# 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!

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

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 $CD(K,\infty)$? –  mathsnail 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. –  Robert Haslhofer Apr 20 '12 at 18:28