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 $W 1, p (R n)$ 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 $u(x)\in C^\infty_c(R^n)$ , infinitely differentiable function with compact support. Introduce $u λ (x): = u (λ x)$ . 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 $u λ$ 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.

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Category: Sobolev spaces

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