Pairing an Integrable Function on the Torus with 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 cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the integral over $\mathbb{T}^{n}$, the class $\mathcal{L}^{1}(\mathbb{T}^{n})$, the space $L^{1}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic classes $C_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$ and the restriction $\varphi|_{Q}$ are the ones fixed there. Every member of $C^{\infty}_{\mathrm{per}}$ belongs to $C_{\mathrm{per}}$, a smooth map on $\mathbb{R}^{n}$ being continuous by claim 3 of Euclidean Space Is Open in Itself and periodicity being the same condition for the two classes by Lattice-Periodic Functions and the Periodic Function Classes (iii).

Let $\varphi\in C_{\mathrm{per}}$. By Elementary Properties of Lattice-Periodic Functions (iii) there is a real number $M$ with $0\le M$ and $|\varphi(x)|\le M$ for every $x\in\mathbb{R}^{n}$, and by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i), used with the exponent $1$, the restriction $\varphi|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$ and belongs to $\mathcal{L}^{1}(\mathbb{T}^{n})$. Then the following hold.

1. (The pairing)(i) Let $w\in\mathcal{L}^{1}(\mathbb{T}^{n})$. Then the pointwise product $w\,(\varphi|_{Q})$ belongs to $\mathcal{L}^{1}(\mathbb{T}^{n})$, and for every real number $M$ with $0\le M$ and $|\varphi(x)|\le M$ for every $x\in\mathbb{R}^{n}$,

$$\Bigl|\int_{\mathbb{T}^{n}}w\,(\varphi|_{Q})\,dx\Bigr|\le M\int_{\mathbb{T}^{n}}|w|\,dx .$$

2. (Linearity in each factor)(ii) Let $w,w'\in\mathcal{L}^{1}(\mathbb{T}^{n})$, let $\varphi'\in C_{\mathrm{per}}$ and let $c\in\mathbb{R}$. Then $w+c\,w'\in\mathcal{L}^{1}(\mathbb{T}^{n})$ by Minkowski's Inequality (ii), the map $\varphi+c\,\varphi'$ lies in $C_{\mathrm{per}}$ by Elementary Properties of Lattice-Periodic Functions (i), and

$$\int_{\mathbb{T}^{n}}(w+c\,w')\,(\varphi|_{Q})\,dx=\int_{\mathbb{T}^{n}}w\,(\varphi|_{Q})\,dx+c\int_{\mathbb{T}^{n}}w'\,(\varphi|_{Q})\,dx ,$$ $$\int_{\mathbb{T}^{n}}w\,\bigl((\varphi+c\,\varphi')|_{Q}\bigr)\,dx=\int_{\mathbb{T}^{n}}w\,(\varphi|_{Q})\,dx+c\int_{\mathbb{T}^{n}}w\,(\varphi'|_{Q})\,dx .$$

3. (Dependence on the class only)(iii) Let $w,w'\in\mathcal{L}^{1}(\mathbb{T}^{n})$ satisfy $[w]=[w']$ in $L^{1}(\mathbb{T}^{n})$, that is, $w=w'$ $\lambda_{Q}$-almost everywhere on $Q$ by The Lebesgue Space Lᵖ(μ) (i). Then

$$\int_{\mathbb{T}^{n}}w\,(\varphi|_{Q})\,dx=\int_{\mathbb{T}^{n}}w'\,(\varphi|_{Q})\,dx .$$

4. (The test integrals determine the class)(iv) Let $w,w'\in\mathcal{L}^{1}(\mathbb{T}^{n})$ satisfy

$$\int_{\mathbb{T}^{n}}w\,(\varphi|_{Q})\,dx=\int_{\mathbb{T}^{n}}w'\,(\varphi|_{Q})\,dx\qquad\text{for every }\varphi\in C^{\infty}_{\mathrm{per}} .$$

Then $w=w'$ $\lambda_{Q}$-almost everywhere on $Q$, so that $[w]=[w']$ in $L^{1}(\mathbb{T}^{n})$; and if moreover $w$ and $w'$ lie in $\mathcal{L}^{q}(\mathbb{T}^{n})$ for a real number $q$ with $1\le q$, then $[w]=[w']$ in $L^{q}(\mathbb{T}^{n})$ as well.