The One-Dimensional Slice Bound for a Continuously Differentiable Periodic Function

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 $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, and the integers $\mathbb{Z}$ are the ones fixed there. Let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, let $\mathcal{B}(\mathbb{R})$ be its Borel $\sigma$-algebra and let $\lambda$ be Lebesgue measure on it; integrals with respect to $\lambda$ are those of Measure Spaces and the Lebesgue Integral: Standing Notation §integral, formed for the measure space $(\mathbb{R},\mathcal{B}(\mathbb{R}),\lambda)$. Let $[0,1]$ be the closed interval determined by $0$ and $1$, and let $|t|$ denote the absolute value of a real number $t$. The notation $x[i{:}s]$ for the point of $\mathbb{R}^{n}$ obtained from $x$ by replacing its $i$th coordinate by $s$, the absolute value $|w|$ of a map $w:\mathbb{R}^{n}\to\mathbb{R}$, the zero extensions $\tilde{w}_{i,x}$ and the slice averages $P_{i}w$ are those of The Slice Average of a Continuous Periodic Function.

Let $u\in C^{1}_{\mathrm{per}}$ and let $i\in[n]$. Then $\partial_{i}u\in C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions §derivative, the maps $|u|$ and $|\partial_{i}u|$ lie in $C_{\mathrm{per}}$ by The Slice Average of a Continuous Periodic Function §closure, and so does

by Elementary Properties of Lattice-Periodic Functions §algebra; moreover $0\le\Theta_{i}(x)$ for every $x\in\mathbb{R}^{n}$, by claim 1 of Properties of the Absolute Value in an Ordered Field and claim 2 of Elementary Arithmetic in an Ordered Field. The slice average $P_{i}\Theta_{i}$ is therefore defined and lies in $C_{\mathrm{per}}$, by The Slice Average of a Continuous Periodic Function §defined.

1. (The pointwise bound)¶ For every $x\in\mathbb{R}^{n}$,