Proof of Transfer of a Partial Derivative in Periodic Convolution

Each result cited below is universally quantified over the data appearing in its own statement, and is applied here to the data named in the statement of this lemma. Integrals over $\mathbb{T}^{n}$ are written with the integration variable named $y$, and for $w\in C_{\mathrm{per}}$ we use the abbreviation $\int_{\mathbb{T}^{n}}w\,dy=\int_{\mathbb{T}^{n}}w|_{Q}\,dy$ of Vanishing Mean of a Derivative and Integration by Parts on the Torus.

Step 1. Proof of claim 1.

Since $\psi$ is smooth it is of class $C^{1}$ on $\mathbb{R}^{n}$ by claim 2 of Euclidean Space Is Open in Itself, so $\partial_{i}\psi$ is continuous and vanishes at every $y$ with $R<\lVert y\rVert$, as recorded in claim 1; thus $\partial_{i}\psi$ satisfies the hypotheses placed on a kernel in The Periodised Kernel of a Periodic Convolution, as does $\psi$. Let $\Psi$ be the periodised kernel of $\psi$ and let $\Psi^{(i)}$ be that of $\partial_{i}\psi$; for $x\in\mathbb{R}^{n}$ let $\Psi_{x}$ and $\Psi^{(i)}_{x}$ denote the restrictions to $Q$ of $\Psi(x,\cdot)$ and $\Psi^{(i)}(x,\cdot)$.

Fix $x\in\mathbb{R}^{n}$. By The Periodised Kernel of a Periodic Convolution (iii) the map $\Psi(x,\cdot)$ lies in $C^{\infty}_{\mathrm{per}}$, hence in $C^{1}_{\mathrm{per}}$, a smooth map on $\mathbb{R}^{n}$ being of class $C^{1}$ by claim 2 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). By The Periodised Kernel of a Periodic Convolution (v), applied to the kernel $\psi$ and the function $(\partial_{i}\varphi)|_{Q}$,

$$\bigl(\psi\star((\partial_{i}\varphi)|_{Q})\bigr)(x)=\int_{\mathbb{T}^{n}}\Psi_{x}\,(\partial_{i}\varphi)|_{Q}\,dy=\int_{\mathbb{T}^{n}}\Psi(x,\cdot)\,(\partial_{i}\varphi)\,dy ,$$

the second equality because the restriction of a pointwise product is the product of the restrictions. Both $\Psi(x,\cdot)$ and $\varphi$ lie in $C^{1}_{\mathrm{per}}$, so Vanishing Mean of a Derivative and Integration by Parts on the Torus (ii), applied with $g=\Psi(x,\cdot)$ and $h=\varphi$, gives

$$\int_{\mathbb{T}^{n}}\bigl(\partial_{i}\Psi(x,\cdot)\bigr)\,\varphi\,dy=-\int_{\mathbb{T}^{n}}\Psi(x,\cdot)\,(\partial_{i}\varphi)\,dy .$$

By The Periodised Kernel of a Periodic Convolution (vii), $\partial_{i}\Psi(x,\cdot)=-\Psi^{(i)}(x,\cdot)$ as maps on $\mathbb{R}^{n}$, so the left-hand side equals $-\int_{\mathbb{T}^{n}}\Psi^{(i)}(x,\cdot)\,\varphi\,dy$ by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, used with the scalar $-1$. Combining the two displays and cancelling the common sign,

$$\bigl(\psi\star((\partial_{i}\varphi)|_{Q})\bigr)(x)=\int_{\mathbb{T}^{n}}\Psi^{(i)}_{x}\,\varphi|_{Q}\,dy=\bigl((\partial_{i}\psi)\star(\varphi|_{Q})\bigr)(x),$$

the second equality by The Periodised Kernel of a Periodic Convolution (v) applied to the kernel $\partial_{i}\psi$ and the function $\varphi|_{Q}$. Finally, $\psi$ is of class $C^{k}$ on $\mathbb{R}^{n}$ for every natural number $k$ by Smooth Functions on an Open Subset of ℝⁿ, so Properties of Periodic Convolution on the Torus (i), applied with the kernel $\psi$ and the function $\varphi|_{Q}$, gives $\partial_{i}(\psi\star(\varphi|_{Q}))=(\partial_{i}\psi)\star(\varphi|_{Q})$. As $x$ was arbitrary, claim 1 follows.

