Proof of Properties of Periodic Convolution on the Torus

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 theorem. Throughout, $\tilde{u}$ is the periodic extension of $u$, and by Periodic Convolution and Mollification on the Torus (i) the map $\psi\star u$ is the convolution of $\psi$ with $\tilde{u}$ in the sense of that lemma; as recorded in Periodic Convolution and Mollification on the Torus, the pair $(\psi,\tilde{u})$ satisfies the hypotheses of that lemma, so all of its claims are available here. We write $\bar{B}(z,r)$ for the closed ball in $\mathbb{R}^{n}$ and use that the Euclidean distance satisfies $d_{E}(a,b)=\lVert a-b\rVert$, 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.

Step 1. Proof of claim 1.

By Convolution of a Locally Integrable Function with a Compactly Supported Kernel (ii) the map $\psi\star u$ is continuous on $\mathbb{R}^{n}$.

Periodicity. Let $x\in\mathbb{R}^{n}$ and $m\in\mathbb{Z}^{n}$, and let $g:\mathbb{R}^{n}\to\mathbb{R}$ be given by $g(y)=\psi(x+m-y)\,\tilde{u}(y)$, which is measurable and integrable by Convolution of a Locally Integrable Function with a Compactly Supported Kernel (i) applied at the point $x+m$. By claim 3 of Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ, applied with $a=m$, the map $z\mapsto g(z+m)$ is integrable and

$$\int_{\mathbb{R}^{n}}g(z+m)\,d\lambda_{n}(z)=\int_{\mathbb{R}^{n}}g\,d\lambda_{n}=(\psi\star u)(x+m).$$

For every $z$ we have $g(z+m)=\psi(x-z)\,\tilde{u}(z+m)=\psi(x-z)\,\tilde{u}(z)$, the second equality because $\tilde{u}$ is $\mathbb{Z}^{n}$-periodic by The Periodic Extension of a Function on the Unit Cell (i). Hence the left-hand side is $(\psi\star u)(x)$, and $(\psi\star u)(x+m)=(\psi\star u)(x)$. So $\psi\star u$ is $\mathbb{Z}^{n}$-periodic, and being continuous it lies in $C_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes (iii).

Smoothness. Let $k$ be a natural number and let $\psi$ be of class $C^{k}$ on $\mathbb{R}^{n}$; then $\psi$ is of class $C^{1}$ by claim 2 of Euclidean Space Is Open in Itself. By Convolution of a Locally Integrable Function with a Compactly Supported Kernel (iv) each $\partial_{i}\psi$ is continuous and vanishes at every $y$ with $R<\lVert y\rVert$, so $\partial_{i}\psi$ may serve as the kernel in Periodic Convolution and Mollification on the Torus (i) and $(\partial_{i}\psi)\star u$ is the periodic convolution of $\partial_{i}\psi$ with $u$; the same clause gives

$$\partial_{i}(\psi\star u)=(\partial_{i}\psi)\star u\qquad(i\in[n]).$$

By Convolution of a Locally Integrable Function with a Compactly Supported Kernel (v) the map $\psi\star u$ is of class $C^{k}$ on $\mathbb{R}^{n}$; it is $\mathbb{Z}^{n}$-periodic by the previous paragraph, hence lies in $C^{k}_{\mathrm{per}}$ by Lattice-Periodic Functions and the Periodic Function Classes (iii). If $\psi$ is smooth then $\psi\star u$ is smooth by the same clause, and therefore lies in $C^{\infty}_{\mathrm{per}}$.

Step 2. Proof of claim 2.

