Sobolev conjugate

The Sobolev conjugate of

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 $u\_lambda(x):=u(lambda\; x)$. 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 $u\_lambda$ 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 $1-n/p+n/q\; ot\; =\; 0$, 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.

ee also

*Sergei Lvovich Sobolev