Proof of Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws

Each result cited is universally quantified over the data in its own statement. Write $\lambda=\mathrm{law}(\hat{X})$, $\nu=\mathrm{law}(\hat{Y})$ and $D=\lVert\hat{X}-\hat{Y}\rVert_{2}$.

Step 0 (tools). (a) Tuple algebra. Fix a tracial W*-probability space $(K,N,\Psi)$. By Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing and Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space, its $L^{2}$ $d$-tuples lie in the complex Hilbert space $K^{d}$, sums and real multiples are those of $K^{d}$, the pairing $\langle\cdot,\cdot\rangle_{2}$ is the inner product of $K^{d}$ and takes real values on $L^{2}$ $d$-tuples, and $\lVert\cdot\rVert_{2}$ is the norm of $K^{d}$; the difference $Z-W$ of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples is $Z+(-1)W$. By conjugate symmetry and linearity in the second argument (Complex Inner Product Space, conditions 1, 2 and 3) and realness, $\langle\cdot,\cdot\rangle_{2}$ is symmetric and real bilinear on $L^{2}$ $d$-tuples, so for all $L^{2}$ $d$-tuples $Z,W$,

Moreover $|\langle Z,W\rangle_{2}|\le\lVert Z\rVert_{2}\lVert W\rVert_{2}$ by Cauchy-Schwarz Inequality in a Complex Inner Product Space, and $\lVert tZ\rVert_{2}=|t|\,\lVert Z\rVert_{2}$ and $\lVert Z+W\rVert_{2}\le\lVert Z\rVert_{2}+\lVert W\rVert_{2}$ by claim 2 of The Induced Norm is a Norm, and Induces a Metric.

(b) $\Phi$ on pairs. For $L^{2}$ $d$-tuples $X,Y$ of $(K,N,\Psi)$, Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling gives $\mathrm{pr}^{1}_{\#}\mathrm{law}(X,Y)=\mathrm{law}(X)$, $\mathrm{pr}^{2}_{\#}\mathrm{law}(X,Y)=\mathrm{law}(Y)$ and $\mathcal{I}(\mathrm{law}(X,Y))=\lVert X-Y\rVert_{2}^{2}$, and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments gives $\widehat{M}(\mathrm{law}(Z))=\lVert Z\rVert_{2}^{2}$. Writing $g(Z)=\log(1+\lVert Z\rVert_{2}^{2})$, so that $G(\mathrm{law}(Z))=g(Z)$, and using the lifts,

If $X'$ is a further $L^{2}$ $d$-tuple of $(K,N,\Psi)$, the pairs $(X',Y)$ and $(X,Y)$ are $L^{2}$ $2d$-tuples (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations) whose difference has coordinates $X'_{j}-X_{j}$ ($j\in[d]$) followed by $d$ zeros, so its $L^{2}$ norm is $\lVert X'-X\rVert_{2}$ by the definition of the norm in Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz gives $\widehat{W}_{2}(\mathrm{law}(X',Y),\mathrm{law}(X,Y))\le\lVert X'-X\rVert_{2}$; likewise $\widehat{W}_{2}(\mathrm{law}(X,Y'),\mathrm{law}(X,Y))\le\lVert Y'-Y\rVert_{2}$.

(c) Logarithm. For reals $s,s'\ge0$, put $t=(1+s')/(1+s)>0$. Then $\log(1+s')=\log(1+s)+\log t$ by the product rule in The Natural Logarithm, and $\log t\le t-1=(s'-s)/(1+s)$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log. For $L^{2}$ $d$-tuples $Z,Z'$ of $(K,N,\Psi)$, apply this with $s=\lVert Z\rVert_{2}^{2}$, $s'=\lVert Z'\rVert_{2}^{2}$; by (E) with $W=Z'-Z$, $s'-s=2\langle Z,Z'-Z\rangle_{2}+\lVert Z'-Z\rVert_{2}^{2}$, and $1/(1+s)\le1$, so

(d) Norms are law data. If $Z,Z'$ are $L^{2}$ $d$-tuples (of possibly different spaces) with $\mathrm{law}(Z)=\mathrm{law}(Z')$, then $\lVert Z\rVert_{2}=\lVert Z'\rVert_{2}$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments and Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field (claim 3). If $\mathrm{law}(X,P)=\mathrm{law}(X',P')$, then $\mathrm{law}(X)=\mathrm{law}(X')$ and $\mathrm{law}(P)=\mathrm{law}(P')$ by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling.

