The Sobolev conjugate of p for
, where n is space dimensionality, is
![{\displaystyle p^{*}={\frac {pn}{n-p}}>p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2afb5d806dfb7bc82ea763a5b26ea8eb98770d50)
This is an important parameter in the Sobolev inequalities.
Motivation[edit]
A question arises whether u from the Sobolev space
belongs to
for some q > p. More specifically, when does
control
? 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 (*)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f12a7f0f71b8589f4f176bfe9d40e466a3545d54)
can not be true for arbitrary q. Consider
, infinitely differentiable function with compact support. Introduce
. 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/369dca4793ee6162abf7ce281ea4191134e9e0b8)
The inequality (*) for
results in the following inequality for
![{\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})}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c96f67df198406c1dd188182f0bef5e0bc97429b)
If
then by letting
going to zero or infinity we obtain a contradiction. Thus the inequality (*) could only be true for
,
which is the Sobolev conjugate.
See also[edit]
References[edit]