Sobolev Conjugate

The Sobolev conjugate of ''p'' for 1\leq p , where ''n'' is space dimensionality, is
:$p^*=\frac>p$
This is an important parameter in the Sobolev inequalities.

Motivation

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

:$\, u\, _\leq C(p,q)\, Du\, _ \qquad \qquad (*)$

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

:$\begin \, u_\lambda \, _^q &= \int_, u(\lambda x), ^qdx=\frac\int_, u(y), ^qdy=\lambda^\, u\, _^q \\ \, Du_\lambda\, _^p &= \int_, \lambda Du(\lambda x), ^pdx=\frac\int_, Du(y), ^pdy=\lambda^ \, Du \, _^p \end$

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

:$\, u\, _\leq \lambda^C(p,q)\, Du\, _$

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

:$q=\frac$,

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

Sobolev spaces