Proof of The Lebesgue Bound for Periodic Convolution and Convergence of Mollifications

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. Let $\Psi$ be the periodised kernel of $\psi$, and for $x,y\in\mathbb{R}^{n}$ let $\Psi_{x}$ and $\Psi^{y}$ be the restrictions to $Q$ of $\Psi(x,\cdot)$ and of $\Psi(\cdot,y)$, each in $\mathcal{L}^{1}(\mathbb{T}^{n})$ with integral of absolute value at most $K$ by that clause. We use repeatedly that for an integrable $g$ one has $-|g|\le g\le|g|$ pointwise by claim 3 of Properties of the Absolute Value in an Ordered Field, hence by 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,

$$\Bigl|\int g\,d\mu\Bigr|\le\int|g|\,d\mu. \tag{$\ast$}$$

We also use that $\lambda_{Q}(Q)=1$, from The Flat Torus: Standing Notation (iii).

Step 0. A measurable nonnegative integrand on the product.

Let $u\in\mathcal{L}^{p}(\mathbb{T}^{n})$. By The Periodic Extension of a Function on the Unit Cell (iv) the map $u$ lies in $\mathcal{L}^{1}(\mathbb{T}^{n})$, so The Periodised Kernel of a Periodic Convolution (v) applies and gives

$$(\psi\star u)(x)=\int_{\mathbb{T}^{n}}\Psi_{x}\,u\,dy\qquad(x\in\mathbb{R}^{n}).$$

The second projection $(x,y)\mapsto y$ from $Q\times Q$ to $Q$ is measurable with respect to $\mathcal{B}_{Q}\otimes\mathcal{B}_{Q}$ and $\mathcal{B}_{Q}$, since the preimage of $A\in\mathcal{B}_{Q}$ is the measurable rectangle $Q\times A$, a member of the product $\sigma$-algebra by Measurable Rectangles and the Product σ-Algebra. Hence $(x,y)\mapsto u(y)$ is measurable with respect to $\mathcal{B}_{Q}\otimes\mathcal{B}_{Q}$. The restriction of $\Psi$ to $Q\times Q$ is measurable with respect to the same $\sigma$-algebra by The Periodised Kernel of a Periodic Convolution (vi). Consequently, by Arithmetic of Measurable Real- and [0,∞]-Valued Maps and claim 4 of that lemma, the maps

$$(x,y)\mapsto|\Psi(x,y)|\,|u(y)|\qquad\text{and}\qquad(x,y)\mapsto|\Psi(x,y)|\,\bigl(|u(y)|\bigr)^{p}$$

are measurable with respect to $\mathcal{B}_{Q}\otimes\mathcal{B}_{Q}$ and take values in $[0,\infty)$, the second using Power-Integrable Functions and the p-Seminorm (i) for measurability of $|u|^{p}$. The measure $\lambda_{Q}$ is finite, hence $\sigma$-finite, so Tonelli's and Fubini's Theorems applies to $(Q,\mathcal{B}_{Q},\lambda_{Q})$ in both variables.

Step 1. Proof of claim 1.

That $(\psi\star u)|_{Q}$ lies in $\mathcal{L}^{p}(\mathbb{T}^{n})$ is Properties of Periodic Convolution on the Torus (iv). It remains to prove the inequality. By the representation of step 0, by $(\ast)$, and by claim 4 of Properties of the Absolute Value in an Ordered Field,

$$|(\psi\star u)(x)|\le\int_{\mathbb{T}^{n}}|\Psi(x,y)|\,|u(y)|\,dy\qquad(x\in Q). \tag{1}$$

The case $p=1$. Integrating (1) over $Q$, using claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion for monotonicity and then the Tonelli identity of Tonelli's and Fubini's Theorems to exchange the order,

$$\int_{\mathbb{T}^{n}}\bigl|(\psi\star u)|_{Q}\bigr|\,dx\le\int_{Q}\Bigl(\int_{Q}|\Psi(x,y)|\,|u(y)|\,d\lambda_{Q}(y)\Bigr)d\lambda_{Q}(x)=\int_{Q}|u(y)|\Bigl(\int_{Q}|\Psi^{y}|\,d\lambda_{Q}\Bigr)d\lambda_{Q}(y).$$

By The Periodised Kernel of a Periodic Convolution (iv) the inner integral is at most $K$, so by monotonicity again the right-hand side is at most $K\lVert u\rVert_{1}$. Since $\lVert u\rVert_{1}=\int_{Q}|u|\,d\lambda_{Q}$ by Power-Integrable Functions and the p-Seminorm (iii) and Properties of Real Powers (ii), this is the asserted inequality for $p=1$.

