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The '''Sobolev conjugate''' of ''p'' for <math>1\leq p <n</math>, where ''n'' is space dimensionality, is
:<math> p^*=\frac{pn}{n-p}>p</math>
This is an important parameter in the [[Sobolev inequality|Sobolev inequalities]].
==Motivation==
A question arises whether ''u'' from the [[Sobolev space]] <math>W^{1,p}(\R^n)</math> belongs to <math>L^q(\R^n)</math> for some ''q'' > ''p''. More specifically, when does <math>\|Du\|_{L^p(\R^n)}</math> control <math>\|u\|_{L^q(\R^n)}</math>? It is easy to check that the following inequality

:<math>\|u\|_{L^q(\R^n)}\leq C(p,q)\|Du\|_{L^p(\R^n)} \qquad \qquad (*)</math>    

can not be true for arbitrary ''q''. Consider <math>u(x)\in C^\infty_c(\R^n)</math>, infinitely differentiable function with compact support. Introduce <math>u_\lambda(x):=u(\lambda x)</math>. We have that:

:<math>\begin{align}
\|u_\lambda \|_{L^q(\R^n)}^q &= \int_{\R^n}|u(\lambda x)|^qdx=\frac{1}{\lambda^n}\int_{\R^n}|u(y)|^qdy=\lambda^{-n}\|u\|_{L^q(\R^n)}^q \\
\|Du_\lambda\|_{L^p(\R^n)}^p &= \int_{\R^n}|\lambda Du(\lambda x)|^pdx=\frac{\lambda^p}{\lambda^n}\int_{\R^n}|Du(y)|^pdy=\lambda^{p-n} \|Du \|_{L^p(\R^n)}^p
\end{align}</math>

The inequality (*) for <math>u_\lambda</math> results in the following inequality for <math>u</math>

:<math>\|u\|_{L^q(\R^n)}\leq \lambda^{1-\frac{n}{p}+\frac{n}{q}}C(p,q)\|Du\|_{L^p(\R^n)}</math>

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

:<math>q=\frac{pn}{n-p}</math>,

which is the Sobolev conjugate.

==See also==
*Sergei Lvovich Sobolev
* [[Conjugate index|conjugate index]] 

==References==
* Lawrence C. Evans. Partial differential equations. [[Graduate Studies in Mathematics]], Vol 19. American Mathematical Society. 1998.  {{ISBN|0-8218-0772-2}}

[[Category:Sobolev spaces]]

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