The Lebesgue Bound for Periodic Convolution and Convergence of Mollifications

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$ and a real number $p$ with $1\le p$; the cell $Q$, the measure space $(Q,\mathcal{B}_{Q},\lambda_{Q})$, the classes $\mathcal{L}^{p}(\mathbb{T}^{n})$ and $L^{p}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic classes $C_{\mathrm{per}}$ and $C^{\infty}_{\mathrm{per}}$, the restriction $v|_{Q}$, and $\lambda_{n}$ and the Euclidean norm $\lVert\,\cdot\,\rVert$ are the ones fixed there. Let $\lVert\,\cdot\,\rVert_{r}$ denote the $L^{r}$ seminorm of $(Q,\mathcal{B}_{Q},\lambda_{Q})$ for a real number $r$ with $1\le r$, and let $\psi\star u$ denote the periodic convolution. Continuity and smoothness of a map on $\mathbb{R}^{n}$ are read as in Periodic Convolution and Mollification on the Torus.

Throughout, $R$ is a real number with $0<R$, the map $\psi:\mathbb{R}^{n}\to\mathbb{R}$ is continuous with $\psi(y)=0$ for every $y\in\mathbb{R}^{n}$ satisfying $R<\lVert y\rVert$, and $K$ denotes the integral of $|\psi|$ over $\mathbb{R}^{n}$, a nonnegative real number by that clause. Then the following hold.

1. (The Lebesgue bound)(i) Let $u\in\mathcal{L}^{p}(\mathbb{T}^{n})$. Then $(\psi\star u)|_{Q}\in\mathcal{L}^{p}(\mathbb{T}^{n})$ and

$$\bigl\lVert(\psi\star u)|_{Q}\bigr\rVert_{p}\le K\,\lVert u\rVert_{p}.$$

2. (Convergence of the mollifications)(ii) Let $\delta\in\mathbb{R}$ with $0<\delta$, let $\rho$ be a mollifier kernel of radius $\delta$ on $\mathbb{R}^{n}$, let $u\in\mathcal{L}^{p}(\mathbb{T}^{n})$, and for a real number $\varepsilon$ with $0<\varepsilon$ let $\rho_{\varepsilon}\star u$ be the mollification of $u$ of parameter $\varepsilon$ by $\rho$. Then for every real number $\eta$ with $0<\eta$ there is a real number $\varepsilon_{0}$ with $0<\varepsilon_{0}$ such that

$$\bigl\lVert(\rho_{\varepsilon}\star u)|_{Q}-u\bigr\rVert_{p}\le\eta\qquad\text{for every real }\varepsilon\text{ with }0<\varepsilon<\varepsilon_{0}.$$

Consequently the sequence whose $k$th term is the class of $(\rho_{1/k}\star u)|_{Q}$, for $k$ a natural number, converges to $[u]$ in $L^{p}(\mathbb{T}^{n})$.

3. (Smooth periodic functions are dense)(iii) The set $\{[w|_{Q}]:w\in C^{\infty}_{\mathrm{per}}\}$ is dense in $L^{p}(\mathbb{T}^{n})$.