The case $1<p$. Let $q$ be the conjugate exponent of $p$, so $1/p+1/q=1$. Fix $x\in Q$ and put, on $Q$,

$$a=|\Psi_{x}|^{1/q},\qquad b=|\Psi_{x}|^{1/p}\,|u| .$$

By Properties of Real Powers (vi) we have $a^{q}=|\Psi_{x}|$ and $b^{p}=|\Psi_{x}|\,|u|^{p}$, the latter also using Properties of Real Powers (iii). Both $a$ and $b$ are measurable by Power-Integrable Functions and the p-Seminorm (i) and Arithmetic of Measurable Real- and [0,∞]-Valued Maps. Now $\int_{Q}a^{q}\,d\lambda_{Q}=\int_{Q}|\Psi_{x}|\,d\lambda_{Q}\le K<\infty$, so $a\in\mathcal{L}^{q}(\mathbb{T}^{n})$ with $\lVert a\rVert_{q}\le K^{1/q}$ by Power-Integrable Functions and the p-Seminorm (iii) and Properties of Real Powers (v); and $\Psi(x,\cdot)$ is bounded, say by $M'$, being a member of $C_{\mathrm{per}}$ by The Periodised Kernel of a Periodic Convolution (iii) and bounded by Elementary Properties of Lattice-Periodic Functions (iii), so $\int_{Q}b^{p}\,d\lambda_{Q}\le M'\int_{Q}|u|^{p}\,d\lambda_{Q}<\infty$ and $b\in\mathcal{L}^{p}(\mathbb{T}^{n})$.

The pointwise product satisfies $a\,b=|\Psi_{x}|^{1/q}|\Psi_{x}|^{1/p}|u|=|\Psi_{x}|\,|u|$, by Properties of Real Powers (iv) with $1/q+1/p=1$, followed by Properties of Real Powers (ii), which gives $t^{1}=t$ for nonnegative $t$. Applying Hölder's Inequality (i) to the conjugate pair $q,p$ and the functions $a\in\mathcal{L}^{q}$, $b\in\mathcal{L}^{p}$,

$$\int_{Q}|\Psi_{x}|\,|u|\,d\lambda_{Q}=\lVert a\,b\rVert_{1}\le\lVert a\rVert_{q}\,\lVert b\rVert_{p}\le K^{1/q}\Bigl(\int_{Q}|\Psi_{x}|\,|u|^{p}\,d\lambda_{Q}\Bigr)^{1/p}.$$

Combining with (1) and raising to the power $p$, which preserves the inequality by Properties of Real Powers (v), and using Properties of Real Powers (iii) and Properties of Real Powers (vi),

$$\bigl(|(\psi\star u)(x)|\bigr)^{p}\le K^{p/q}\int_{Q}|\Psi(x,y)|\,|u(y)|^{p}\,d\lambda_{Q}(y)\qquad(x\in Q).$$

Integrating over $x\in Q$, using monotonicity and then the Tonelli identity of Tonelli's and Fubini's Theorems and The Periodised Kernel of a Periodic Convolution (iv) as in the case $p=1$,

$$\int_{Q}\bigl|(\psi\star u)|_{Q}\bigr|^{p}\,d\lambda_{Q}\le K^{p/q}\int_{Q}|u(y)|^{p}\Bigl(\int_{Q}|\Psi^{y}|\,d\lambda_{Q}\Bigr)d\lambda_{Q}(y)\le K^{p/q}\cdot K\cdot\bigl(\lVert u\rVert_{p}\bigr)^{p}.$$

