I am studying inequalities in Sobolev spaces. In particular, this one:
For all $m\in\mathbb{Z}^+\cup{0}$, there exists $c>0$ such that, for all $u,v\in L^{\infty}\left(\mathbb{R}^n\right)\cap H^m\left(\mathbb{R}^n\right)$, $$ \Vert uv \Vert_m\le c\left\{\left|u\right|_{L^{\infty}}\Vert D^mv\Vert_0+\Vert D^mu\Vert_0\left|v\right|_{L^{\infty}}\right\} $$
I know what means $D^{\alpha}$ if $\alpha=\left(\alpha_1,\dots,\alpha_n\right)$, but what does $D^mv$ mean if $m\in\mathbb{N}$?
Thanks a lot!
$D^mv(x)$ consists of all partial derivatives of order $n$. It is a linear map (often called higher-order tensor) from $(\mathbb R^n)^m$ into $\mathbb R$.