The linear test and the third-moment tensor - The KLS theorem and its methods

Part of the moment-map mechanism, Chapter The moment map: CMH and the linear test; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 16.2 and Corollary 16.2 as Theorem D12.1 and Corollary D12.1, under hypotheses $(\mathrm H)$ of Definition D12.1, which the manuscript hypotheses imply. The Stein identity for quadratic tests shows that the mixed tensor $\E_\mu[\tau_{ij}X_k]$ is half the third-moment tensor. Projecting each column $\tau a$ onto the span of $\mathbf 1,X_1,\dots,X_n$ then gives a Pythagorean decomposition, from which the corollary follows. The third-derivative form (c) is proved only under one additional stated hypothesis.

Lemma D12.1, via Lemma D12.2, shows that the manuscript hypotheses imply $(\mathrm H)$ .
Lemma D12.4, using Lemma D12.3, shows that $\nabla\varphi$ is a $C^1$ diffeomorphism between open sets of full measure. So $\tau$ is defined $\mu$ -a.e. and satisfies the transport identity between $\mu$ and $\nu$ .
Proposition D12.1 proves the Stein identity (D12.8) for polynomials of degree at most two. The proof uses the divergence theorem on balls and shell selection (Lemma D12.5). Its consequences (Corollary D12.2) are $\E_\mu\tau=\Id$ and $T_{ijk}=N_{ijk}+N_{ikj}$ .
Lemma D12.6: a symmetry argument applied to step 3 shows that $N$ is totally symmetric, so $N=\tfrac12T$ .
Theorem (a) and (b): the $L^2(\mu)$ projection onto $\mathrm{span}\{\mathbf 1,X_b\}$ , with coefficients taken from step 4, yields (D12.4).
Corollary: since $\E_\mu\abs{\tau a}^2=a^\top\E_\mu[\tau^2]a$ , step 5 gives the directional bound. Products of centered exponentials (Example D12.1) attain the constant for $c=2$ .
Part (c) is Proposition D12.2, proved under $\varphi\in C^3$ and $\varphi_{ijk}\in L^1(\nu)$ (Remark D12.1).

Scope. This dossier proves Lemma 16.2 and Corollary 16.2 of Section The linear test: the necessary linear-sector condition, verbatim, as Theorem D12.1 and Corollary D12.1 below. It is self-contained: no ledger node is used, and in particular nothing is taken from the uncertified dossier for Lemma 16.1.

What this dossier does not prove. It proves nothing about Conjecture 16.1, Conjecture 16.2 or Conjecture 0.1 beyond the exact implication stated in Corollary D12.1; no bound on $\kappa_n$ of (0.13); no statement about the trace-upgrade cluster (conj:trace-upgrade, high-rank conj:stein-weighted, conj:product-alignment), which is not opened here. The final sentence of the manuscript lemma, which identifies the third-moment tensor with $\E_\nu[\partial_{ijk}\varphi]$ , is proved under one additional explicitly stated hypothesis (Proposition D12.2 and Remark D12.1); everything else is proved under the manuscript’s hypotheses exactly.

1. Setting, conventions, and the refined statements¶

Throughout, $n\ge1$ , $\inner{\cdot}{\cdot}$ and $\abs{\cdot}$ are the Euclidean inner product and norm on $\R^n$ , $\norm{\cdot}_{\HS}$ is the Hilbert–Schmidt norm on matrices and on 3-tensors, and $\mathrm{Sym}_n$ is the space of real symmetric $n\times n$ matrices. For a $C^2$ function $\varphi$ we write $\varphi_i=\partial_i\varphi$ , $\varphi_{ij}=\partial_{ij}\varphi$ , $H=D^2\varphi=(\varphi_{ij})$ , and, when $\varphi\in C^3$ , $\varphi_{ijk}=\partial_{ijk}\varphi$ . $B_R=\{\abs y<R\}$ , $\partial B_R$ its boundary, $\sigma_R$ the $(n-1)$ -dimensional surface measure on $\partial B_R$ (for $n=1$ , counting measure on $\{\pm R\}$ ), and $\mathbf n(y)=y/\abs y$ the outward unit normal. Repeated indices are not summed unless a sum sign is written.

