Proof of Splitting a Bounded First Variation into a Square-Integrable Potential Force and a Finite Free Fisher Information

Each result cited below is universally quantified over the data in its own statement, and is applied to the data named at the point of use. The rules of Elementary Order Arithmetic in an Ordered Field, Elementary Arithmetic in an Ordered Field and Properties of the Absolute Value in an Ordered Field for adding, multiplying and scaling inequalities and for absolute values are used without further mention, as are the rules for limits of sums, products and quotients of convergent real sequences and the fact that a non-strict inequality between the terms of convergent real sequences passes to their limits (limits being those of Limit of a Sequence of Real Numbers).

Conventions. As in One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative, a point of $\mathbb{R}^{2}$ is written $(x,y)$, and the coordinate projections $\mathrm{pr}_{1}(x,y)=x$, $\mathrm{pr}_{2}(x,y)=y$ are Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §projections; by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §product the image measures of $\mu\boxtimes\mu$ under $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ are $\mu$. $\lambda_{1}$ is the Lebesgue measure on $\mathcal{B}(\mathbb{R})$. Integrals against $\mu$ and $\mu\boxtimes\mu$ are those of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §measures, where it is recorded that bounded Borel functions are integrable against them. Null means of $\mu\boxtimes\mu$-measure $0$. Dominated convergence always refers to The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §dominated. A function $f:\mathbb{R}\to\mathbb{R}$ is Lipschitz with constant $L$, for a real $L\ge0$, if $|f(x)-f(y)|\le L|x-y|$ for all $x,y$ (Lipschitz Map Between Metric Spaces for the absolute-value metric); it is then continuous by A Lipschitz Map is Uniformly Continuous, hence Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation §borel-maps, and compositions of Borel maps are Borel by the same clause. For $n\in\mathbb{N}$ with $n\ge1$ let $T_{n}(t)=\max(-n,\min(t,n))$; checking the cases $t<-n$, $|t|\le n$, $t>n$ one finds that $T_{n}$ is nondecreasing, Lipschitz with constant $1$, $T_{n}(t)^{2}=\min(t^{2},n^{2})\le t\,T_{n}(t)$, and $|T_{n}(t)|\le n$.

Fix $V$, $a$, $\mu$ as in the statement, and let $C_{V}$ be a constant as in Confining Potentials on the Real Line §slope.

Step 0 (Standing facts).

(0-) Differentiability implies continuity. If $f:\mathbb{R}\to\mathbb{R}$ is differentiable at $x_{0}$ in the sense of Derivative at an Interior Point, then, taking $\varepsilon=1$ there, there is $\delta>0$ with $|f(x_{0}+h)-f(x_{0})|\le(|f'(x_{0})|+1)|h|$ whenever $0<|h|<\delta$; hence $f$ is continuous at $x_{0}$. A function differentiable at every point is therefore continuous, and so is its restriction to any closed interval.

(0a) The diagonal. As $\mu\in\mathcal{D}_{\log}$, $\mu\in\mathcal{P}_{2}(\mathbb{R})$ and $\mu(\{x\})=0$ for every $x$ (The Logarithmic Energy of a Probability Measure on the Real Line §energy). The diagonal $\Delta=\{(x,x):x\in\mathbb{R}\}$ is Borel by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §borel and null by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §diagonal.

