Sobolev inequality

A Sobolev inequality bounds a function in one $L^q$ norm by its weak derivatives in another $L^p$ norm. In three dimensions, the first-order case $\Vert u{\Vert }_{L^6}\le C\Vert \nabla u{\Vert }_{L^2}$ is the endpoint estimate behind the continuous embedding $H^1\hookrightarrow L^6$.

For $1\le p<n$, the Sobolev conjugate exponent $p^{\ast }$ is defined byThe first-order Sobolev inequality maps one derivative in $L^p({\mathbb{R}}^n)$ to the function in $L^{p^{\ast }}({\mathbb{R}}^n)$.