The Chain Rule for Weak Derivatives on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, 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 $(Q,\mathcal{B}_{Q},\lambda_{Q})$. Weak partial derivatives of classes and the notation $\partial_{j}U$ are those of The Weak Partial Derivative on the Torus, and $H^{1}(\mathbb{T}^{n})$ is the Sobolev space fixed there. 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).

Let $\mathbb{R}$ be the real numbers and let $|\cdot|$ be the absolute value on $\mathbb{R}$. Differentiability of a function from $\mathbb{R}$ to $\mathbb{R}$ at a point, and the derivative there, are those of that definition; they apply at every point because by claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line the set $\mathbb{R}$ is an interval every point of which is an interior point of it. For $\phi:\mathbb{R}\to\mathbb{R}$ differentiable at every point of $\mathbb{R}$, $\phi'$ denotes the function from $\mathbb{R}$ to $\mathbb{R}$ whose value at $s$ is the derivative of $\phi$ at $s$. We regard $\mathbb{R}$ also as the Euclidean space $\mathbb{R}^{1}$, which is open in itself by claim 1 of Euclidean Space Is Open in Itself; being of class $C^{1}$ on $\mathbb{R}^{1}$ is as defined there, and $\partial_{1}$ denotes the partial derivative with respect to the first variable. For $\psi:\mathbb{R}\to\mathbb{R}$ and $v:Q\to\mathbb{R}$, $\psi\circ v$ denotes the function from $Q$ to $\mathbb{R}$ whose value at $x$ is $\psi(v(x))$, and for $v,w:Q\to\mathbb{R}$ the pointwise product $v\,w$ is the function whose value at $x$ is $v(x)\,w(x)$.

Let $M\in\mathbb{R}$ satisfy $0\le M$ and let $\phi:\mathbb{R}\to\mathbb{R}$ be of class $C^{1}$ on $\mathbb{R}^{1}$; then $\phi$ is differentiable at every point of $\mathbb{R}$ and $\phi'(s)=\partial_{1}\phi(s)$ for every $s\in\mathbb{R}$, by claim 2 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line. Suppose further that

$$|\phi'(s)|\le M\qquad\text{for every }s\in\mathbb{R}.$$

Then the following hold.

1. (Lipschitz bound)(i) For all $s,t\in\mathbb{R}$,

$$|\phi(s)-\phi(t)|\le M\,|s-t| .$$

2. (The composed class)(ii) Let $U\in L^{2}(\mathbb{T}^{n})$ and let $u$ be a representative of $U$. Then $\phi\circ u\in\mathcal{L}^{2}(\mathbb{T}^{n})$; and if $\tilde{u}$ is a further representative of $U$, then $\phi\circ u$ and $\phi\circ\tilde{u}$ agree almost everywhere, so that the class $[\phi\circ u]\in L^{2}(\mathbb{T}^{n})$ depends only on $U$. We write $\phi(U)$ for that class.

3. (The chain rule)(iii) Let $U\in H^{1}(\mathbb{T}^{n})$, let $u$ be a representative of $U$ and, for $j\in[n]$, let $g_{j}$ be a representative of $\partial_{j}U$. Then for every $j\in[n]$ the pointwise product $(\phi'\circ u)\,g_{j}$ belongs to $\mathcal{L}^{2}(\mathbb{T}^{n})$ and

$$\bigl\lVert(\phi'\circ u)\,g_{j}\bigr\rVert_{2}\le M\,\lVert g_{j}\rVert_{2} .$$

Moreover $\phi(U)\in H^{1}(\mathbb{T}^{n})$ and

$$\partial_{j}\phi(U)=\bigl[(\phi'\circ u)\,g_{j}\bigr]\qquad\text{for every }j\in[n].$$