(0b) Difference quotients. For $\varphi:\mathbb{R}\to\mathbb{R}$ let $Q_{\varphi}:\mathbb{R}^{2}\to\mathbb{R}$ be $Q_{\varphi}(x,y)=\frac{\varphi(x)-\varphi(y)}{x-y}$ if $x\ne y$ and $Q_{\varphi}(x,x)=0$. Then $Q_{\varphi}(x,y)=Q_{\varphi}(y,x)$ and $Q_{b\varphi+b'\varphi'}=bQ_{\varphi}+b'Q_{\varphi'}$ for reals $b,b'$. If $\varphi$ is Borel, so is $Q_{\varphi}$: with $\ell$ the logarithmic kernel of The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure, Borel by The Logarithmic Kernel on the Real Line: Borel Measurability, a Linear Lower Bound, and the Null Diagonal of an Atomless Measure §borel, and $\exp$ differentiable (claim 3 of Basic Properties of the Exponential Function), hence continuous by (0-) and Borel, the function $w(x,y)=(x-y)\exp(\ell(x,y))^{2}$ is Borel by claims 2 and 3 of Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions; for $x\ne y$, $\exp(\ell(x,y))=\exp(-\log|x-y|)=1/|x-y|$ by claim 2 of Basic Properties of the Exponential Function and The Natural Logarithm, so $w(x,y)=1/(x-y)$, while $w(x,x)=0$; hence $Q_{\varphi}=(\varphi\circ\mathrm{pr}_{1}-\varphi\circ\mathrm{pr}_{2})\,w$ is Borel.

(0c) Bounds on quotients. If $\varphi$ is Lipschitz with constant $L$, then $|Q_{\varphi}|\le L$, so $Q_{\varphi}$ is bounded and Borel, hence $\mu\boxtimes\mu$-integrable. If $\varphi$ is differentiable at every point with continuous derivative bounded in absolute value by $L$, then off $\Delta$ the function $Q_{\varphi}$ is the function $F$ of The Difference Quotient of a Function with Bounded Continuous Derivative is Bounded, Symmetric and Continuous on the Plane, so $|Q_{\varphi}|\le L$ by The Difference Quotient of a Function with Bounded Continuous Derivative is Bounded, Symmetric and Continuous on the Plane §bound and $\varphi$ is Lipschitz with constant $L$. By One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives this applies to every $g\in C_{c}^{\infty}(\mathbb{R})$ (with $g'$) and to the derivative $\psi'$ of every $\psi\in C_{c}^{\infty}(\mathbb{R})$ (with $(\psi')'=\Delta\psi$). A test function $g$ is continuous by (0-) and compactly supported, hence bounded by claim 1 of A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Bounded and Integrable.

(0d) The hypothesis in terms of $Q$. For $\psi\in C_{c}^{\infty}(\mathbb{R})$ the function $F_{\psi}$ of One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §quotient equals $Q_{\psi'}$ off the null set $\Delta$; both are integrable, so $\int F_{\psi}\,d(\mu\boxtimes\mu)=\int Q_{\psi'}\,d(\mu\boxtimes\mu)$ by The Lebesgue Integral and Null Sets: Almost-Everywhere Comparison, Markov's Inequality, and Dominated Convergence Almost Everywhere §comparison. Also $\lVert\nabla\psi\rVert_{\mu}^{2}=\int(\psi')^{2}\,d\mu$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives.

(0e) Kernels. Fix a mollifier kernel $\rho$ of radius $1$ on $\mathbb{R}=\mathbb{R}^{1}$ (claim 2 of Existence of Mollifier Kernels of Every Radius). For real $s>0$ let $\rho_{s}(y)=s^{-1}\rho(s^{-1}y)$, a mollifier kernel of radius $s$ by Rescaling a Mollifier Kernel: smooth, nonnegative, zero at every $y$ with $|y|>s$, with $\int\rho_{s}\,d\lambda_{1}=1$. By claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set and Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space, $\rho_{s}\in C_{c}^{\infty}(\mathbb{R})$. By (0c) there are reals $B_{\rho},L_{\rho}\ge0$ with $|\rho|\le B_{\rho}$ and $|Q_{\rho}|\le L_{\rho}$; then $|\rho_{s}|\le B_{\rho}/s$, and since $Q_{\rho_{s}}(x,y)=s^{-2}Q_{\rho}(x/s,y/s)$ for $x\ne y$, $|Q_{\rho_{s}}|\le L_{\rho}/s^{2}$.

