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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