Definition D12.2 (Third-moment tensor)

Under $(\mathrm H2)$ put, exactly as in Section The linear test: the necessary linear-sector condition,

T_{ijk}:=\E_\mu[X_iX_jX_k],\qquad T_3(\mu):=(T_{ijk})_{i,j,k},\qquad T_3(a):=\E_\mu\bigl[\inner Xa\,X\otimes X\bigr]\in\mathrm{Sym}_n\quad(a\in\R^n),

(D12.1)

so that $T_3(a)_{ib}=\sum_kT_{kib}\,a_k$ . All entries are finite by $\E_\mu\abs X^3<\infty$ , $T_{ijk}$ is symmetric in $(i,j,k)$ , $T_3(a)$ is symmetric, and $\norm{T_3(a)}_{\HS}^2=\sum_{i,b}T_3(a)_{ib}^2$ .

The canonical Stein kernel is defined in Section The Stein kernel and the $H^{-1}$ inequality as $\tau_\mu=H\circ(\nabla\varphi)^{-1}$ , (4.19). Under $(\mathrm H)$ alone $\nabla\varphi$ need not be injective on all of $\R^n$ ; Lemma D12.4 shows that it is injective on an open set of full $\nu$ -measure whose image is an open set of full $\mu$ -measure, so that (4.19) defines $\tau$ $\mu$ -almost everywhere, which is all that $L^2(\mu)$ statements need. In the compact-target regular class of Theorem 4.1 the two open sets are $\R^n$ and $\operatorname{int}P$ and nothing is new.

Theorem D12.1 (refined form of Lemma 16.2)

Assume $(\mathrm H)$ and let $\tau$ be the canonical Stein kernel of Lemma D12.4. Then every entry $\tau_{ij}$ lies in $L^2(\mu)$ , and:

(a) $\E_\mu[\tau]=\Id$ , and for all $i,j,k$ ,

\E_\mu[\tau_{ij}X_k]=\E_\nu[\varphi_{ij}\varphi_k]=\tfrac12\,\E_\mu[X_iX_jX_k];

(D12.2)

in particular $\bigl(\E_\nu[\varphi_{ij}\varphi_k]\bigr)_{i,j,k}$ is totally symmetric.

(b) For every $a\in\R^n$ the column $\tau a$ decomposes in $L^2(\mu;\R^n)$ as

\tau a=a+\tfrac12\,T_3(a)\,X+v_a,\qquad \E_\mu[v_a]=0,\qquad \E_\mu[v_a\otimes X]=0,

(D12.3)

the three summands $a$ , $\tfrac12T_3(a)X$ , $v_a$ being pairwise orthogonal in $L^2(\mu;\R^n)$ , and consequently

\E_\mu\abs{\tau a}^2=\abs a^2+\tfrac14\norm{T_3(a)}_{\HS}^2+\E_\mu\abs{v_a}^2 .

(D12.4)

(c) If in addition $\varphi\in C^3(\R^n)$ and $\varphi_{ijk}\in L^1(\nu)$ for all $i,j,k$ , then $\E_\nu[\varphi_{ijk}]=\tfrac12\,\E_\mu[X_iX_jX_k]$ .

Part (b) is the display of Lemma 16.2; part (a) is the source-coordinate content of its last sentence, and part (c) is that sentence’s third-derivative form (see Remark D12.1).

Corollary D12.1 (refined form of Corollary 16.2)

Assume $(\mathrm H)$ . Then $\E_\mu[\tau^2]=\E_\nu[H^2]$ is a finite symmetric positive semidefinite matrix and, for every unit vector $a$ ,

\norm{T_3(a)}_{\HS}^2\le4\bigl(a^\top\E_\mu[\tau^2]\,a-1\bigr) \le4\bigl(\lmax(\E_\mu[\tau^2])-1\bigr).

