The Sobolev Embedding of the First Sobolev Space of the Torus into the Sixth Lebesgue Space in Dimensions at Most Three

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$ and $n\le3$; the cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ together with the notation $\int_{\mathbb{T}^{n}}v\,dx$, the classes $\mathcal{L}^{t}(\mathbb{T}^{n})$ and the spaces $L^{t}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, for a real number $t$ with $1\le t$, the periodic classes $C_{\mathrm{per}}$, $C^{1}_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$, the partial derivatives $\partial_{j}$, and the restriction $v|_{Q}$ are the ones fixed there. Let $\lVert\,\cdot\,\rVert_{t}$ denote the $L^{t}$ seminorm of $(Q,\mathcal{B}_{Q},\lambda_{Q})$. Weak partial derivatives of classes and the notation $\partial_{j}U$ are those of The Weak Partial Derivative on the Torus, and $H^{1}(\mathbb{T}^{n})$ together with its norm $\lVert\,\cdot\,\rVert_{H^{1}}$ is the Sobolev space fixed there. Then the following hold.

1. (The bound for continuously differentiable periodic functions)(i) Let $u\in C^{1}_{\mathrm{per}}$. Then $[u|_{Q}]$ belongs to $H^{1}(\mathbb{T}^{n})$, by Elementary Properties of the Weak Partial Derivative on the Torus (i) together with The Sobolev Space of Once Weakly Differentiable Square-Integrable Classes on the Torus (i), the restriction $u|_{Q}$ belongs to $\mathcal{L}^{6}(\mathbb{T}^{n})$, by Elementary Properties of Lattice-Periodic Functions (iii) and Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i), and

$$\lVert u|_{Q}\rVert_{6}\le5\,\bigl\lVert[u|_{Q}]\bigr\rVert_{H^{1}} .$$

2. (The embedding)(ii) Let $U\in H^{1}(\mathbb{T}^{n})$ and let $v$ be a representative of $U$, that is, a member of $\mathcal{L}^{2}(\mathbb{T}^{n})$ with $[v]=U$. Then $v\in\mathcal{L}^{6}(\mathbb{T}^{n})$ and

$$\lVert v\rVert_{6}\le5\,\lVert U\rVert_{H^{1}} .$$

Since any two representatives of $U$ agree almost everywhere, they determine one and the same class in $L^{6}(\mathbb{T}^{n})$, by The Lebesgue Space Lᵖ(μ) (i); so this assigns to each $U\in H^{1}(\mathbb{T}^{n})$ a well-defined element of $L^{6}(\mathbb{T}^{n})$, whose norm is at most $5\lVert U\rVert_{H^{1}}$.