(e) The Ekeland inequality. By hypothesis, $\Phi(\gamma)\le\Phi(\hat{\gamma})+\delta\,\widehat{W}_{2}(\gamma,\hat{\gamma})$ for every $\gamma\in\Sigma^{2}_{2d}$.

Clause 1. Put $s=\lVert\hat{X}\rVert_{2}^{2}$ and $\kappa=2\beta\varepsilon/(1+s)$, so $q=(\kappa/\varepsilon)\hat{X}$. Let $\pi_{u}=\mathrm{law}(\hat{X},p+q)$; by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling, $\mathrm{pr}^{1}_{\#}\pi_{u}=\lambda$, so $\pi_{u}$ is a plan at $\lambda$. Let $T=(B,0)$ be the affine datum from $2d$ to $2d$ variables with, for $i,j\in[d]$, $B_{ij}=1$ if $i=j$ and $0$ otherwise, $B_{i,d+j}=0$, $B_{d+i,j}=1+\kappa$ if $i=j$ and $0$ otherwise, and $B_{d+i,d+j}=-\varepsilon$ if $i=j$ and $0$ otherwise. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, for all $L^{2}$ $d$-tuples $X,P$ of any tracial W*-probability space, $T(X,P)=(X,(1+\kappa)X-\varepsilon P)$ (the terms with coefficient $0$ vanish). Since $\varepsilon p=\hat{X}-\hat{Y}$ and $\varepsilon q=\kappa\hat{X}$, we get $(1+\kappa)\hat{X}-\varepsilon(p+q)=\hat{Y}$, so $T(\hat{X},p+q)=(\hat{X},\hat{Y})$.

Let $\eta>0$ and put $r=\eta\,(1/(2\varepsilon)+\beta)^{-1}>0$. Let $(K,N,\Psi)$ be a tracial W*-probability space and $X,P,X'$ be $L^{2}$ $d$-tuples of it with $\mathrm{law}(X,P)=\pi_{u}$ and $\lVert X'-X\rVert_{2}<r$. Put $Y=(1+\kappa)X-\varepsilon P$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward (twice),

By (d), $\mathrm{law}(X)=\lambda$ and $\lVert X\rVert_{2}^{2}=s$; and $X-Y=\varepsilon P-\kappa X$, so

By (e) with $\gamma=\mathrm{law}(X',Y)$ and (b), $\Phi(\mathrm{law}(X',Y))\le\Phi(\mathrm{law}(X,Y))+\delta\lVert X'-X\rVert_{2}$. Expanding both sides by (F) and cancelling $v_{N}(Y)$ and $\beta g(Y)$,

By (E) with $Z=X-Y$ and $W=X'-X$, $\lVert X'-Y\rVert_{2}^{2}-\lVert X-Y\rVert_{2}^{2}=2\langle X-Y,X'-X\rangle_{2}+\lVert X'-X\rVert_{2}^{2}$; by (L) with $Z=X$, $Z'=X'$, $\beta(g(X')-g(X))\le\langle\frac{2\beta}{1+s}X,X'-X\rangle_{2}+\beta\lVert X'-X\rVert_{2}^{2}$. Adding, using bilinearity, (P) and $u_{N}(X)=u(\lambda)$,

Since $\lVert X'-X\rVert_{2}<r$, $(\frac{1}{2\varepsilon}+\beta)\lVert X'-X\rVert_{2}^{2}\le\eta\lVert X'-X\rVert_{2}$, whence $u_{N}(X')\le u(\lambda)+\langle P,X'-X\rangle_{2}+(\delta+\eta)\lVert X'-X\rVert_{2}$. Thus $\pi_{u}$ is a plan superdifferential of $u$ at $\lambda$ with slack $\delta$ (Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §super), and $\pi_{u}\in J^{+}_{\delta}u(\lambda)$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §superjet.