Since $p/q+1=p(1-1/p)+1=p$, Properties of Real Powers (ii), which identifies $K$ with $K^{1}$, together with Properties of Real Powers (iv) gives $K^{p/q}\cdot K=K^{p}$, so the right-hand side is $\bigl(K\lVert u\rVert_{p}\bigr)^{p}$ by Properties of Real Powers (iii). Taking the power $1/p$ of both sides, which preserves the inequality by Properties of Real Powers (v), and using Power-Integrable Functions and the p-Seminorm (iii) together with Properties of Real Powers (vi), we obtain $\lVert(\psi\star u)|_{Q}\rVert_{p}\le K\lVert u\rVert_{p}$.

Step 2. Proof of claim 2.

By Rescaling a Mollifier Kernel the map $\rho_{\varepsilon}$ is a mollifier kernel of radius $\varepsilon\delta$ on $\mathbb{R}^{n}$. By the nonnegativity and unit-mass conditions of Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$, together with claim 1 of Properties of the Absolute Value in an Ordered Field, the constant $K$ attached to the kernel $\rho_{\varepsilon}$ by The Periodised Kernel of a Periodic Convolution (i) equals $\int_{\mathbb{R}^{n}}\rho_{\varepsilon}\,d\lambda_{n}=1$.

Let $\eta\in\mathbb{R}$ with $0<\eta$. The numbers $\eta/3$ and $\eta/2$ are positive by claim 8 of Elementary Order Arithmetic in an Ordered Field and claim 7 of the same lemma. By Continuous Periodic Functions are Power-Integrable and Dense on the Torus (iv) the set of classes $[w|_{Q}]$ with $w\in C_{\mathrm{per}}$ is dense in $L^{p}(\mathbb{T}^{n})$, so by Characterization of the Closure in a Metric Space by Open Balls the open ball of centre $[u]$ and radius $\eta/3$ in $L^{p}(\mathbb{T}^{n})$ meets it: there is $w\in C_{\mathrm{per}}$ with $\lVert[w|_{Q}]-[u]\rVert_{L^{p}(\mathbb{T}^{n})}<\eta/3$, that is, $\lVert w|_{Q}-u\rVert_{p}<\eta/3$ by The Lebesgue Space Lᵖ(μ) (iii).

By A Continuous Lattice-Periodic Function is Uniformly Continuous (i), applied to $w$ with the tolerance $\eta/3$, there is a real $r$ with $0<r$ such that $|w(x)-w(z)|\le\eta/3$ whenever $\lVert x-z\rVert\le r$. Put $\varepsilon_{0}=r/\delta$, a positive real by claim 7 of Elementary Order Arithmetic in an Ordered Field, and let $\varepsilon$ be real with $0<\varepsilon<\varepsilon_{0}$, so that $\varepsilon\delta<r$.

By Minkowski's Inequality (i), applied twice,

$$\bigl\lVert(\rho_{\varepsilon}\star u)|_{Q}-u\bigr\rVert_{p}\le \bigl\lVert(\rho_{\varepsilon}\star u)|_{Q}-\bigl(\rho_{\varepsilon}\star(w|_{Q})\bigr)|_{Q}\bigr\rVert_{p} +\bigl\lVert\bigl(\rho_{\varepsilon}\star(w|_{Q})\bigr)|_{Q}-w|_{Q}\bigr\rVert_{p} +\bigl\lVert w|_{Q}-u\bigr\rVert_{p}.$$

For the first summand, $w|_{Q}\in\mathcal{L}^{p}(\mathbb{T}^{n})$ by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i), and Properties of Periodic Convolution on the Torus (iii) with the scalar $-1$ gives $\rho_{\varepsilon}\star u-\rho_{\varepsilon}\star(w|_{Q})=\rho_{\varepsilon}\star(u-w|_{Q})$; by claim 1 of this theorem, applied to the kernel $\rho_{\varepsilon}$ whose constant is $1$, that summand is at most $\lVert u-w|_{Q}\rVert_{p}<\eta/3$.

