Integrability of Products of Four Sobolev Classes 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 $1\le n$ and $n\le3$; the cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the integral over $\mathbb{T}^{n}$, and the classes $\mathcal{L}^{t}(\mathbb{T}^{n})$ and spaces $L^{t}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, for a real number $t$ with $1\le t$, are the ones fixed there. Let $\lVert\,\cdot\,\rVert_{t}$ denote the $L^{t}$ seminorm of $(Q,\mathcal{B}_{Q},\lambda_{Q})$. A representative of a class $U\in L^{t}(\mathbb{T}^{n})$ is a member $u$ of $\mathcal{L}^{t}(\mathbb{T}^{n})$ with $[u]=U$; two representatives of one class agree almost everywhere by The Lebesgue Space Lᵖ(μ) (i). Weak partial derivatives of classes are those of The Weak Partial Derivative on the Torus, and $H^{1}(\mathbb{T}^{n})$, with its norm $\lVert\,\cdot\,\rVert_{H^{1}}$, is the Sobolev space fixed there; $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ is the inner product of $L^{2}(\mathbb{T}^{n})$. For $s\in\mathbb{R}$ and a natural number $m$, $s^{m}$ is the $m$-th power of $s$; for a map $v$ defined on $Q$, the pointwise power $v^{m}$ is the map on $Q$ whose value at $y$ is $(v(y))^{m}$, and the pointwise product of finitely many maps on $Q$ is formed valuewise. Write $5=1+1+1+1+1$ and $625=5^{4}$. Then the following hold.

1. (Products of four Sobolev classes)(i) Let $U_{1},U_{2},U_{3},U_{4}\in H^{1}(\mathbb{T}^{n})$ and, for each $k$, let $u_{k}$ be a representative of $U_{k}$. Then the pointwise product $u_{1}u_{2}u_{3}u_{4}$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$,

$$\Bigl|\int_{\mathbb{T}^{n}}u_{1}u_{2}u_{3}u_{4}\,dx\Bigr|\le625\,\lVert U_{1}\rVert_{H^{1}}\,\lVert U_{2}\rVert_{H^{1}}\,\lVert U_{3}\rVert_{H^{1}}\,\lVert U_{4}\rVert_{H^{1}},$$

and the value of this integral is the same for every choice of representatives $u_{1},u_{2},u_{3},u_{4}$ of $U_{1},U_{2},U_{3},U_{4}$.

2. (Fourth powers)(ii) Let $U\in H^{1}(\mathbb{T}^{n})$ and let $u$ be a representative of $U$. Then the pointwise fourth power $u^{4}$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$,

$$\Bigl|\int_{\mathbb{T}^{n}}u^{4}\,dx\Bigr|\le625\,\bigl(\lVert U\rVert_{H^{1}}\bigr)^{4},$$

and the value of this integral is the same for every representative $u$ of $U$.

3. (The cube pairing)(iii) Let $U,W\in H^{1}(\mathbb{T}^{n})$, let $u$ be a representative of $U$ and let $w$ be a representative of $W$. Then the pointwise cube $u^{3}$ lies in $\mathcal{L}^{2}(\mathbb{T}^{n})$ and the class $[u^{3}]\in L^{2}(\mathbb{T}^{n})$ is the same for every representative $u$ of $U$. Moreover the pointwise product $u^{3}w$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$ and

$$\bigl\langle[u^{3}],W\bigr\rangle_{L^{2}}=\int_{\mathbb{T}^{n}}u^{3}w\,dx .$$