# Sobolev conjugate

The Sobolev conjugate of p for ${\displaystyle 1\leq p, where n is space dimensionality, is

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

This is an important parameter in the Sobolev inequalities.

## Motivation

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

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

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

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

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

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

If ${\displaystyle 1-n/p+n/q\not =0}$, then by letting ${\displaystyle \lambda }$ going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for

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

which is the Sobolev conjugate.