(D12.5)

In particular, if $\mathcal C$ is any class of measures satisfying $(\mathrm H)$ with $\E_\mu[\tau^2]\preceq c\,\Id$ for every member, then $c\ge1$ and every directional third moment in $\mathcal C$ satisfies $\norm{T_3(a)}_{\HS}\le2\sqrt{c-1}$ for all unit $a$ : the constant $c=4$ of (16.14) gives $2\sqrt3$ and the constant $c=2$ of (16.15) gives 2, which the products of centered exponentials of Example D12.1 attain in every direction. Conversely, if a measure satisfying $(\mathrm H)$ has $\norm{T_3(a)}_{\HS}>2\sqrt3$ for some unit $a$ , then $a^\top\E_\nu[H^2]a>4$ , so $\E_\nu[H^2]\not\preceq4\Id$ and $\mu$ violates (16.14).

2. Preliminaries¶

Proof D12.2

Let $D=\{V<\infty\}$ , a convex set. Since $\int e^{-V}>0$ , $D$ has positive Lebesgue measure, hence nonempty interior (a convex set with empty interior lies in an affine hyperplane). Fix $y_0\in\operatorname{int}D$ ; $V$ is finite and continuous on a neighbourhood of $y_0$ and has a subgradient $g$ at $y_0$ : $V(y)\ge V(y_0)+\inner g{y-y_0}$ for all $y$ . Put $m_0=V(y_0)$ and $K=\{V\le m_0+1\}$ , a convex set containing a ball $B_\delta(y_0)$ , $\delta>0$ , and of finite Lebesgue measure, since $e^{-(m_0+1)}\mathrm{Leb}(K)\le\int e^{-V}$ . If $z\in K$ then $\operatorname{conv}(B_\delta(y_0)\cup\{z\})\subset K$ contains the cone with apex $z$ over the $(n-1)$ -disc of radius $\delta$ centred at $y_0$ and orthogonal to $z-y_0$ , whose volume is $\frac1n\omega_{n-1}\delta^{n-1}\abs{z-y_0}$ ( $\omega_{n-1}$ the volume of the unit ball of $\R^{n-1}$ , $\omega_0=1$ ). Hence $\abs{z-y_0}\le\rho_0:=n\,\mathrm{Leb}(K)/(\omega_{n-1}\delta^{n-1})$ for every $z\in K$ . Put $\rho=\rho_0+1$ ; every $z$ with $\abs{z-y_0}=\rho$ satisfies $V(z)>m_0+1$ .

Let $\abs{y-y_0}\ge\rho$ and $V(y)<\infty$ . With $t=\rho/\abs{y-y_0}\in(0,1]$ and $z=y_0+t(y-y_0)$ , convexity gives $V(z)\le tV(y)+(1-t)m_0$ , so $V(y)\ge m_0+(V(z)-m_0)/t\ge m_0+\abs{y-y_0}/\rho$ ; the same bound is trivial when $V(y)=+\infty$ . For $\abs{y-y_0}<\rho$ the subgradient inequality gives $V(y)\ge m_0-\abs g\rho$ . Hence $V(y)\ge\abs{y-y_0}/\rho-C_1$ for all $y$ , with $C_1=\abs g\rho+\abs{m_0}+1$ , and therefore $V(y)\ge c\abs y-C$ with $c=1/\rho$ , $C=C_1+\abs{y_0}/\rho$ . Finally $e^{\eps\abs y}e^{-V(y)}\le e^{C}e^{-(c-\eps)\abs y}$ is integrable for $\eps<c$ , and $\abs y^k\le k!\,\eps^{-k}e^{\eps\abs y}$ .

Proof D12.3

$Z$ is closed, so $F(Z)$ is $\sigma$ -compact and it suffices to treat $Z\cap Q_0$ for a closed cube $Q_0$ of side 1. Let $L=\max_{Q_0}\norm{DF}_{\op}$ and $\omega(d)=\sup\{\norm{DF(y)-DF(y')}_{\op}:y,y'\in Q_0,\ \abs{y-y'}\le d\}$ , so that $\omega(d)\to0$ as $d\to0$ by uniform continuity of $DF$ on $Q_0$ . Partition $Q_0$ into $N^n$ closed subcubes of side $1/N$ and diameter $d=\sqrt n/N$ . Let $Q$ be a subcube meeting $Z$ at some $y_0$ . For $y\in Q$ ,

F(y)-F(y_0)-DF(y_0)(y-y_0)=\int_0^1\bigl(DF(y_0+s(y-y_0))-DF(y_0)\bigr)(y-y_0)\dd s

