# Sobolev conjugate

The Sobolev conjugate of p for $1\leq p, 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 $W^{{1,p}}(R^{n})$ belongs to $L^{q}(R^{n})$ for some q>p. More specifically, when does $\|Du\|_{{L^{p}(R^{n})}}$ control $\|u\|_{{L^{q}(R^{n})}}$? It is easy to check that the following inequality

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

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

$\|u_{\lambda }\|_{{L^{q}(R^{n})}}^{q}=\int _{{R^{n}}}|u(\lambda x)|^{q}dx={\frac {1}{\lambda ^{n}}}\int _{{R^{n}}}|u(y)|^{q}dy=\lambda ^{{-n}}\|u\|_{{L^{q}(R^{n})}}^{q}$
$\|Du_{\lambda }\|_{{L^{p}(R^{n})}}^{p}=\int _{{R^{n}}}|\lambda Du(\lambda x)|^{p}dx={\frac {\lambda ^{p}}{\lambda ^{n}}}\int _{{R^{n}}}|Du(y)|^{p}dy=\lambda ^{{p-n}}\|Du\|_{{L^{p}(R^{n})}}^{p}$

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

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

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.

## References

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