Gagliardo-Nirenberg inequality 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!
 A: The mother of all Gagliardo-Nirenberg inequalities is
$$
\Vert u\Vert_{L^{\frac{n}{n-1}}(\mathbb R^n)}\le c_n \Vert \nabla u\Vert_{L^{1}(\mathbb R^n)},
\tag {GN}
$$
where $c_n$ depends only on $n$ and $u$ runs say in $C^1_c(\mathbb R^n)$. Applying this to $u=v^2$, you get with $p=\frac{2n}{n-1}$
$$\Vert v\Vert_{L^p}^2=
\left(\int\vert v\vert^{\frac{2n}{n-1}} dx\right)^{\frac{n-1}{n}}\le c_n
\int2 \vert v\vert\vert\nabla v\vert dx\le 2c_n\Vert v\Vert_{L^p}\Vert \nabla v\Vert_{L^{p'}},
$$
which is
$
\Vert v\Vert_{L^{\frac{2n}{n-1}}}\le 2c_n\Vert \nabla v\Vert_{L^{\frac{2n}{n+1}}}.
$
Manipulations of the same type (apply (GN) to $u=v^\alpha$)
induce the Sobolev inclusions
$$
\dot W^{s,p}\subset \dot W^{t,q},\quad s>t,\ p< q,\quad (s-t)/n=1/p-1/q
$$
with $p,q\in(1,+\infty)$. Derivatives are somehow a convertible currency that you can exchange against a fixed amount of $L^p$ regularity according to the exchange rate displayed above, but $L^p$ regularity is a non-convertible currency which cannot buy derivatives.