(0f) Mollifying a Lipschitz function. Let $f:\mathbb{R}\to\mathbb{R}$ be Lipschitz with constant $K$ and $0<\varepsilon\le1$. The convolution $f*\rho_{\varepsilon}$ is defined on all of $\mathbb{R}$, $(f*\rho_{\varepsilon})(x)=\int f(x-y)\rho_{\varepsilon}(y)\,\lambda_{1}(dy)$, the integrand being integrable by claim 1 of The Convolution Integrand is Continuous, Compactly Supported and Integrable. (f1) It is smooth by claim 2 of Convolution with a $C^k$ Kernel is of Class $C^k$. (f2) $|(f*\rho_{\varepsilon})(x)-f(x)|\le K\varepsilon$: by claim 2 of Linearity and Monotonicity of the Lebesgue Integral and $\int\rho_{\varepsilon}\,d\lambda_{1}=1$, the difference is $\int(f(x-y)-f(x))\rho_{\varepsilon}(y)\,\lambda_{1}(dy)$, whose integrand is at most $K|y|\rho_{\varepsilon}(y)\le K\varepsilon\rho_{\varepsilon}(y)$ in absolute value, $\rho_{\varepsilon}$ vanishing where $|y|>\varepsilon$. (f3) $f*\rho_{\varepsilon}$ is Lipschitz with constant $K$, the integrand of $(f*\rho_{\varepsilon})(x)-(f*\rho_{\varepsilon})(x')$ being at most $K|x-x'|\rho_{\varepsilon}(y)$ in absolute value. (f4) If $|f|\le B$ then $|f*\rho_{\varepsilon}|\le B$, likewise. (f5) If $A>0$ and $f(t)=0$ whenever $|t|>A$, then $(f*\rho_{\varepsilon})(x)=0$ whenever $|x|>A+1$ (for each $y$, either $\rho_{\varepsilon}(y)=0$ or $|x-y|\ge|x|-\varepsilon>A$), so $f*\rho_{\varepsilon}\in C_{c}^{\infty}(\mathbb{R})$ by claim 2 of Compact Support on $\mathbb{R}^n$ Means Vanishing Outside a Bounded Set and Test Functions on Euclidean Space, Their Gradient Maps and Laplacians §space.

(0g) Cutoffs. For real $R\ge1$ let $\chi_{R}$ be the cutoff of Scaled Cutoffs and the Second-Moment Test Functions: Uniform Derivative Bounds and Agreement on a Ball §cutoff with $q=1$ and $M_{1}\ge0$ the constant there: $\chi_{R}$ is smooth and compactly supported, $0\le\chi_{R}\le1$, $\chi_{R}(x)=1$ for $|x|\le R$, $\chi_{R}(x)=0$ for $|x|\ge2R$, $|\partial_{1}\chi_{R}|\le M_{1}/R$. So $\chi_{R}\in C_{c}^{\infty}(\mathbb{R})$ with $\chi_{R}'=\partial_{1}\chi_{R}$ (One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives), and by (0c) $|Q_{\chi_{R}}|\le M_{1}/R$.

Step 1 (Properties of $V$).

(V1) $V$ and $V'$ are differentiable at every point (Confining Potentials on the Real Line §confining), hence continuous by (0-) and Borel; $V''$ is continuous.

(V2) $V'$ is nondecreasing. Let $y<x$ and $q=\frac{V(x)-V(y)}{x-y}$. For $t\in(0,1)$, convexity (Convex Real-Valued Function on a Convex Subset of $\mathbb{R}^n$ with $n=1$) gives $V(y+t(x-y))\le tV(x)+(1-t)V(y)$, i.e. $\frac{V(y+h)-V(y)}{h}\le q$ for every $h\in(0,x-y)$; and $V(x+t(y-x))\le tV(y)+(1-t)V(x)$, i.e. $\frac{V(x+h)-V(x)}{h}\ge q$ for every $h\in(y-x,0)$ (dividing by $h<0$). By Derivative at an Interior Point, for every $\eta>0$ such quotients come within $\eta$ of $V'(y)$, respectively $V'(x)$; hence $V'(y)\le q\le V'(x)$. Consequently $Q_{V'}\ge0$ everywhere.