Suppose $[u']=[u]$ in $L^{p}(\mathbb{T}^{n})$, that is, $u=u'$ $\lambda_{Q}$-almost everywhere on $Q$ by The Lebesgue Space Lᵖ(μ) (i). Both $u$ and $u'$ are measurable with respect to $\mathcal{B}_{Q}$ by Power-Integrable Functions and the p-Seminorm (ii), so The Periodic Extension of a Function on the Unit Cell (iii) gives $\tilde{u}=\tilde{u}'$ $\lambda_{n}$-almost everywhere on $\mathbb{R}^{n}$. Fix $x\in\mathbb{R}^{n}$. Then $\psi(x-y)\tilde{u}(y)=\psi(x-y)\tilde{u}'(y)$ for almost every $y$, and both maps are measurable and integrable by Convolution of a Locally Integrable Function with a Compactly Supported Kernel (i). By the last assertion of The Lebesgue Integral and Null Sets (iii) their integrals coincide, that is, $(\psi\star u)(x)=(\psi\star u')(x)$. As $x$ was arbitrary, claim 2 follows.

Step 3. Proof of claim 3.

By Minkowski's Inequality (ii) the set $\mathcal{L}^{p}(\mathbb{T}^{n})$ is a linear subspace of the space of real-valued maps on $Q$, so $u+c\,u'\in\mathcal{L}^{p}(\mathbb{T}^{n})$. Its periodic extension is $\tilde{u}+c\,\tilde{u}'$ by The Periodic Extension of a Function on the Unit Cell (ii). The map $\psi+c\,\psi'$ is continuous by claim 5 of Continuity of Sums and Products of Real-Valued Functions on a Metric Space and vanishes at every $y$ with $R<\lVert y\rVert$, since $\psi$ and $\psi'$ both do. Both identities of claim 3 are now exactly the two identities of Convolution of a Locally Integrable Function with a Compactly Supported Kernel (iii), read with $w=\tilde{u}$ and $w'=\tilde{u}'$.

Step 4. Proof of claim 4.

The kernel is bounded. Put $S=\{y\in\mathbb{R}^{n}:\psi(y)\ne0\}$. If $R<\lVert y\rVert$ then $\psi(y)=0$, so $S\subseteq\bar{B}(0,R)$. The support of $\psi$ is the closure of $S$, that is, the intersection of all closed sets containing $S$; since $\bar{B}(0,R)$ is closed and bounded by Elementary Properties of the Closed Ball in a Metric Space, that support is closed and bounded, hence compact by The Heine-Borel Theorem in $\mathbb{R}^n$, and Compactness of Closed Balls and the Unit Sphere, so $\psi$ is compactly supported and therefore bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable. Fix a real $M$ with $0\le M$ and $|\psi(y)|\le M$ for every $y$.

A finite index set. By The Archimedean Property of the Real Numbers choose a natural number $N$ with $R+1\le N$, and put

$$G=\{k\in\mathbb{Z}^{n}:-N\le k_{i}\le N\ \text{for every}\ i\in[n]\}.$$

Let $J=\{j\in\mathbb{Z}:-N\le j\le N\}$. The initial segment $[2N+1]$ is finite by claim 1 of Basic Properties of Finite Sets, and the map sending $l\in[2N+1]$ to $l-N-1$ is onto $J$: its values are integers by claim 2 of Arithmetic, Order, Discreteness and Intervals of the Integers and satisfy $-N\le l-N-1\le N$ because $1\le l\le 2N+1$, while a given $j\in J$ is the value at $l=j+N+1$, which lies in $[2N+1]$ and is a natural number by claim 1 of Arithmetic, Order, Discreteness and Intervals of the Integers. So $J$ is finite by claim 4 of Basic Properties of Finite Sets, and $G$, the set of $n$-tuples with all entries in $J$, is finite by claim 3 of Finiteness of Cartesian Products, Tuple Sets, and Permutation Sets. Let $c$ be its number of elements.

