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

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.
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