Transfer of a Partial Derivative in Periodic Convolution

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 periodic classes $C_{\mathrm{per}}$, $C^{1}_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$, the restriction $v|_{Q}$, and the Euclidean norm $\lVert\,\cdot\,\rVert$ are the ones fixed there, as are the partial derivatives $\partial_{i}$. A map $\mathbb{R}^{n}\to\mathbb{R}$ is called continuous, and smooth, as in Periodic Convolution and Mollification on the Torus, and $\psi\star u$ denotes the periodic convolution.

Throughout, $R$ is a real number with $0<R$, the map $\psi:\mathbb{R}^{n}\to\mathbb{R}$ is smooth with $\psi(y)=0$ for every $y\in\mathbb{R}^{n}$ satisfying $R<\lVert y\rVert$, and $\varphi\in C^{1}_{\mathrm{per}}$. Every member of $C^{1}_{\mathrm{per}}$ belongs to $C_{\mathrm{per}}$, since a map of class $C^{1}$ on $\mathbb{R}^{n}$ is continuous by claim 3 of Euclidean Space Is Open in Itself and periodicity is the same condition for both classes by Lattice-Periodic Functions and the Periodic Function Classes (iii); and $\partial_{i}\varphi\in C_{\mathrm{per}}$ for every $i\in[n]$ by Elementary Properties of Lattice-Periodic Functions (ii). Hence $\varphi|_{Q}$ and $(\partial_{i}\varphi)|_{Q}$ lie in $\mathcal{L}^{1}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i), used with the exponent $1$, so that the periodic convolutions written below are defined. Then the following hold.

1. (Transfer of a partial derivative)(i) Let $i\in[n]$. The map $\partial_{i}\psi$ is continuous and vanishes at every $y$ with $R<\lVert y\rVert$, by Convolution of a Locally Integrable Function with a Compactly Supported Kernel (iv), so it may be taken as a kernel in its own right. As maps on $\mathbb{R}^{n}$,

$$\psi\star\bigl((\partial_{i}\varphi)|_{Q}\bigr)=(\partial_{i}\psi)\star(\varphi|_{Q})=\partial_{i}\bigl(\psi\star(\varphi|_{Q})\bigr).$$

2. (Uniform approximation together with the first derivatives)(ii) Let $\delta$ be a real number with $0<\delta$, let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$, and for a real number $\varepsilon$ with $0<\varepsilon$ let $\rho_{\varepsilon}$ be its rescaling, a mollifier kernel of radius $\varepsilon\delta$ by that lemma. Then for every real number $\eta$ with $0<\eta$ there is a real number $\varepsilon_{0}$ with $0<\varepsilon_{0}$ such that for every real number $\varepsilon$ with $0<\varepsilon<\varepsilon_{0}$, every $x\in\mathbb{R}^{n}$ and every $i\in[n]$,

$$\bigl|\bigl(\rho_{\varepsilon}\star(\varphi|_{Q})\bigr)(x)-\varphi(x)\bigr|\le\eta \qquad\text{and}\qquad \bigl|\partial_{i}\bigl(\rho_{\varepsilon}\star(\varphi|_{Q})\bigr)(x)-\partial_{i}\varphi(x)\bigr|\le\eta .$$