(D12.6)

has norm at most $\omega(d)d$ , while $DF(y_0)(y-y_0)$ lies in the range of $DF(y_0)$ , a linear subspace of dimension at most $n-1$ , hence in some hyperplane $E'$ , and has norm at most $Ld$ . Splitting the remainder into its components along $E'$ and $E'^{\perp}$ , $F(Q)-F(y_0)$ is contained in the cylinder $\{e+r:\ e\in E',\ \abs e\le(L+\omega(d))d,\ r\perp E',\ \abs r\le\omega(d)d\}$ , whose volume is $2\omega_{n-1}\bigl((L+\omega(d))d\bigr)^{n-1}\omega(d)d$ . Summing over at most $N^n$ subcubes,

\mathrm{Leb}\bigl(F(Z\cap Q_0)\bigr)\le N^n\cdot2\omega_{n-1}(L+\omega(d))^{n-1}\omega(d)\,d^n =2\omega_{n-1}n^{n/2}(L+\omega(d))^{n-1}\,\omega(d)\xrightarrow[N\to\infty]{}0 .

(D12.7)

Lemma D12.4 (Structure of the moment map; the canonical Stein kernel)

Assume $(\mathrm H1)$ and $\mu=(\nabla\varphi)_\#\nu\ll\mathrm{Leb}$ . Let $G=\{y\in\R^n:\det H(y)>0\}$ and $\Omega=\nabla\varphi(G)$ . Then:

(i) if $y_1\ne y_2$ and $\nabla\varphi(y_1)=\nabla\varphi(y_2)$ , then $H(y_1)(y_2-y_1)=0$ ; in particular $\nabla\varphi$ is injective on $G$ and $(\nabla\varphi)^{-1}(\Omega)=G$ ;

(ii) $G$ is open with $\nu(G)=1$ , $\Omega$ is open with $\mu(\Omega)=1$ , and $\nabla\varphi|_G:G\to\Omega$ is a $C^1$ diffeomorphism;

(iii) the function $\tau:\R^n\to\mathrm{Sym}_n$ defined by $\tau:=H\circ(\nabla\varphi|_G)^{-1}$ on $\Omega$ and $\tau:=\Id$ on $\R^n\setminus\Omega$ is Borel, continuous and positive semidefinite on $\Omega$ , and satisfies $\tau\circ\nabla\varphi=H$ on $G$ , hence $\nu$ -almost everywhere. It is the canonical Stein kernel (4.19), and any two versions of it agree $\mu$ -a.e.;

(iv) (transport) for every Borel $\Phi:\R^n\times\mathrm{Sym}_n\to[0,\infty]$ , $\E_\mu\bigl[\Phi(X,\tau(X))\bigr]=\E_\nu\bigl[\Phi(\nabla\varphi,H)\bigr]$ ; the same holds for real-valued $\Phi$ whenever either side is absolutely convergent, and then both are.

Proof D12.4

(i) Put $p=\nabla\varphi(y_1)=\nabla\varphi(y_2)$ and $w=y_2-y_1\ne0$ . The subgradient inequalities $\varphi(y_2)\ge\varphi(y_1)+\inner pw$ and $\varphi(y_1)\ge\varphi(y_2)-\inner pw$ add to $0\ge0$ , so both are equalities. The function $g(t)=\varphi(y_1+tw)-\varphi(y_1)-t\inner pw$ , $t\in\R$ , is $C^2$ , convex, nonnegative (by the subgradient inequality at $y_1$ ), and $g(0)=g(1)=0$ ; by convexity $g\equiv0$ on $[0,1]$ . Hence $g''(0)=\inner{H(y_1)w}{w}=0$ (the two-sided second derivative exists because $\varphi\in C^2$ , and it is computed from the right, where $g$ vanishes identically). Since $H(y_1)\succeq0$ , $\inner{H(y_1)w}w=0$ forces $H(y_1)w=0$ . Thus a point of $G$ shares its gradient with no other point: $\nabla\varphi$ is injective on $G$ , and a point outside $G$ cannot map into $\Omega$ , i.e. $(\nabla\varphi)^{-1}(\Omega)=G$ .