The covering. Fix $x\in\mathbb{R}^{n}$ and let $a\in\mathbb{Z}^{n}$ have $i$th coordinate $\lfloor x_{i}\rfloor$, the integer part. Let $y\in\bar{B}(x,R)$ and let $m\in\mathbb{Z}^{n}$ be the unique lattice vector with $y-m\in Q$, given by The Half-Open Unit Cell Tiles Euclidean Space (ii); by The Half-Open Unit Cell Tiles Euclidean Space (iii), $m_{i}=\lfloor y_{i}\rfloor$ for every $i$. By claim 4 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square we have $|y_{i}-x_{i}|\le\lVert y-x\rVert\le R$. From $\lfloor y_{i}\rfloor\le y_{i}<\lfloor y_{i}\rfloor+1$ and the same for $x_{i}$ we get

$$m_{i}-a_{i}=\lfloor y_{i}\rfloor-\lfloor x_{i}\rfloor<y_{i}-(x_{i}-1)\le R+1\le N,\qquad m_{i}-a_{i}>(y_{i}-1)-x_{i}\ge-R-1\ge-N,$$

so $k=m-a$ lies in $G$ and $y\in Q+m=Q+(a+k)$. Hence $\bar{B}(x,R)\subseteq\bigcup_{k\in G}\bigl(Q+(a+k)\bigr)$.

The bound. The map $|\tilde{u}|$ is the periodic extension of $|u|$, and $|u|$ is integrable with respect to $\lambda_{Q}$ by The Periodic Extension of a Function on the Unit Cell (iv) together with Integrable Functions and Their Lebesgue Integral. By The Periodic Extension of a Function on the Unit Cell (v), applied to $|u|$ with $h=a+k$,

$$\int_{\mathbb{R}^{n}}\mathbf{1}_{Q+(a+k)}\,|\tilde{u}|\,d\lambda_{n}=\int_{\mathbb{T}^{n}}|u|\,dx=\lVert u\rVert_{1}\qquad(k\in G).$$

If $y\notin\bar{B}(x,R)$ then $R<\lVert x-y\rVert$ and $\psi(x-y)=0$; and by the covering, for $y\in\bar{B}(x,R)$ we have $\mathbf{1}_{Q+(a+k)}(y)=1$ for at least one $k\in G$. Since all terms are nonnegative, it follows that

$$|\psi(x-y)|\,|\tilde{u}(y)|\le M\sum_{k\in G}\mathbf{1}_{Q+(a+k)}(y)\,|\tilde{u}(y)|\qquad\text{for every }y\in\mathbb{R}^{n}.$$

Integrating and using claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion for monotonicity and for the finitely many summands,

$$\int_{\mathbb{R}^{n}}|\psi(x-y)\,\tilde{u}(y)|\,d\lambda_{n}(y)\le M\,c\,\lVert u\rVert_{1}\le M\,c\,\lVert u\rVert_{p},$$

the last step by The Periodic Extension of a Function on the Unit Cell (iv). Finally, for an integrable $g$ we have $-|g|\le g\le|g|$ pointwise by claim 3 of Properties of the Absolute Value in an Ordered Field, so claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and claim 6 of Properties of the Absolute Value in an Ordered Field give $\bigl|\int g\,d\lambda_{n}\bigr|\le\int|g|\,d\lambda_{n}$; applied to $g(y)=\psi(x-y)\tilde{u}(y)$ this yields

$$|(\psi\star u)(x)|\le M\,c\,\lVert u\rVert_{p}.$$

Put $C=M\,c$, a nonnegative real number determined by $n$, $R$ and $\psi$ alone. As $x$ was arbitrary, the first assertion of claim 4 holds.

Consequence. The map $\psi\star u$ is continuous by claim 1, hence measurable with respect to $\mathcal{B}(\mathbb{R}^{n})$ by claims 3(a) and 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, so for every Borel subset $A$ of $\mathbb{R}$ the set $\{y\in Q:(\psi\star u)(y)\in A\}$ equals the intersection of $\{x\in\mathbb{R}^{n}:(\psi\star u)(x)\in A\}\in\mathcal{B}(\mathbb{R}^{n})$ with $Q\in\mathcal{B}(\mathbb{R}^{n})$; that intersection lies in $\mathcal{B}(\mathbb{R}^{n})$ and is contained in $Q$, hence lies in $\mathcal{B}_{Q}$. So the restriction $(\psi\star u)|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$. Writing $t=C\lVert u\rVert_{p}$, we have $|(\psi\star u)|_{Q}|\le t$ pointwise on $Q$, hence $\bigl||(\psi\star u)|_{Q}|\bigr|^{p}\le t^{p}$ by Properties of Real Powers (v), and therefore