(V3) $V$ is bounded below. By Confining Potentials on the Real Line §superquadratic with $M=1$ there is $K_{1}>0$ with $x^{2}\le V(x)$, so $V(x)\ge0$, whenever $|x|\ge K_{1}$. By (V1) and Extreme Value Theorem on a Closed Interval the restriction of $V$ to $[-K_{1},K_{1}]$ attains a minimum $v_{0}$. With $b=\max(0,-v_{0})$ we get $V\ge-b$ on $\mathbb{R}$.

(V4) Local Lipschitz bound for $V'$. For real $A>0$, (V1) and Extreme Value Theorem on a Closed Interval give a real $L_{A}\ge0$ with $|V''|\le L_{A}$ on $[-A,A]$. For $y<x$ in $[-A,A]$, Mean Value Theorem on a Closed Real Interval applied to the restriction of $V'$ to $[y,x]$ gives $c\in(y,x)$ with $V'(x)-V'(y)=V''(c)(x-y)$, so $|V'(x)-V'(y)|\le L_{A}|x-y|$.

(V5) An integrable majorant for $Q_{V'}$. Let $C'$ be the constant of Confining Potentials on the Real Line §curvature for $\varepsilon=1$ and let $H(x,y)=|V(x)|+|V(y)|+b+|C'|$. Then $0\le Q_{V'}\le H$ everywhere. On $\Delta$ this is clear. Let $x\ne y$, say $y<x$ (both sides are symmetric). As in (V4) there is $c\in(y,x)$ with $Q_{V'}(x,y)=V''(c)\le|V(c)|+C'$. Writing $c=tx+(1-t)y$ with $t=\frac{c-y}{x-y}\in(0,1)$, convexity gives $V(c)\le tV(x)+(1-t)V(y)\le|V(x)|+|V(y)|$; if $V(c)\ge0$ this bounds $|V(c)|$, and if $V(c)<0$ then $|V(c)|=-V(c)\le b$ by (V3). Hence $Q_{V'}(x,y)\le H(x,y)$. The function $H$ is Borel and nonnegative, and as $V$ is $\mu$-integrable, the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation §pushforward for $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ and claim 1 of Linearity and Monotonicity of the Lebesgue Integral give

Clause 1. By (V1), $V'$ is Borel, and $|V'|\le C_{V}(1+|V|)$ by Confining Potentials on the Real Line §slope (so $C_{V}\ge0$). By claim 1 of Linearity and Monotonicity of the Lebesgue Integral, $\int|V'|\,d\mu\le C_{V}(1+\int|V|\,d\mu)<\infty$, so $V'$ is $\mu$-integrable. For $\psi\in C_{c}^{\infty}(\mathbb{R})$, $\psi'$ is Borel and bounded by some $L_{\psi}$ (One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives), so $V'\psi'$ is Borel with $|V'\psi'|\le L_{\psi}|V'|$, hence integrable.

Clause 2. Let $C\ge0$ be as in the hypothesis. For $\varphi:\mathbb{R}\to\mathbb{R}$ Lipschitz and bounded put

both integrals existing by Clause 1 and (0c). $\Lambda$ is linear by (0b) and claim 2 of Linearity and Monotonicity of the Lebesgue Integral. By (0c) and (0d) the hypothesis says: $\psi'$ is Lipschitz and bounded, and $\Lambda(\psi')^{2}\le C^{2}\int(\psi')^{2}\,d\mu$ for every $\psi\in C_{c}^{\infty}(\mathbb{R})$.

