Sobolev conjugate

The Sobolev conjugate of p for $$1\leq p p$$

This is an important parameter in the Sobolev inequalities.

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


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

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{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}$$

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


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

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


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

which is the Sobolev conjugate.