Sobolev conjugate

The Sobolev conjugate of p for $1\leq p <n$, 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 $\|Du\|_{L^p(\mathbb{R}^n)}$ control $\|u\|_{L^q(\mathbb{R}^n)}$? It is easy to check that the following inequality

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

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

$\|u_\lambda\|_{L^q(\mathbb{R}^n)}^q=\int_{\mathbb{R}^n}|u(\lambda x)|^qdx=\frac{1}{\lambda^n}\int_{\mathbb{R}^n}|u(y)|^qdy=\lambda^{-n}\|u\|_{L^q(\mathbb{R}^n)}^q$

$\|Du_\lambda\|_{L^p(\mathbb{R}^n)}^p=\int_{\mathbb{R}^n}|\lambda Du(\lambda x)|^pdx=\frac{\lambda^p}{\lambda^n}\int_{\mathbb{R}^n}|Du(y)|^pdy=\lambda^{p-n}\|Du\|_{L^p(\mathbb{R}^n)}^p$

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

$\|u\|_{L^q(\mathbb{R}^n)}\leq \lambda^{1-n/p+n/q}C(p,q)\|Du\|_{L^p(\mathbb{R}^n)}$

If $1-n/p+n/q\not = 0$, 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

References

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