(A) Every test function. Let $g\in C_{c}^{\infty}(\mathbb{R})$, so $g$ is Lipschitz and bounded, say $|g|\le B_{g}$, by (0c). We show $\Lambda(g)^{2}\le C^{2}\int g^{2}\,d\mu$. Let $c=\int g\,d\lambda_{1}$ (Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function §integrable). For $j\in\mathbb{N}$ with $j\ge1$, $\rho_{j}$ is a nonnegative test function of unit mass as in claim 2 of Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function, so by Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function §correction and Test Functions on the Real Line: Unit Mass, Large Mass, Mean-Zero Correction, and the Primitive of a Mean-Zero Test Function §primitive there is $\psi_{j}\in C_{c}^{\infty}(\mathbb{R})$ with $\psi_{j}'=g-c\rho_{j}$. Hence

By (0e), $|\Lambda(\rho_{j})|\le\frac{B_{\rho}}{j}\int|V'|\,d\mu+a\frac{L_{\rho}}{j^{2}}\to0$. Also $(g-c\rho_{j})^{2}\to g^{2}$ pointwise with $(g-c\rho_{j})^{2}\le(B_{g}+|c|B_{\rho})^{2}$, so $\int(g-c\rho_{j})^{2}\,d\mu\to\int g^{2}\,d\mu$ by dominated convergence. Letting $j\to\infty$ gives the claim.

(B) Compactly supported Lipschitz functions. Let $\varphi$ be Lipschitz with constant $L_{\varphi}$ and let $A>0$ with $\varphi(t)=0$ for $|t|>A$. Then $|\varphi|\le B_{\varphi}:=L_{\varphi}(2A+1)$ (compare with $\varphi(A+1)=0$). We show $\Lambda(\varphi)^{2}\le C^{2}\int\varphi^{2}\,d\mu$. For $k\in\mathbb{N}$, $k\ge1$, let $g_{k}=\varphi*\rho_{1/k}$; by (0f), $g_{k}\in C_{c}^{\infty}(\mathbb{R})$, $|g_{k}|\le B_{\varphi}$, $g_{k}$ is Lipschitz with constant $L_{\varphi}$, and $|g_{k}-\varphi|\le L_{\varphi}/k$. By (A), $\Lambda(g_{k})^{2}\le C^{2}\int g_{k}^{2}\,d\mu$. As $k\to\infty$: $\int V'g_{k}\,d\mu\to\int V'\varphi\,d\mu$ by dominated convergence with majorant $B_{\varphi}|V'|$; $Q_{g_{k}}\to Q_{\varphi}$ pointwise (off $\Delta$ by pointwise convergence of $g_{k}$, on $\Delta$ trivially) with $|Q_{g_{k}}|\le L_{\varphi}$ by (0c), so $\int Q_{g_{k}}\,d(\mu\boxtimes\mu)\to\int Q_{\varphi}\,d(\mu\boxtimes\mu)$; and $\int g_{k}^{2}\,d\mu\to\int\varphi^{2}\,d\mu$ with majorant $B_{\varphi}^{2}$. Letting $k\to\infty$ gives the claim.

(C) Truncations of $V'$. Fix $n\in\mathbb{N}$, $n\ge1$, and let $u_{n}=T_{n}\circ V'$: continuous, $|u_{n}|\le n$, nondecreasing by (V2), and $u_{n}^{2}\le V'u_{n}$ pointwise. For $x>y$, $0\le T_{n}(V'(x))-T_{n}(V'(y))\le V'(x)-V'(y)$ since $T_{n}$ is nondecreasing and Lipschitz with constant $1$; dividing by $x-y$ and using symmetry and (V5),

so $Q_{u_{n}}$, Borel by (0b), is integrable with $\int Q_{u_{n}}\,d(\mu\boxtimes\mu)\le K_{V}$. By (V4), $u_{n}$ is Lipschitz with constant $L_{A}$ on each $[-A,A]$. For $j\in\mathbb{N}$, $j\ge1$, let $\varphi_{j}=\chi_{j}u_{n}$. It vanishes where $|x|\ge2j$, and it is Lipschitz with constant $L_{2j}+nM_{1}/j$: for $|x|,|y|\le2j$, $\varphi_{j}(x)-\varphi_{j}(y)=\chi_{j}(x)(u_{n}(x)-u_{n}(y))+u_{n}(y)(\chi_{j}(x)-\chi_{j}(y))$, bounded by $(L_{2j}+nM_{1}/j)|x-y|$ using (0g); for $|x|\le2j<|y|$, $|\varphi_{j}(x)-\varphi_{j}(y)|=|u_{n}(x)|\,|\chi_{j}(x)-\chi_{j}(y)|\le nM_{1}j^{-1}|x-y|$ as $\chi_{j}(y)=0$; symmetrically; and both values vanish if $|x|,|y|>2j$. By (B), and $|\varphi_{j}|\le|u_{n}|$,

