Sobolev conjugate

A question arises whether u from the Sobolev space ${\displaystyle W^{1,p}(\mathbb {R} ^{n})}$ belongs to ${\displaystyle L^{q}(\mathbb {R} ^{n})}$ for some q > p. More specifically, when does ${\displaystyle \|Du\|_{L^{p}(\mathbb {R} ^{n})}}$ control ${\displaystyle \|u\|_{L^{q}(\mathbb {R} ^{n})}}$? It is easy to check that the following inequality

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

can not be true for arbitrary q. Consider ${\displaystyle u(x)\in C_{c}^{\infty }(\mathbb {R} ^{n})}$, infinitely differentiable function with compact support. Introduce ${\displaystyle u_{\lambda }(x):=u(\lambda x)}$. We have that:

${\displaystyle {\begin{aligned}\|u_{\lambda }\|_{L^{q}(\mathbb {R} ^{n})}^{q}&=\int _{\mathbb {R} ^{n}}|u(\lambda x)|^{q}dx={\frac {1}{\lambda ^{n}}}\int _{\mathbb {R} ^{n}}|u(y)|^{q}dy=\lambda ^{-n}\|u\|_{L^{q}(\mathbb {R} ^{n})}^{q}\\\|Du_{\lambda }\|_{L^{p}(\mathbb {R} ^{n})}^{p}&=\int _{\mathbb {R} ^{n}}|\lambda Du(\lambda x)|^{p}dx={\frac {\lambda ^{p}}{\lambda ^{n}}}\int _{\mathbb {R} ^{n}}|Du(y)|^{p}dy=\lambda ^{p-n}\|Du\|_{L^{p}(\mathbb {R} ^{n})}^{p}\end{aligned}}}$

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

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

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

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

which is the Sobolev conjugate.

• Sergei Lvovich Sobolev

• Lawrence C. Evans. Partial differential equations. Graduate Studies in Mathematics, Vol 19. American Mathematical Society. 1998. ISBN 0-8218-0772-2