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

\|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 $u(x)\in C_{c}^{\infty }(\mathbb {R} ^{n})$ , infinitely differentiable function with compact support. Introduce $u_{\lambda }(x):=u(\lambda x)$ . We have that:

{\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 $u_{\lambda }$ results in the following inequality for $u$

\|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 $1-{\frac {n}{p}}+{\frac {n}{q}}\neq 0,$ 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.

Motivation

See also

References