Proof of Vanishing Mean of a Derivative and Integration by Parts on the Torus

Each result cited below is universally quantified over the data appearing in its own statement, and is applied to the data named here. For a point $x\in\mathbb{R}^{n}$, an index $i\in[n]$ and a real $t$, write $x\oplus t$ for the point obtained from $x$ by replacing its $i$th coordinate by $x_{i}+t$; thus $x\oplus t=x+t\,e$, where $e$ is the point whose $i$th coordinate is $1$ and whose other coordinates are $0$. Note that $\lVert(x\oplus t)-x\rVert=|t|$ by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square.

Claim 1. Let $g\in C^{1}_{\mathrm{per}}$ and $i\in[n]$. A function of class $C^{1}$ on $\mathbb{R}^{n}$ is continuous, by clause 2 of the calculus setting, so $g\in C_{\mathrm{per}}$. By Elementary Properties of Lattice-Periodic Functions (ii) we also have $\partial_{i}g\in C_{\mathrm{per}}$, and by Elementary Properties of Lattice-Periodic Functions (iii) there are nonnegative reals $M_{0}$ and $M_{1}$ with $|g(x)|\le M_{0}$ and $|\partial_{i}g(x)|\le M_{1}$ for every $x\in\mathbb{R}^{n}$.

Work on the measure space $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}),\lambda_{n})$, and let $U$ be the open interval with endpoints $-1$ and $1$. Define $F:U\times\mathbb{R}^{n}\to\mathbb{R}$ by

$$F(t,x)=\mathbf{1}_{Q}(x)\,g(x\oplus t).$$

We verify the three hypotheses of Differentiation under the Integral Sign.

(i) Fix $t\in U$. The map $x\mapsto x\oplus t$ satisfies $d_{E}(x\oplus t,y\oplus t)=\lVert x-y\rVert=d_{E}(x,y)$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, so $x\mapsto g(x\oplus t)$ is continuous on $\mathbb{R}^{n}$: given $x^{1}$ and a positive $\varepsilon$, the $\delta$ furnished by continuity of $g$ at $x^{1}\oplus t$ serves in Continuous Map Between Metric Spaces. It is therefore measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ and $\mathcal{B}(\mathbb{R})$ by claim 3(a) of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets together with claim 5 there, and multiplying by $\mathbf{1}_{Q}$ preserves measurability exactly as in the proof of Continuous Periodic Functions are Power-Integrable and Dense on the Torus (ii). Moreover $|F(t,\cdot)|\le M_{0}\mathbf{1}_{Q}$, whose integral is $M_{0}\lambda_{n}(Q)=M_{0}$ by The Integral of an Indicator Function is the Measure of the Set, The Half-Open Unit Cell Tiles Euclidean Space (i) and claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion; so $F(t,\cdot)$ is integrable by the criterion of Integrable Functions and Their Lebesgue Integral.

(ii) Fix $x\in\mathbb{R}^{n}$. If $x\notin Q$ then $F(\,\cdot\,,x)$ is identically $0$ on $U$, hence differentiable at every point of $U$ with derivative $0$, since all its difference quotients vanish. If $x\in Q$ then $F(t,x)=g(x\oplus t)$, and for $t\in U$ and a nonzero real $h$ small enough that $t+h\in U$ we have $(x\oplus t)\oplus h=x\oplus(t+h)$, so the difference quotient

$$\frac{F(t+h,x)-F(t,x)}{h}=\frac{g\bigl((x\oplus t)\oplus h\bigr)-g(x\oplus t)}{h}$$

is exactly the difference quotient whose limit as $h$ tends to $0$ defines the partial derivative $\partial_{i}g(x\oplus t)$, which exists because $g$ is of class $C^{1}$ on $\mathbb{R}^{n}$. Since every point of $U$ is an interior point of $U$, this limit is also the derivative of $F(\,\cdot\,,x)$ at $t$. In both cases $D_{1}F(t,x)=\mathbf{1}_{Q}(x)\,\partial_{i}g(x\oplus t)$.

