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 periodicity being the same condition for the two classes by Lattice-Periodic Functions and the Periodic Function Classes (iii), and $\partial_{i}u\in C_{\mathrm{per}}$ for every $i\in[n]$ by Elementary Properties of Lattice-Periodic Functions (ii). Hence, by Elementary Properties of Lattice-Periodic Functions (iii) and Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i), 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

$$A_{i}=\lVert u|_{Q}\rVert_{1}+\bigl\lVert(\partial_{i}u)|_{Q}\bigr\rVert_{1},$$

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

Then the following hold.

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

$$|u(x)|\le A_{1}\qquad\text{for every }x\in\mathbb{R}^{n}.$$

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

$$\bigl(\lVert u|_{Q}\rVert_{2}\bigr)^{2}\le A_{1}A_{2}.$$

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

$$\bigl(\lVert u|_{Q}\rVert_{3/2}\bigr)^{3/2}\le\bigl(A_{1}A_{2}A_{3}\bigr)^{1/2}.$$