Sobolev conjugate

The Sobolev conjugate of p for $1\le p<n$, where n is space dimensionality, is

$p^{*}=\frac{pn}{n-p}>p$

This is an important parameter in the Sobolev inequalities.

Motivation

A question arises whether u from the Sobolev space $W^{1,p}({\mathbb{ℝ}}^n)$ belongs to $L^q({\mathbb{ℝ}}^n)$ for some q > p. More specifically, when does $\Vert Du{\Vert }_{L^p({\mathbb{ℝ}}^n)}$ control $\Vert u{\Vert }_{L^q({\mathbb{ℝ}}^n)}$? It is easy to check that the following inequality

$\Vert u{\Vert }_{L^q({\mathbb{ℝ}}^n)}\le C(p,q)\Vert Du{\Vert }_{L^p({\mathbb{ℝ}}^n)}(*)$

can not be true for arbitrary q. Consider $u(x)\in C_c^{\mathrm{\infty }}({\mathbb{ℝ}}^n)$, infinitely differentiable function with compact support. Introduce $u_{\lambda }(x):=u(\lambda x)$. We have that:

$\begin{array}{rl}\Vert u_{\lambda }{\Vert }_{L^q({\mathbb{ℝ}}^n)}^q& =\underset{{\mathbb{ℝ}}^n}{\int }|u(\lambda x){|}^{q}dx=\frac{1}{{\lambda }^n}\underset{{\mathbb{ℝ}}^n}{\int }|u(y){|}^{q}dy={\lambda }^{-n}\Vert u{\Vert }_{L^q({\mathbb{ℝ}}^n)}^q\\ \Vert D{u}_{\lambda }{\Vert }_{L^p({\mathbb{ℝ}}^n)}^p& =\underset{{\mathbb{ℝ}}^n}{\int }|\lambda Du(\lambda x){|}^{p}dx=\frac{{\lambda }^p}{{\lambda }^n}\underset{{\mathbb{ℝ}}^n}{\int }|Du(y){|}^{p}dy={\lambda }^{p-n}\Vert Du{\Vert }_{L^p({\mathbb{ℝ}}^n)}^p\end{array}$

The inequality (*) for $u_{\lambda }$ results in the following inequality for $u$

$\Vert u{\Vert }_{L^q({\mathbb{ℝ}}^n)}\le {\lambda }^{1-\frac{n}{p}+\frac{n}{q}}C(p,q)\Vert Du{\Vert }_{L^p({\mathbb{ℝ}}^n)}$

If $1-\frac{n}{p}+\frac{n}{q}\ne 0,$ then by letting $\lambda$ going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for

$q=\frac{pn}{n-p}$,

which is the Sobolev conjugate.

See also

• Sergei Lvovich Sobolev
• conjugate index

References

• Lawrence C. Evans. Partial differential equations. Graduate Studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN 0-8218-0772-2