(iii) $|D_{1}F(t,x)|\le M_{1}\mathbf{1}_{Q}(x)$ for all $t\in U$ and $x\in\mathbb{R}^{n}$, and $M_{1}\mathbf{1}_{Q}$ is integrable as in (i).

By Differentiation under the Integral Sign the function $\Phi(t)=\int_{\mathbb{R}^{n}}F(t,x)\,d\lambda_{n}(x)$ is differentiable at every point of $U$ with

$$\Phi'(t)=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}(x)\,\partial_{i}g(x\oplus t)\,d\lambda_{n}(x),\qquad\text{so}\qquad\Phi'(0)=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,\partial_{i}g\,d\lambda_{n}.$$

$\Phi$ is constant on $U$. Fix $t\in U$ and put $h=t\,e$, so that $x\oplus t=x+h$. The map $\mathbf{1}_{Q}g$ is integrable by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (ii), so by The Half-Open Unit Cell Tiles Euclidean Space (v), applied to the measurable $\mathbb{Z}^{n}$-periodic map $g$, the map $\mathbf{1}_{Q+h}g$ is integrable with

$$\int_{\mathbb{R}^{n}}\mathbf{1}_{Q+h}\,g\,d\lambda_{n}=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,g\,d\lambda_{n}=\Phi(0).$$

For every $x\in\mathbb{R}^{n}$ we have $x+h\in Q+h$ exactly when $x\in Q$, so

$$(\mathbf{1}_{Q+h}\,g)(x+h)=\mathbf{1}_{Q}(x)\,g(x+h)=F(t,x).$$

Claim 3 of Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ, applied with $a=h$ to the integrable map $\mathbf{1}_{Q+h}g$, therefore gives

$$\Phi(t)=\int_{\mathbb{R}^{n}}(\mathbf{1}_{Q+h}g)(x+h)\,d\lambda_{n}(x)=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q+h}\,g\,d\lambda_{n}=\Phi(0).$$

Since $\Phi$ is constant on $U$, every difference quotient of $\Phi$ at $0$ is $0$, so $\Phi'(0)=0$ by Derivative of a Real Function on an Interval. Hence $\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,\partial_{i}g\,d\lambda_{n}=0$, and Continuous Periodic Functions are Power-Integrable and Dense on the Torus (ii), applied to $\partial_{i}g\in C_{\mathrm{per}}$, turns this into

$$\int_{\mathbb{T}^{n}}\partial_{i}g\,dx=\int_{\mathbb{R}^{n}}\mathbf{1}_{Q}\,\partial_{i}g\,d\lambda_{n}=0 .$$

Claim 2. Let $g,h\in C^{1}_{\mathrm{per}}$ and $i\in[n]$. The product $gh$ lies in $C^{1}_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions (i), and by claim 1 of Constants, Affine Functions, Sums, Products and Quotients of Cᵏ Functions, applied on the open set $\mathbb{R}^{n}$,

$$\partial_{i}(gh)=(\partial_{i}g)\,h+g\,(\partial_{i}h)$$

as maps on $\mathbb{R}^{n}$. By Elementary Properties of Lattice-Periodic Functions (ii) the maps $\partial_{i}g$ and $\partial_{i}h$ lie in $C_{\mathrm{per}}$, so by Elementary Properties of Lattice-Periodic Functions (i) the products $(\partial_{i}g)h$ and $g(\partial_{i}h)$ lie in $C_{\mathrm{per}}$, and their restrictions to $Q$ lie in $\mathcal{L}^{1}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i). Restricting the displayed identity to $Q$ and integrating, claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives

$$\int_{\mathbb{T}^{n}}\partial_{i}(gh)\,dx=\int_{\mathbb{T}^{n}}(\partial_{i}g)\,h\,dx+\int_{\mathbb{T}^{n}}g\,(\partial_{i}h)\,dx .$$

The left-hand side is $0$ by claim 1 applied to $gh$, so the two integrals on the right are negatives of each other, which is the assertion.