(ii) $G$ is open by continuity of $\det H$ . $Z:=\R^n\setminus G=\{\det H=0\}$ is the critical set of the $C^1$ map $\nabla\varphi$ , so $\nabla\varphi(Z)$ is Lebesgue-null by Lemma D12.3, hence $\mu$ -null by $\mu\ll\mathrm{Leb}$ , hence $0=\mu(\nabla\varphi(Z))=\nu\bigl((\nabla\varphi)^{-1}(\nabla\varphi(Z))\bigr)\ge\nu(Z)$ . So $\nu(G)=1$ and $\mu(\Omega)=\nu((\nabla\varphi)^{-1}(\Omega))=\nu(G)=1$ by (i). On $G$ the map $\nabla\varphi$ is $C^1$ , injective, with invertible differential $H$ ; by the inverse function theorem it is open, so $\Omega$ is open, and its inverse is $C^1$ on $\Omega$ .

(iii) Continuity on $\Omega$ follows from (ii); Borel measurability on $\R^n$ from that and from $\Omega$ being open; positive semidefiniteness from convexity of $\varphi$ . For $y\in G$ , $\tau(\nabla\varphi(y))=H\bigl((\nabla\varphi|_G)^{-1}(\nabla\varphi(y))\bigr)=H(y)$ . The formula (4.19) is exactly $H\circ(\nabla\varphi)^{-1}$ on the set $\Omega$ where the inverse is defined, and $\mu(\R^n\setminus\Omega)=0$ .

(iv) For $y\in G$ , $\Phi(\nabla\varphi(y),\tau(\nabla\varphi(y)))=\Phi(\nabla\varphi(y),H(y))$ by (iii), so the two functions agree $\nu$ -a.e. and $\E_\nu[\Phi(\nabla\varphi,\tau\circ\nabla\varphi)]=\E_\nu[\Phi(\nabla\varphi,H)]$ ; the left side equals $\E_\mu[\Phi(X,\tau(X))]$ by the definition of the pushforward. The real-valued case follows by applying this to $\abs\Phi$ , $\Phi^+$ and $\Phi^-$ .

Proof D12.5

Put $F=F_1+\dots+F_m$ and $S(R)=\int_{\partial B_R}Fe^{-\varphi}\dd\sigma_R\in[0,\infty]$ . By Tonelli’s theorem in polar coordinates, $S$ is Borel on $(0,\infty)$ and $\int_0^\infty S(R)\dd R=\int_{\R^n}Fe^{-\varphi}\dd y<\infty$ . For each $k\ge1$ the set $\{R\ge k:S(R)\le1/k\}$ has positive Lebesgue measure, since otherwise $S>1/k$ a.e. on $[k,\infty)$ and $S\notin L^1$ ; pick $R_k$ in it. Then $R_k\ge k$ and $0\le\int_{\partial B_{R_k}}F_le^{-\varphi}\dd\sigma_{R_k}\le S(R_k)\le1/k$ .

3. The Stein identity for polynomials of degree at most two¶

Proof D12.6

Choose $C_f<\infty$ with $\abs{f(x)}\le C_f(1+\abs x^2)$ and $\abs{\partial_jf(x)}\le C_f(1+\abs x)$ for all $x$ and $j$ . Integrability. $\abs{X_if(X)}\le C_f\abs X(1+\abs X^2)\in L^1(\mu)$ by $(\mathrm H2)$ . Next, $\E_\mu[\tau_{ij}^2]=\E_\nu[\varphi_{ij}^2]\le\E_\nu\norm H_{\HS}^2<\infty$ by Lemma D12.4(iv) and $(\mathrm H3)$ , so $\abs{\tau_{ij}\partial_jf}\le C_f\abs{\tau_{ij}}(1+\abs X)\le C_f\bigl(\tfrac12\tau_{ij}^2+1+\abs X^2\bigr)\in L^1(\mu)$ , using $\E_\mu\abs X^2=\Tr\Id=n$ . By Lemma D12.4(iv) the source integrands $\varphi_if(\nabla\varphi)$ and $\varphi_{ij}(\partial_jf)(\nabla\varphi)$ are in $L^1(\nu)$ with the same integrals.

