One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative

In the setting of Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation, in dimension $d=1$. As in Borel Sigma-Algebra on Euclidean Space and the preamble of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets, the real line $\mathbb{R}$ is identified with the Euclidean space $\mathbb{R}^{1}$, a point of $\mathbb{R}^{1}$ being read as its sole coordinate, so that $\mathcal{B}(\mathbb{R}^{1})=\mathcal{B}(\mathbb{R})$, $\mathcal{P}(\mathbb{R}^{1})=\mathcal{P}(\mathbb{R})$, $\mathcal{P}_{2}(\mathbb{R}^{1})=\mathcal{P}_{2}(\mathbb{R})$, $L^{2}(\Omega;\mathbb{R}^{1})=L^{2}(\Omega;\mathbb{R})$, $\mathbb{R}^{1+1}=\mathbb{R}^{2}$, and the Euclidean distance of $\mathbb{R}^{1}$ is the absolute-value metric $d_{\mathbb{R}}$ by $\mathbb{R}^1$ with the Euclidean Distance Is Isometric to the Real Line. A point of $\mathbb{R}^{2}$ is written $(x,y)$ with $x,y\in\mathbb{R}$, meaning the point $\iota(x,y)$ with the concatenation map $\iota=\iota^{1,1}$, so that $\mu\boxtimes\nu\in\mathcal{P}(\mathbb{R}^{2})$ for $\mu,\nu\in\mathcal{P}(\mathbb{R})$. Differentiability of a function $\mathbb{R}\to\mathbb{R}$ at a point is that of Derivative of a Real Function on an Interval, every real number being an interior point of the interval $\mathbb{R}$ by claim 1 of One-Dimensional Derivatives, Partial Derivatives, and Smoothness on the Real Line; for $f:\mathbb{R}\to\mathbb{R}$ differentiable at every point, $f':\mathbb{R}\to\mathbb{R}$ is its derivative. Continuity of a map from $\mathbb{R}$ or from $\mathbb{R}^{2}$ into $\mathbb{R}$ is continuity for the Euclidean distances, as in Differential Calculus and Convexity on Euclidean Open Sets: Standing Notation (iv); bounded is bounded; and Borel and integrable are as in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (ii) and Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii). For each test function $\psi\in C_{c}^{\infty}(\mathbb{R})$, with gradient map $\nabla\psi$ and Laplacian $\Delta\psi$ as fixed there, $\psi'$ denotes its derivative, which exists by claim 2, and $F_{\psi}:\mathbb{R}^{2}\to\mathbb{R}$ is the function

$$F_{\psi}(x,y)=\frac{\psi'(x)-\psi'(y)}{x-y}\ \ \text{if }x\ne y,\qquad F_{\psi}(x,x)=\Delta\psi(x).$$

1. (Scalars)(i) For all $x,y\in\mathbb{R}=\mathbb{R}^{1}$, $x\cdot y=xy$, $\lVert x\rVert=|x|$ and $\lVert x\rVert^{2}=x^{2}$. Consequently, for $\mu\in\mathcal{P}(\mathbb{R})$, the space $L^{2}(\mu;\mathbb{R})$ consists of the classes of the Borel maps $\xi:\mathbb{R}\to\mathbb{R}$ with $\int_{\mathbb{R}}\xi^{2}\,d\mu<\infty$, and for $\xi,\eta\in L^{2}(\mu;\mathbb{R})$

$$\langle\xi,\eta\rangle_{\mu}=\int_{\mathbb{R}}\xi\,\eta\,d\mu,\qquad \lVert\xi\rVert_{\mu}^{2}=\int_{\mathbb{R}}\xi^{2}\,d\mu .$$

2. (Derivatives)(ii) Every $\psi\in C_{c}^{\infty}(\mathbb{R})$ is differentiable at every point of $\mathbb{R}$ with $\psi'=\partial_{1}\psi=\nabla\psi$; $\psi'$ is differentiable at every point of $\mathbb{R}$ with $(\psi')'=\partial_{1}\partial_{1}\psi=\Delta\psi$; and $\psi'$ and $\Delta\psi$ are continuous and bounded. In particular, for $\mu\in\mathcal{P}(\mathbb{R})$ and $\xi\in L^{2}(\mu;\mathbb{R})$,

$$\langle\xi,\nabla\psi\rangle_{\mu}=\int_{\mathbb{R}}\xi\,\psi'\,d\mu,\qquad \lVert\nabla\psi\rVert_{\mu}^{2}=\int_{\mathbb{R}}(\psi')^{2}\,d\mu .$$

3. (Difference quotient of the derivative)(iii) For every $\psi\in C_{c}^{\infty}(\mathbb{R})$, the function $F_{\psi}$ is continuous and Borel, $F_{\psi}(x,y)=F_{\psi}(y,x)$ for all $x,y\in\mathbb{R}$, and $|F_{\psi}(x,y)|\le L$ for all $x,y\in\mathbb{R}$ whenever $L$ is a real number with $|\Delta\psi(t)|\le L$ for every $t\in\mathbb{R}$; such an $L\ge0$ exists by claim 2. Consequently $F_{\psi}$ is integrable with respect to every $\rho\in\mathcal{P}(\mathbb{R}^{2})$, and $\bigl|\int_{\mathbb{R}^{2}}F_{\psi}\,d\rho\bigr|\le L$ for every such $L$.

4. (Linearity)(iv) For all $\psi,\phi\in C_{c}^{\infty}(\mathbb{R})$ and $a,b\in\mathbb{R}$, the function $a\psi+b\phi$ is a test function by The Gradient of a Test Function is Bounded and Square-Integrable, and Its Laplacian Bounded and Integrable, Against Every Probability Measure (iii), $F_{a\psi+b\phi}(x,y)=a\,F_{\psi}(x,y)+b\,F_{\phi}(x,y)$ for all $x,y\in\mathbb{R}$, and hence, for every $\rho\in\mathcal{P}(\mathbb{R}^{2})$,

$$\int_{\mathbb{R}^{2}}F_{a\psi+b\phi}\,d\rho=a\int_{\mathbb{R}^{2}}F_{\psi}\,d\rho+b\int_{\mathbb{R}^{2}}F_{\phi}\,d\rho .$$