq⩽K‖f‖αLp‖Φ‖βLr,
where Φ(x) is an arbitrary majorant of the function f(l)(x), x∈(−∞,∞), k⩽l. The set of parameters p,q,r,k,l for which the inequalities (1) hold is described. Various generalizations of these inequalities are given.
Bibliography: 22 titles.
\Bibitem{Gab76}
\by V.~N.~Gabushin
\paper Inequalities between derivatives in $L_p$-metrics for $0<p\leqslant\infty$
\jour Math. USSR-Izv.
\yr 1976
\vol 10
\issue 4
\pages 823--844
\mathnet{http://mi.mathnet.ru/eng/im2208}
\crossref{https://doi.org/10.1070/IM1976v010n04ABEH001817}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=430772}
\zmath{https://zbmath.org/?q=an:0332.46017}
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https://doi.org/10.1070/IM1976v010n04ABEH001817
https://www.mathnet.ru/eng/im/v40/i4/p869
This publication is cited in the following 6 articles:
Vitalii V. Arestov, “Approximation of differentiation operators by bounded linear operators in lebesgue spaces on the axis and related problems in the spaces of $(p,q)$-multipliers and their predual spaces”, Ural Math. J., 9:2 (2023), 4–27
P. Yu. Glazyrina, N. S. Payuchenko, “O neravenstve Kolmogorova dlya pervoi i vtoroi proizvodnykh na osi i periode”, Tr. IMM UrO RAN, 28, no. 2, 2022, 84–95
N. S. Payuchenko, “Reduktsiya neravenstva Kolmogorova dlya polozhitelnoi srezki vtoroi proizvodnoi na osi k neravenstvu dlya vypuklykh funktsii na otrezke”, Sib. elektron. matem. izv., 18:2 (2021), 1625–1638
A. V. Gasnikov, “Time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity”, Izv. Math., 73:6 (2009), 1111–1148
A. I. Zvyagintsev, “Strict inequalities for the derivatives of functions satisfying certain boundary conditions”, Math. Notes, 62:5 (1997), 596–606
V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems”, Russian Math. Surveys, 51:6 (1996), 1093–1126
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