I've read on another topic that general interpolation result from Gagliardo-Nirenberg inequality can be read as follow :

\begin{equation} \|D^ju\|^1_{L^p} \leq C \|D^mu\|^a_{L^r} \|u\|^{1-a}_{L^q} \end{equation}

with some relations between $a$, $r$, $q$ and $p$, $j$ and $m$.

Does this inequality stands in $\mathbb{R}^n$ ? More precisely, I would make sure that $C$ only depends on $f$, and not of its support.

Thanks for any help!