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 $u\in H_0^1(0,1)$, write $u(x)=(1-x){\int }_0^x{u}^{\prime }-x{\int }_x^1{u}^{\prime }$. The squared $L^2$ norm of this piecewise constant test kernel is $x(1-x)$, so the Cauchy-Schwarz inequality gives the displayed pointwise estimate. Integrating against a nonnegative weight $w$ gives $\int w|u{|}^2\le [\int wx(1-x)]\int |u^{\prime }{|}^2$. In particular $w=x^2$ gives the constant $1/20$, useful for controlling a negative quadratic potential by derivative energy. The estimate is sharp at each fixed evaluation point, whose equality profile is a tent function, although its integrated constant need not be sharp.

For a continuously differentiable function $F$ on an interval $I=[t-\delta /2,t+\delta /2]$,To prove it, average $|F(t){|}^2-|F(u){|}^2$ over $u\in I$ using the fundamental theorem of calculus. On either half of $I$, the resulting derivative kernel has absolute value at most $1/2$. Since $|(|F{|}^2)\prime |\le 2|FF\prime |$, the displayed bound follows.

On a bounded three-dimensional domain, interpolation between $L^2$ and the Sobolev inequality $H^1\hookrightarrow L^6$ gives

This spatial form of Hardy's inequality controls a singular weight by derivatives. For $d\ge 3$ and $u\in H^1({\mathbb{R}}^d)$,

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 $L^{p^{\ast }}({\mathbb{R}}^n)$.