Now let $j\to\infty$. $\int V'\varphi_{j}\,d\mu\to\int V'u_{n}\,d\mu$ by dominated convergence (majorant $n|V'|$, and $\varphi_{j}(x)=u_{n}(x)$ once $j\ge|x|$). For $x\ne y$, $Q_{\varphi_{j}}(x,y)=\chi_{j}(x)Q_{u_{n}}(x,y)+u_{n}(y)Q_{\chi_{j}}(x,y)$, so by (1) and (0g) $|Q_{\varphi_{j}}|\le H+nM_{1}$, an integrable majorant; and $Q_{\varphi_{j}}(x,y)=Q_{u_{n}}(x,y)$ once $j\ge\max(|x|,|y|)$. Hence $\int Q_{\varphi_{j}}\,d(\mu\boxtimes\mu)\to\int Q_{u_{n}}\,d(\mu\boxtimes\mu)$ by dominated convergence. With $\Lambda(u_{n}):=\int V'u_{n}\,d\mu-a\int Q_{u_{n}}\,d(\mu\boxtimes\mu)$ we obtain $\Lambda(u_{n})^{2}\le C^{2}X_{n}^{2}$, where $X_{n}\ge0$ and $X_{n}^{2}=\int u_{n}^{2}\,d\mu$; so $\Lambda(u_{n})\le CX_{n}$. Since $a>0$ and $\int Q_{u_{n}}\le K_{V}$, using $u_{n}^{2}\le V'u_{n}$ and claim 2 of Linearity and Monotonicity of the Lebesgue Integral,

hence $\int u_{n}^{2}\,d\mu\le C^{2}+2aK_{V}$ for every $n$.

(D) Conclusion. For every $x$, $u_{n}(x)^{2}=\min(V'(x)^{2},n^{2})$ is nondecreasing in $n$ with limit $V'(x)^{2}$. By Monotone Convergence Theorem, $\int(V')^{2}\,d\mu=\lim_{n}\int u_{n}^{2}\,d\mu\le C^{2}+2aK_{V}<\infty$. As $V'$ is Borel, its class lies in $L^{2}(\mu;\mathbb{R})$ by One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §scalars; write $\lVert V'\rVert_{\mu}$ for its norm. Let $\psi\in C_{c}^{\infty}(\mathbb{R})$. By One-Dimensional Test Functions: Scalars, Derivatives, and the Difference Quotient of the Derivative §derivatives, $\int V'\psi'\,d\mu=\langle V',\nabla\psi\rangle_{\mu}$, and by The Cauchy-Schwarz Inequality in a Real Inner Product Space in the real Hilbert space $L^{2}(\mu;\mathbb{R})$ (Square-Integrable Vector Fields Against a Probability Measure on Euclidean Space, and Test Functions: Standing Notation §l2mu), $|\int V'\psi'\,d\mu|\le\lVert V'\rVert_{\mu}\lVert\nabla\psi\rVert_{\mu}$. With the hypothesis,

Dividing by $a>0$, the nonnegative real number $(C+\lVert V'\rVert_{\mu})/a$ is a constant as in Finite Free Fisher Information, the Free Score and the Free Fisher Information of a Probability Measure on the Real Line §finite; since $\mu\in\mathcal{P}_{2}(\mathbb{R})$ by (0a), $\mu\in\mathcal{P}_{2}^{\Phi^{*}}(\mathbb{R})$.