The Slice Average of a Continuous 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 class $C_{\mathrm{per}}$, Euclidean space $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, the integer lattice $\mathbb{Z}^{n}$, 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 class $\mathcal{L}^{1}(\mathbb{T}^{n})$, the restriction $v|_{Q}$, and the power $t^{a}$ of a nonnegative real number $t$ with positive real exponent $a$ 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 Measurable Functions and the Lebesgue Integral: Standing Notation (v), 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 $t$. For maps $v,w:\mathbb{R}^{n}\to\mathbb{R}$ we write $vw$ for their pointwise product, $|w|$ for the map $x\mapsto|w(x)|$, and, when $0\le w(x)$ for every $x\in\mathbb{R}^{n}$ and $a$ is a positive real number, $w^{a}$ for the map $x\mapsto(w(x))^{a}$.

For $x\in\mathbb{R}^{n}$, $i\in[n]$ and $s\in\mathbb{R}$ let $x[i{:}s]$ denote the point of $\mathbb{R}^{n}$ whose $i$th coordinate is $s$ and whose $k$th coordinate is $x_{k}$ for every $k\in[n]$ with $k\ne i$; thus $x[i{:}x_{i}]=x$, and $x[i{:}s]$ depends only on $s$ and on the coordinates $x_{k}$ with $k\ne i$. For $j\in[n]$, a map $v:\mathbb{R}^{n}\to\mathbb{R}$ is said to be free of the $j$th coordinate if $v(x[j{:}s])=v(x)$ for every $x\in\mathbb{R}^{n}$ and every $s\in\mathbb{R}$.

Then the following hold.

1. (The slice average)(i) Let $i\in[n]$ and let $w\in C_{\mathrm{per}}$. For every $x\in\mathbb{R}^{n}$ the map $s\mapsto w(x[i{:}s])$ is continuous from $(\mathbb{R},d_{\mathbb{R}})$ to $(\mathbb{R},d_{\mathbb{R}})$, and the map $\tilde{w}_{i,x}:\mathbb{R}\to\mathbb{R}$ taking the value $w(x[i{:}s])$ at $s\in[0,1]$ and the value $0$ at every $s\in\mathbb{R}$ with $s\notin[0,1]$ is measurable with respect to $\mathcal{B}(\mathbb{R})$ and integrable with respect to $\lambda$. Setting

$$(P_{i}w)(x)=\int_{\mathbb{R}}\tilde{w}_{i,x}\,d\lambda\qquad(x\in\mathbb{R}^{n})$$

therefore defines a map $P_{i}w:\mathbb{R}^{n}\to\mathbb{R}$, called the slice average of $w$ in the $i$th coordinate, and $P_{i}w\in C_{\mathrm{per}}$. If $0\le w(x)$ for every $x\in\mathbb{R}^{n}$, then $0\le(P_{i}w)(x)$ for every $x\in\mathbb{R}^{n}$.

2. (Free coordinates)(ii) Let $i\in[n]$ and let $w\in C_{\mathrm{per}}$. Then $P_{i}w$ is free of the $i$th coordinate. If in addition $j\in[n]$ and $w$ is free of the $j$th coordinate, then $P_{i}w$ is free of the $j$th coordinate.

3. (Factoring out a free factor)(iii) Let $i\in[n]$ and let $\mathbf{1}$ denote the map on $\mathbb{R}^{n}$ with constant value $1$. Then $\mathbf{1}\in C_{\mathrm{per}}$ and $P_{i}\mathbf{1}=\mathbf{1}$. If $v,w\in C_{\mathrm{per}}$ and $v$ is free of the $i$th coordinate, then $vw\in C_{\mathrm{per}}$ and

$$P_{i}(vw)=v\,P_{i}w ;$$

in particular $P_{i}v=v$.

4. (Cauchy--Schwarz for the slice average)(iv) Let $i\in[n]$ and let $v,w\in C_{\mathrm{per}}$ satisfy $0\le v(x)$ and $0\le w(x)$ for every $x\in\mathbb{R}^{n}$. Then $vw\in C_{\mathrm{per}}$ and $0\le(vw)(x)$ for every $x\in\mathbb{R}^{n}$, so that $(vw)^{1/2}$ is defined; it lies in $C_{\mathrm{per}}$, and for every $x\in\mathbb{R}^{n}$

$$\Bigl(P_{i}\bigl[(vw)^{1/2}\bigr]\Bigr)(x)\le\bigl((P_{i}v)(x)\bigr)^{1/2}\,\bigl((P_{i}w)(x)\bigr)^{1/2}.$$

5. (Absolute values and powers)(v) Let $w\in C_{\mathrm{per}}$. Then $|w|\in C_{\mathrm{per}}$. If moreover $0\le w(x)$ for every $x\in\mathbb{R}^{n}$ and $a$ is a positive real number, then $w^{a}\in C_{\mathrm{per}}$ and $0\le w^{a}(x)$ for every $x\in\mathbb{R}^{n}$. If $j\in[n]$ and $w$ is free of the $j$th coordinate, then $|w|$ is free of the $j$th coordinate, and so is $w^{a}$ in the case just described.

6. (A map free of every coordinate is constant)(vi) Let $v:\mathbb{R}^{n}\to\mathbb{R}$ be free of the $j$th coordinate for every $j\in[n]$. Then there is a real number $c$ with $v(x)=c$ for every $x\in\mathbb{R}^{n}$.

7. (The slice average preserves the integral over the cell)(vii) Suppose that $n\le3$. Let $i\in[n]$ and let $w\in C_{\mathrm{per}}$ satisfy $0\le w(x)$ for every $x\in\mathbb{R}^{n}$. Then $w|_{Q}$ and $(P_{i}w)|_{Q}$ belong to $\mathcal{L}^{1}(\mathbb{T}^{n})$ and

$$\int_{\mathbb{T}^{n}}(P_{i}w)|_{Q}\,dx=\int_{\mathbb{T}^{n}}w|_{Q}\,dx .$$