Geodesic
Inequalities between derivatives in $L_p$-metrics for $0$
Izvestiya. Mathematics , Tome 10 (1976) no. 4, pp. 823-844

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We consider inequalities of the form \begin{equation} \|f^{(k)}\|_{L_q}\leqslant K\|f\|^\alpha_{L_p}\|\Phi\|^\beta_{L_r}, \tag{1} \end{equation} 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. 
@article{IM2_1976_10_4_a9,
     author = {V. N. Gabushin},
     title = {Inequalities between derivatives in $L_p$-metrics for $0<p\leqslant\infty$},
     journal = {Izvestiya. Mathematics },
     pages = {823--844},
     publisher = {mathdoc},
     volume = {10},
     number = {4},
     year = {1976},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1976_10_4_a9/}
}
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V. N. Gabushin. Inequalities between derivatives in $L_p$-metrics for $0