Integration by parts on balls. The vector field $W=f(\nabla\varphi)e^{-\varphi}e_i$ is $C^1$ on $\R^n$ because $\varphi\in C^2$ , with

\Div W=\partial_i\bigl[f(\nabla\varphi)e^{-\varphi}\bigr] =\Bigl[\sum_j(\partial_jf)(\nabla\varphi)\,\varphi_{ji}-f(\nabla\varphi)\,\varphi_i\Bigr]e^{-\varphi}.

(D12.9)

The divergence theorem on $B_R$ and $\varphi_{ji}=\varphi_{ij}$ give, for every $R>0$ ,

\sum_j\int_{B_R}\varphi_{ij}(\partial_jf)(\nabla\varphi)\dd\nu-\int_{B_R}\varphi_if(\nabla\varphi)\dd\nu =\int_{\partial B_R}f(\nabla\varphi)\,\mathbf n_i\,e^{-\varphi}\dd\sigma_R .

(D12.10)

The right side is bounded in absolute value by $C_f\int_{\partial B_R}(1+\abs{\nabla\varphi}^2)e^{-\varphi}\dd\sigma_R$ , and $\int(1+\abs{\nabla\varphi}^2)\dd\nu=1+\E_\mu\abs X^2=1+n<\infty$ . Apply Lemma D12.5 with $F_1=1+\abs{\nabla\varphi}^2$ and let $R=R_k\to\infty$ in (D12.10): the right side tends to 0, and each integral on the left converges to the integral over $\R^n$ by dominated convergence, since the integrands are in $L^1(\nu)$ . This is the source-coordinate identity; (D12.8) follows by Lemma D12.4(iv).

4. Total symmetry of the mixed tensor¶

Proof D12.8

The two expressions for $N_{ijk}$ agree by Lemma D12.4(iv), and $\abs{N_{ijk}}\le(\E_\mu\tau_{ij}^2)^{1/2}(\E_\mu X_k^2)^{1/2}<\infty$ by Cauchy–Schwarz, $(\mathrm H3)$ and isotropy. Symmetry in the first two indices, $N_{ijk}=N_{jik}$ , is the symmetry of $H$ . Corollary D12.2(ii) reads $T_{ijk}=N_{ijk}+N_{ikj}$ ; the same identity with the roles of $i$ and $j$ exchanged reads $T_{jik}=N_{jik}+N_{jki}$ . Since $T_{ijk}=T_{jik}$ and $N_{ijk}=N_{jik}$ , subtracting gives $N_{ikj}=N_{jki}$ , and applying first-pair symmetry to the right side, $N_{ikj}=N_{kji}$ . Relabelling $(i,k,j)\to(a,b,c)$ this is $N_{abc}=N_{cba}$ : $N$ is invariant under the transposition of its first and third indices. Together with invariance under the transposition of the first two indices, $N$ is invariant under the group these two transpositions generate, namely all of $S_3$ . Finally $T_{ijk}=N_{ijk}+N_{ikj}=2N_{ijk}$ .

5. Proof of Theorem D12.1¶

Proof D12.9 (Proof of Theorem D12.1)

$\tau_{ij}\in L^2(\mu)$ was shown in the proof of Proposition D12.1.

(a) is Corollary D12.2(i) and Lemma D12.6.

(b) By isotropy, the functions $\mathbf 1,X_1,\dots,X_n$ are orthonormal in $L^2(\mu)$ : $\E_\mu[\mathbf 1^2]=1$ , $\E_\mu[X_b]=0$ , $\E_\mu[X_bX_c]=\delta_{bc}$ . Let $P$ be the orthogonal projection of $L^2(\mu)$ onto their span $\mathcal V$ . For $u\in L^2(\mu)$ , $Pu=\E_\mu[u]\,\mathbf 1+\sum_b\E_\mu[uX_b]\,X_b$ , and $u-Pu\perp\mathcal V$ , i.e.\ $\E_\mu[u-Pu]=0$ and $\E_\mu[(u-Pu)X_b]=0$ for all $b$ . Fix $a\in\R^n$ and apply this to $u_i:=(\tau a)_i=\sum_k\tau_{ik}a_k\in L^2(\mu)$ . By (a),