$$\int_{Q}\bigl|(\psi\star u)|_{Q}\bigr|^{p}\,d\lambda_{Q}\le t^{p}\,\lambda_{Q}(Q)=t^{p}<\infty$$

by claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion and The Integral of an Indicator Function is the Measure of the Set, using $\lambda_{Q}(Q)=1$ from The Flat Torus: Standing Notation (iii). So $(\psi\star u)|_{Q}\in\mathcal{L}^{p}(\mathbb{T}^{n})$ by Power-Integrable Functions and the p-Seminorm (ii), and by Power-Integrable Functions and the p-Seminorm (iii) together with Properties of Real Powers (v) and Properties of Real Powers (vi),

$$\bigl\lVert(\psi\star u)|_{Q}\bigr\rVert_{p}\le(t^{p})^{1/p}=t=C\,\lVert u\rVert_{p}.$$

Step 5. Proof of claim 5.

Let $\rho$ be a mollifier kernel of radius $\delta$ and let $w\in C_{\mathrm{per}}$, so that $w|_{Q}\in\mathcal{L}^{p}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i) and the periodic extension of $w|_{Q}$ is $w$ itself by The Periodic Extension of a Function on the Unit Cell (i). Fix $x\in\mathbb{R}^{n}$.

By claim 3 of Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ, applied with $a=x$ to the integrable map $\rho$ of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$, the map $y\mapsto\rho(x-y)$ is integrable with

$$\int_{\mathbb{R}^{n}}\rho(x-y)\,d\lambda_{n}(y)=\int_{\mathbb{R}^{n}}\rho\,d\lambda_{n}=1,$$

the last equality being the unit-mass condition of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$. The map $y\mapsto\rho(x-y)w(y)$ is integrable by Convolution of a Locally Integrable Function with a Compactly Supported Kernel (i), so by claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion,

$$\bigl(\rho\star(w|_{Q})\bigr)(x)-w(x)=\int_{\mathbb{R}^{n}}\rho(x-y)\,w(y)\,d\lambda_{n}(y)-w(x)\int_{\mathbb{R}^{n}}\rho(x-y)\,d\lambda_{n}(y)=\int_{\mathbb{R}^{n}}\rho(x-y)\bigl(w(y)-w(x)\bigr)\,d\lambda_{n}(y).$$

Let $y\in\mathbb{R}^{n}$. If $\delta<\lVert x-y\rVert$ then $\rho(x-y)=0$ by the support condition of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$, so the integrand vanishes; otherwise $\lVert x-y\rVert\le\delta$ and $|w(y)-w(x)|\le\eta$ by hypothesis. Since $0\le\rho(x-y)$ by the nonnegativity condition of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$, in both cases

$$\bigl|\rho(x-y)\bigl(w(y)-w(x)\bigr)\bigr|=\rho(x-y)\,|w(y)-w(x)|\le\eta\,\rho(x-y),$$

using claims 1 and 4 of Properties of the Absolute Value in an Ordered Field, the first giving $|\rho(x-y)|=\rho(x-y)$ from nonnegativity of the kernel, and claim 5 of Elementary Arithmetic in an Ordered Field. Applying the bound $\bigl|\int g\bigr|\le\int|g|$ established in step 4, then claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion for monotonicity and the scalar $\eta$,

$$\Bigl|\bigl(\rho\star(w|_{Q})\bigr)(x)-w(x)\Bigr|\le\int_{\mathbb{R}^{n}}\eta\,\rho(x-y)\,d\lambda_{n}(y)=\eta .$$

As $x$ was arbitrary, claim 5 follows. $\blacksquare$