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

In the setting of The Flat Torus: Standing Notation. For $u\in C_{\mathrm{per}}$ the restriction $u|_{Q}$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus §member, and we abbreviate

Recall from Elementary Properties of Lattice-Periodic Functions §derivative that $\partial_{i}g\in C_{\mathrm{per}}$ for $g\in C^{1}_{\mathrm{per}}$ and $i\in[n]$, and from Elementary Properties of Lattice-Periodic Functions §algebra that $C_{\mathrm{per}}$ is closed under pointwise products, so that all four integrands below lie in $C_{\mathrm{per}}$ and the abbreviation applies to them. Then the following hold.

1. (Vanishing mean of a derivative)¶ Let $g\in C^{1}_{\mathrm{per}}$ and let $i\in[n]$. Then

2. (Integration by parts)¶ Let $g,h\in C^{1}_{\mathrm{per}}$ and let $i\in[n]$. Then