Sobolev conjugate

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The Sobolev conjugate of 1\leq p <n is

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

This is an important parameter in the Sobolev inequalities.

[edit] Motivation

A question arises whether u from the Sobolev space W1,p(Rn) belongs to Lq(Rn) 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^\infty_c(R^n), infinitely differentiable function with compact support. Introduce uλ(x): = u(λx). We have that

\|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

The inequality (*) for uλ results in the following inequality for u

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

If 1-n/p+n/q\not = 0, then by letting λ 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.

[edit] See also