For the second summand, Properties of Periodic Convolution on the Torus (v), applied to the mollifier kernel $\rho_{\varepsilon}$ of radius $\varepsilon\delta$ and to $w$ with the bound $\eta/3$ — legitimate because $\lVert x-z\rVert\le\varepsilon\delta$ implies $\lVert x-z\rVert\le r$ and hence $|w(x)-w(z)|\le\eta/3$ — gives

$$\bigl|\bigl(\rho_{\varepsilon}\star(w|_{Q})\bigr)(x)-w(x)\bigr|\le\eta/3\qquad(x\in\mathbb{R}^{n}).$$

Writing $t=\eta/3$, the difference $h=\bigl(\rho_{\varepsilon}\star(w|_{Q})\bigr)|_{Q}-w|_{Q}$ therefore satisfies $|h|\le t$ on $Q$, so $|h|^{p}\le t^{p}$ by Properties of Real Powers (v) and

$$\int_{Q}|h|^{p}\,d\lambda_{Q}\le t^{p}\lambda_{Q}(Q)=t^{p}$$

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; hence $\lVert h\rVert_{p}\le(t^{p})^{1/p}=t=\eta/3$ by Power-Integrable Functions and the p-Seminorm (iii), Properties of Real Powers (v) and Properties of Real Powers (vi).

The third summand is less than $\eta/3$. Adding the three bounds gives $\lVert(\rho_{\varepsilon}\star u)|_{Q}-u\rVert_{p}\le\eta$, as asserted.

For the final sentence, let $\eta$ be a positive real and let $\varepsilon_{0}$ be as above. By The Archimedean Property of the Real Numbers there is a natural number $k_{0}$ with $1/\varepsilon_{0}<k_{0}$, hence $1/k_{0}<\varepsilon_{0}$ by claim 7 of Elementary Order Arithmetic in an Ordered Field. For every natural $k$ with $k_{0}\le k$ we have $0<1/k\le1/k_{0}<\varepsilon_{0}$, so

$$\bigl\lVert[(\rho_{1/k}\star u)|_{Q}]-[u]\bigr\rVert_{L^{p}(\mathbb{T}^{n})}=\bigl\lVert(\rho_{1/k}\star u)|_{Q}-u\bigr\rVert_{p}\le\eta ,$$

using The Lebesgue Space Lᵖ(μ) (iii). As $\eta$ was an arbitrary positive real, the sequence of these classes converges to $[u]$ in $L^{p}(\mathbb{T}^{n})$.

Step 3. Proof of claim 3.

By Existence of Mollifier Kernels of Every Radius there is a mollifier kernel $\rho$ of radius $1$ on $\mathbb{R}^{n}$. Let $c\in L^{p}(\mathbb{T}^{n})$ and let $\theta$ be a positive real; choose a representative $u\in\mathcal{L}^{p}(\mathbb{T}^{n})$ with $[u]=c$, which exists by The Lebesgue Space Lᵖ(μ) (i). By claim 2, applied with $\eta=\theta/2$, there is a positive real $\varepsilon$ with $\lVert(\rho_{\varepsilon}\star u)|_{Q}-u\rVert_{p}\le\theta/2<\theta$. The kernel $\rho_{\varepsilon}$ is smooth by Mollifier Kernel of Radius $\delta$ on $\mathbb{R}^n$ and Rescaling a Mollifier Kernel, so $\rho_{\varepsilon}\star u\in C^{\infty}_{\mathrm{per}}$ by Properties of Periodic Convolution on the Torus (i). Hence the open ball of centre $c$ and radius $\theta$ in $L^{p}(\mathbb{T}^{n})$ contains the class $[(\rho_{\varepsilon}\star u)|_{Q}]$, a member of $\{[w|_{Q}]:w\in C^{\infty}_{\mathrm{per}}\}$. As $c$ and $\theta$ were arbitrary, Characterization of the Closure in a Metric Space by Open Balls gives that the closure of that set is all of $L^{p}(\mathbb{T}^{n})$, that is, the set is dense. $\blacksquare$