Step 2. Proof of claim 2.

Let $\eta$ be a real number with $0<\eta$. The maps $\varphi$ and $\partial_{1}\varphi,\dots,\partial_{n}\varphi$ all lie in $C_{\mathrm{per}}$, so A Continuous Lattice-Periodic Function is Uniformly Continuous (i), applied to each of them with the tolerance $\eta$, provides positive real numbers $r_{0},r_{1},\dots,r_{n}$ such that $|\varphi(x)-\varphi(z)|\le\eta$ for all $x,z$ with $\lVert x-z\rVert\le r_{0}$, and $|\partial_{i}\varphi(x)-\partial_{i}\varphi(z)|\le\eta$ for all $x,z$ with $\lVert x-z\rVert\le r_{i}$, for each $i\in[n]$. Let $r$ be the least of $r_{0},r_{1},\dots,r_{n}$; it is a positive real number not exceeding any of them, by claim 9 of Elementary Order Arithmetic in an Ordered Field for two numbers together with an induction over the finitely many indices. Put $\varepsilon_{0}=r\,\delta^{-1}$, a positive real number by claims 7 and 5 of Elementary Order Arithmetic in an Ordered Field.

Let $\varepsilon$ be a real number with $0<\varepsilon<\varepsilon_{0}$. Multiplying by the positive factor $\delta$, claim 10 of Elementary Order Arithmetic in an Ordered Field gives $\varepsilon\delta<\varepsilon_{0}\delta=r$. Consequently any pair $x,z$ with $\lVert x-z\rVert\le\varepsilon\delta$ also satisfies $\lVert x-z\rVert\le r$, hence $\lVert x-z\rVert\le r_{0}$ and $\lVert x-z\rVert\le r_{i}$ for every $i$, so the estimates of the previous paragraph hold at the scale $\varepsilon\delta$.

The rescaled kernel $\rho_{\varepsilon}$ is a mollifier kernel of radius $\varepsilon\delta$, hence smooth and vanishing at every $y$ with $\varepsilon\delta<\lVert y\rVert$ by Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$; so the hypotheses placed on $\psi$ in this lemma hold for $\rho_{\varepsilon}$ with the radius $\varepsilon\delta$ in place of $R$, and claim 1 applies to it. By Properties of Periodic Convolution on the Torus (v), applied with the mollifier kernel $\rho_{\varepsilon}$ of radius $\varepsilon\delta$, the function $w=\varphi$ and the bound $\eta$,

$$\bigl|\bigl(\rho_{\varepsilon}\star(\varphi|_{Q})\bigr)(x)-\varphi(x)\bigr|\le\eta\qquad\text{for every }x\in\mathbb{R}^{n},$$

which is the first estimate of claim 2. Applying the same claim with $w=\partial_{i}\varphi$ gives

$$\bigl|\bigl(\rho_{\varepsilon}\star((\partial_{i}\varphi)|_{Q})\bigr)(x)-\partial_{i}\varphi(x)\bigr|\le\eta\qquad\text{for every }x\in\mathbb{R}^{n},$$

and claim 1, applied with the kernel $\rho_{\varepsilon}$, identifies $\rho_{\varepsilon}\star((\partial_{i}\varphi)|_{Q})$ with $\partial_{i}(\rho_{\varepsilon}\star(\varphi|_{Q}))$. This is the second estimate of claim 2, and $\varepsilon_{0}$ was chosen independently of $x$ and $i$. $\blacksquare$