Clause 2. Put $t=\lVert\hat{Y}\rVert_{2}^{2}$ and $\kappa'=2\beta\varepsilon/(1+t)$, so $q'=(\kappa'/\varepsilon)\hat{Y}$, and let $\pi_{v}=\mathrm{law}(\hat{Y},p-q')$, a plan at $\nu$ as before. Let $T'$ be the affine datum from $2d$ to $2d$ variables with $T'(Y,P)=((1+\kappa')Y+\varepsilon P,\,Y)$ for all $L^{2}$ $d$-tuples $Y,P$ (entries $1+\kappa'$ and $\varepsilon$ on the diagonals of the upper blocks, $1$ on the diagonal of the lower left block, $0$ elsewhere, translation part $0$; Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations). Since $(1+\kappa')\hat{Y}+\varepsilon(p-q')=\hat{Y}+(\hat{X}-\hat{Y})=\hat{X}$, $T'(\hat{Y},p-q')=(\hat{X},\hat{Y})$.

Let $\eta>0$, $r$ as above, and let $Y,P,Y'$ be $L^{2}$ $d$-tuples of a tracial W*-probability space $(K,N,\Psi)$ with $\mathrm{law}(Y,P)=\pi_{v}$ and $\lVert Y'-Y\rVert_{2}<r$. Put $X=(1+\kappa')Y+\varepsilon P$. As in clause 1, $\mathrm{law}(X,Y)=T'_{\#}\pi_{v}=\hat{\gamma}$, $\mathrm{law}(Y)=\nu$, $\lVert Y\rVert_{2}^{2}=t$, and $P=\frac{1}{\varepsilon}(X-Y)-\frac{2\beta}{1+t}Y$. By (e) with $\gamma=\mathrm{law}(X,Y')$, (b) and (F), after cancelling $u_{N}(X)$ and $\beta g(X)$,

By (E) with $Z=X-Y$, $W=-(Y'-Y)$, $\lVert X-Y'\rVert_{2}^{2}-\lVert X-Y\rVert_{2}^{2}=-2\langle X-Y,Y'-Y\rangle_{2}+\lVert Y'-Y\rVert_{2}^{2}$, and by (L), $g(Y')-g(Y)\le\langle\frac{2}{1+t}Y,Y'-Y\rangle_{2}+\lVert Y'-Y\rVert_{2}^{2}$. Hence

so $\pi_{v}\in J^{-}_{\delta}v(\nu)$ by Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §sub and Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws §subjet.

Clause 3. Let $a=\lVert\hat{X}\rVert_{2}\ge0$. From $0\le(a-1)^{2}$ we get $2a\le1+a^{2}$, and trivially $a^{2}\le1+a^{2}$. Hence $\lVert q\rVert_{2}=2\beta a/(1+a^{2})\le\beta$ and $(1+a)\lVert q\rVert_{2}=2\beta a/(1+a^{2})+2\beta a^{2}/(1+a^{2})\le\beta+2\beta=3\beta$. The same computation with $\lVert\hat{Y}\rVert_{2}$ bounds $q'$.

Clause 4. Let $B\ge0$ be real with $|u|\le B$ and $|v|\le B$ on $\Sigma^{2}_{d}$ (Bounded Real-Valued Function on a Set). Since $\mathcal{I}\ge0$ by Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §cost and $G\ge0$ (as $\log(1+s)\ge1-(1+s)^{-1}\ge0$ for $s\ge0$ by The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm §log), $\Phi\le2B$ and $\Phi'\le2B$, so both suprema exist (Approximation Property of the Supremum and the Infimum in $\mathbb{R}$). By the definitions of $\Phi$ and $\Phi'$ and Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §coupling ($\mathcal{I}(\hat{\gamma})=D^{2}$), $\Phi'(\hat{\gamma})=\Phi(\hat{\gamma})+(\frac{1}{2\varepsilon}-\frac{1}{4\varepsilon})D^{2}=\Phi(\hat{\gamma})+\frac{1}{4\varepsilon}D^{2}$. Hence $\frac{1}{4\varepsilon}D^{2}=\Phi'(\hat{\gamma})-\Phi(\hat{\gamma})\le\sup\Phi'-(\sup\Phi-\kappa)$.