\E_\mu[u_i]=\sum_k\delta_{ik}a_k=a_i, \qquad \E_\mu[u_iX_b]=\sum_ka_kN_{ikb}=\tfrac12\sum_ka_kT_{kib}=\tfrac12\,T_3(a)_{ib},

(D12.11)

using total symmetry $N_{ikb}=N_{kib}$ , Lemma D12.6, and Definition D12.2. Hence $Pu_i=a_i\mathbf 1+\tfrac12\sum_bT_3(a)_{ib}X_b=\bigl(a+\tfrac12T_3(a)X\bigr)_i$ . Define $v_a:=\tau a-a-\tfrac12T_3(a)X$ , so that $(v_a)_i=u_i-Pu_i$ . Then $\E_\mu[(v_a)_i]=0$ and $\E_\mu[(v_a)_iX_b]=0$ for all $i,b$ , which is $\E_\mu[v_a]=0$ and $\E_\mu[v_a\otimes X]=0$ . Pairwise orthogonality in $L^2(\mu;\R^n)$ : $\E_\mu\inner a{T_3(a)X}=\inner a{T_3(a)\E_\mu X}=0$ ; $\E_\mu\inner a{v_a}=\inner a{\E_\mu v_a}=0$ ; $\E_\mu\inner{T_3(a)X}{v_a}=\sum_{i,b}T_3(a)_{ib}\E_\mu[X_b(v_a)_i]=0$ . Therefore, expanding $\abs{\tau a}^2=\abs{a+\tfrac12T_3(a)X+v_a}^2$ and taking expectations, the cross terms vanish and

\E_\mu\abs{\tau a}^2=\abs a^2+\tfrac14\E_\mu\abs{T_3(a)X}^2+\E_\mu\abs{v_a}^2, \qquad \E_\mu\abs{T_3(a)X}^2=\sum_{i,b,c}T_3(a)_{ib}T_3(a)_{ic}\E_\mu[X_bX_c]=\norm{T_3(a)}_{\HS}^2,

(D12.12)

which is (D12.4).

(c) is Proposition D12.2 below.

6. Proof of Corollary D12.1, and the saturating example¶

Proof D12.10 (Proof of Corollary D12.1)

Each entry of $\tau^2$ satisfies $\abs{(\tau^2)_{ij}}\le\norm\tau_{\HS}^2$ , which is $\mu$ -integrable by $(\mathrm H3)$ and Lemma D12.4(iv); the same lemma gives $\E_\mu[\tau^2]=\E_\nu[H^2]$ entrywise, and this matrix is symmetric positive semidefinite as an average of squares of symmetric matrices. Since $\tau$ is symmetric, $\abs{\tau a}^2=a^\top\tau^\top\tau a=a^\top\tau^2a$ , so $\E_\mu\abs{\tau a}^2=a^\top\E_\mu[\tau^2]a\le\lmax(\E_\mu[\tau^2])\abs a^2$ . For unit $a$ , (D12.4) gives $\norm{T_3(a)}_{\HS}^2=4\bigl(\E_\mu\abs{\tau a}^2-1-\E_\mu\abs{v_a}^2\bigr)\le4\bigl(a^\top\E_\mu[\tau^2]a-1\bigr)\le4\bigl(\lmax(\E_\mu[\tau^2])-1\bigr)$ .

If $\E_\mu[\tau^2]\preceq c\,\Id$ on a class, then for any member and unit $a$ , $c\ge a^\top\E_\mu[\tau^2]a=\E_\mu\abs{\tau a}^2\ge\abs a^2=1$ by (D12.4), and $\norm{T_3(a)}_{\HS}^2\le4(c-1)$ . With $c=4$ this is $2\sqrt3$ and with $c=2$ it is 2; Example D12.1 shows that the products of centered exponentials satisfy $(\mathrm H)$ , have $\E_\mu[\tau^2]=2\Id$ , and have $\norm{T_3(a)}_{\HS}=2$ for every unit $a$ . Conversely, if $\norm{T_3(a)}_{\HS}>2\sqrt3$ for some unit $a$ , the first inequality gives $a^\top\E_\nu[H^2]a=a^\top\E_\mu[\tau^2]a>1+3=4$ , so $\E_\nu[H^2]\not\preceq4\Id$ , which is the negation of (16.14) for $\mu$ in isotropic position ( $\Sigma=\Id$ ).

