Nash inequality on a compact domain?

I have come across a few papers that make use of the Nash inequality for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the classical Nash inequality on $\mathbb{R}^n$ to that on compact domains so trivial?

I'd really appreciate any references you know of.

EDIT: This is the statement I am looking for

Let $\mathcal{D} = \mathbb{T}^n$ be the unit square in $\mathbb{R}^n$ with periodic boundary conditions. There exist constants $C_1$ and $C_2$ such that such that for $f \in H^1(\mathcal{D})$ then $$||f||_{2}^{1 + \frac{n}{2}} \leq ||f||_1 \left(C_1||f||_{2}^2 + C_2||\nabla f||_2^2\right)^{\frac{n}{4}}$$

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Could you provide a precise statement of the inequality you want a reference for? – Deane Yang Oct 3 at 16:50
In particular, if the functions on the compact domain are assumed to vanish on the boundary, then the inequality on $\mathbb{R}^n$ applies directly. – Deane Yang Oct 3 at 16:58
@Deane: You're right I should have specified that the functions should be periodic. – RadonNikodym Oct 3 at 17:19
You're effectively asking for the Nash inequality on a closed manifold. Cover the manifold by co-ordinate charts, fix a partition of unity subordinate to these charts. The inequality on the manifold then follows by applying the inequality on $\mathbb{R}^n$ on each co-ordinate chart to each compactly supported piece of the function and adding everything up. – Deane Yang Oct 3 at 17:33
The Nash inequality is a particular case of the Gagliardo-Nirenberg inequalities, so you can look up the proofs for that. – Deane Yang Oct 3 at 17:57
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The original Nash inequality in $\mathbb R^d$ is $$\|\nabla f\|_2 \|f\|_1^{2/d} \geq c \|f\|_2^{1+2/d}$$ It is proved in this article: Nash, J. Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 1958 931–954.
Note that the function $f \equiv 1$ fails the original Nash inequality in a compact domain. The extra term you have in yours is a correction for the compact case that is not necessary in the full space $\mathbb R^d$