The Convolution Integrand is Continuous, Compactly Supported and Integrable

Let $n\ge1$ be a natural number, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^n$, let $d$ denote the Euclidean distance, a metric on each Euclidean space, and let $\lambda_n$ be Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$. Write $\bar B(x,r)$ for the closed ball in $(\mathbb{R}^n,d)$.

Let $\Omega\subseteq\mathbb{R}^n$ be open in $(\mathbb{R}^n,d)$ and let $f:\Omega\to\mathbb{R}$ be continuous on $\Omega$ as a map from $(\mathbb{R}^n,d)$ to $(\mathbb{R},d)$. Let $\delta>0$ be a real number and let $\rho:\mathbb{R}^n\to\mathbb{R}$ be continuous on $\mathbb{R}^n$ with $\rho(y)=0$ for every $y\in\mathbb{R}^n$ with $\lVert y\rVert>\delta$.

Let $x\in\mathbb{R}^n$ satisfy $\bar B(x,\delta)\subseteq\Omega$, and let $h_x:\mathbb{R}^n\to\mathbb{R}$ be defined by $h_x(y)=f(x-y)\rho(y)$ for those $y\in\mathbb{R}^n$ with $x-y\in\Omega$, and $h_x(y)=0$ for all other $y$, the difference $x-y$ being that of Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$.

1. (Regularity) $h_x$ is continuous on $\mathbb{R}^n$ and compactly supported; consequently $h_x$ is bounded, measurable with respect to $\mathcal{B}(\mathbb{R}^n)$ and the Borel $\sigma$-algebra of the real line, and integrable with respect to $\lambda_n$.

2. (Bound) The function $\rho$ is integrable with respect to $\lambda_n$, so that $\int_{\mathbb{R}^n}|\rho|\,d\lambda_n$ is a real number; and if $M\ge0$ is a real number with $|f(z)|\le M$ for every $z\in\bar B(x,\delta)$, then $|h_x(y)|\le M|\rho(y)|$ for every $y\in\mathbb{R}^n$ and

$$\Bigl|\int_{\mathbb{R}^n}h_x\,d\lambda_n\Bigr|\le M\int_{\mathbb{R}^n}|\rho|\,d\lambda_n.$$