# sobolev embedding theorem in the smooth metric measure space

we know the sobolev embedding theorem of Saloff-Coste

$\Big(\int_B|F|^{2q}d\mu\Big)^{\frac1q}\le e^{C(1+\sqrt KR)}V^{-2/n}R^2\int_B\Big(|\nabla F|^2+R^{-2}F^2\Big)d\mu$

wtih $Ric\ge-(n-1)K$, for all '$B$' of radius $R$ and volume $V$, $F\in C^{\infty}_0(B)$, $q=n/(n-2)$.

My question is whether this inequality was established in the smooth metric measure space,i.e. $(M,g,e^{-f}d\mu)$ with Bakry-Emery Ricci curvature bouneded below $Ric_f=Ric+Hess f\ge-(n-1)K$?

Thank you!

• I took the liberty of adding some tags from the standard pool. Hope that they are not too inaccurate Apr 20 '12 at 5:07

## 1 Answer

The Sobolev-inequality holds for general metric measure spaces satisfying CD(K,n), in particular for your smooth ones.

See e.g. Theorem 21.15 in Villani's book http://math.univ-lyon1.fr/~villani/Cedrif/B07D.StFlour.pdf

Also note that the $L^2$-version of the Sobolev-inequality follows from the $L^1$-version by inserting a suitable power of the function and using Hölder.

• Thank you for your answer. Can we also obtain the explicit constant on the right hand side the same as Riemannian case? The original proof of Saloff-Coste is based on the heat semigroup. the inequality still holds for $CD(K,\infty)$? Apr 20 '12 at 17:55
• 1) I don't know the explicit constant, but I think you should be able to extract a good estimate for it from Villani's proof. 2) see chapter 25 of Villani's book for heat flow proofs of (related) functional inequalities. 3) With CD(K,$\infty$) you get a logarithmic Sobolev inequality. Apr 20 '12 at 18:28
• The link in the answer is broken. The book referenced (of his many) is Optimal Transport, Old and New. It is now available here: cedricvillani.org/wp-content/uploads/2012/08/preprint-1.pdf and a more stable page to find it may be here: cedricvillani.org/for-mathematicians/surveys-books Mar 30 '16 at 17:37