Sobolev Inequality

We write $C_{comp}^1({\mathbb{R}}^n)$ for the space of $C^1$ compactly supported functions on ${\mathbb{R}}^n$. To understand the size of $u$, the following definition is useful. $S_u(h):=\{x\in {\mathbb{R}}^n\text{ so that }|u(x)|>h\}.$

Sobolev Inequality

If $u\in C_{\text{comp}}^1({\mathbb{R}}^n)$, then
$\Vert u{\Vert }_{L^{\frac{n}{n-1}}}\lesssim \Vert \nabla u{\Vert }_{L^1}.$