The Gagliardo-Nirenberg Inequality on the Torus in Dimensions at Most Three

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the periodic classes $C_{\mathrm{per}}$ and $C^{1}_{\mathrm{per}}$, the partial derivatives $\partial_{i}$, Euclidean space $\mathbb{R}^{n}$, the initial segments $[n]$, the half-open unit cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$ together with the notation $\int_{\mathbb{T}^{n}}v\,dx$, the classes $\mathcal{L}^{t}(\mathbb{T}^{n})$ for a real number $t$ with $1\le t$, the restriction $v|_{Q}$, and the power $\tau^{a}$ of a nonnegative real number $\tau$ with positive real exponent $a$ are the ones fixed there. Let $\lVert\,\cdot\,\rVert_{t}$ denote the $L^{t}$ seminorm of $(Q,\mathcal{B}_{Q},\lambda_{Q})$ and let $|\tau|$ denote the absolute value of a real number $\tau$. The absolute value $|w|$ of a map $w:\mathbb{R}^{n}\to\mathbb{R}$ and the slice averages $P_{i}w$ are those of The Slice Average of a Continuous Periodic Function.

Let $u\in C^{1}_{\mathrm{per}}$. Then $u\in C_{\mathrm{per}}$, a map of class $C^{1}$ being continuous by claim 3 of Euclidean Space is Open in Itself, and $C^k$ Maps are Continuous and periodicity being the same condition for the two classes by Lattice-Periodic Functions and the Periodic Function Classes §classes, and $\partial_{i}u\in C_{\mathrm{per}}$ for every $i\in[n]$ by Elementary Properties of Lattice-Periodic Functions §derivative. Hence, by Elementary Properties of Lattice-Periodic Functions §bounded and Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member, the restrictions $u|_{Q}$ and $(\partial_{i}u)|_{Q}$ are measurable with respect to $\mathcal{B}_{Q}$ and belong to $\mathcal{L}^{t}(\mathbb{T}^{n})$ for every real number $t$ with $1\le t$ and every $i\in[n]$. For $i\in[n]$ put

a nonnegative real number, seminorms taking values in the nonnegative reals by Power-Integrable Functions and the p-Seminorm §seminorm and sums of nonnegative reals being nonnegative by claim 2 of Elementary Arithmetic in an Ordered Field.

Then the following hold.

1. (Dimension one)¶ Suppose $n=1$. Then

2. (Dimension two)¶ Suppose $n=2$. Then

3. (Dimension three)¶ Suppose $n=3$. Then