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 Izv. Akad. Nauk SSSR Ser. Mat., 1976, Volume 40, Issue 4, Pages 869–892 (Mi izv2208)

Inequalities between derivatives in $L_p$-metrics for $0<p\leqslant\infty$

V. N. Gabushin

Abstract: We consider inequalities of the form
$$\|f^{(k)}\|_{L_q}\leqslant K\|f\|^\alpha_{L_p}\|\Phi\|^\beta_{L_r}, \tag{1}$$
where $\Phi(x)$ is an arbitrary majorant of the function $f^{(l)}(x)$, $x\in(-\infty,\infty)$, $k\leqslant l$. 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.

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English version:
Mathematics of the USSR-Izvestiya, 1976, 10:4, 823–844

Bibliographic databases:

UDC: 517.5
MSC: Primary 26A86, 26A84, 46E30; Secondary 46E35

Citation: V. N. Gabushin, “Inequalities between derivatives in $L_p$-metrics for $0<p\leqslant\infty$”, Izv. Akad. Nauk SSSR Ser. Mat., 40:4 (1976), 869–892; Math. USSR-Izv., 10:4 (1976), 823–844

Citation in format AMSBIB
\Bibitem{Gab76} \by V.~N.~Gabushin \paper Inequalities between derivatives in $L_p$-metrics for $0<p\leqslant\infty$ \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1976 \vol 40 \issue 4 \pages 869--892 \mathnet{http://mi.mathnet.ru/izv2208} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=430772} \zmath{https://zbmath.org/?q=an:0332.46017} \transl \jour Math. USSR-Izv. \yr 1976 \vol 10 \issue 4 \pages 823--844 \crossref{https://doi.org/10.1070/IM1976v010n04ABEH001817} 

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This publication is cited in the following articles:
1. V. V. Arestov, “Approximation of unbounded operators by bounded operators and related extremal problems”, Russian Math. Surveys, 51:6 (1996), 1093–1126
2. A. I. Zvyagintsev, “Strict inequalities for the derivatives of functions satisfying certain boundary conditions”, Math. Notes, 62:5 (1997), 596–606
3. 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
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