Example D12.1 (Products of centered exponentials)

Let $\varphi(y)=\sum_{j=1}^n(e^{y_j}-y_j)$ on $\R^n$ . It is smooth and convex, and $e^{-\varphi(y)}=\prod_je^{y_j}e^{-e^{y_j}}$ is the density of $(\log E_1,\dots,\log E_n)$ for independent $E_j\sim\mathrm{Exp}(1)$ , so $\nu$ is a probability measure and $(\mathrm H1)$ holds. $\nabla\varphi(y)=(e^{y_j}-1)_j$ , so $\mu$ is the law of $(E_j-1)_j$ : a product of centered exponentials, which is log-concave with density $e^{-\sum_j(x_j+1)}$ on $(-1,\infty)^n$ , has all moments finite, and is isotropic since $\E E_j=\Var E_j=1$ ; so $(\mathrm H2)$ holds. $H(y)=\diag(e^{y_j})$ , $\E_\nu\norm H_{\HS}^2=\sum_j\E E_j^2=2n<\infty$ ; so $(\mathrm H3)$ holds. Here $\nabla\varphi$ is a diffeomorphism of $\R^n$ onto $(-1,\infty)^n$ and $\tau(x)=\diag(1+x_j)$ on that set. Using $\E E^k=k!$ : $\E_\mu[X_j^3]=\E(E-1)^3=6-6+3-1=2$ , and every $T_{ijk}$ with indices not all equal vanishes by independence and centering, so $T_{ijk}=2\delta_{ij}\delta_{jk}$ , $T_3(a)=\diag(2a_j)$ , and $\norm{T_3(a)}_{\HS}^2=4\abs a^2$ : $\norm{T_3(a)}_{\HS}=2$ for every unit $a$ . Also $\E_\mu[\tau^2]=\diag(\E(1+X_j)^2)=\diag(\E E_j^2)=2\Id$ . Finally $\tau a=(a_j(1+X_j))_j=a+\diag(a_j)X=a+\tfrac12T_3(a)X$ , so $v_a=0$ and (D12.4) reads $2=1+1+0$ . Thus this family attains the bound $2\sqrt{c-1}$ of Corollary D12.1 with $c=2$ in every direction, and has no high-mode term.

7. The third-derivative form¶

Proof D12.11

The vector field $W=\varphi_{ij}e^{-\varphi}e_k$ is $C^1$ since $\varphi\in C^3$ , with $\Div W=(\varphi_{ijk}-\varphi_{ij}\varphi_k)e^{-\varphi}$ . The divergence theorem on $B_R$ gives $\int_{B_R}\varphi_{ijk}\dd\nu-\int_{B_R}\varphi_{ij}\varphi_k\dd\nu=\int_{\partial B_R}\varphi_{ij}\mathbf n_ke^{-\varphi}\dd\sigma_R$ , whose right side is bounded by $\int_{\partial B_R}\norm H_{\HS}e^{-\varphi}\dd\sigma_R$ . Since $\int\norm H_{\HS}\dd\nu\le(\E_\nu\norm H_{\HS}^2)^{1/2}<\infty$ , Lemma D12.5 with $F_1=\norm H_{\HS}$ gives $R_k\to\infty$ along which the right side tends to 0. Both integrands on the left are in $L^1(\nu)$ (the second by Lemma D12.6), so dominated convergence gives $\E_\nu[\varphi_{ijk}]=\E_\nu[\varphi_{ij}\varphi_k]=N_{ijk}=\tfrac12T_{ijk}$ .

Remark D12.1 (Which hypothesis the manuscript’s last sentence uses)

The last sentence of Lemma 16.2, “ $\E_\nu[\partial_{ijk}\varphi]=\tfrac12\E_\mu[X_iX_jX_k]$ ”, presupposes that $\partial_{ijk}\varphi$ exists and is $\nu$ -integrable, which the hypothesis $\varphi\in C^2$ does not provide. This dossier therefore proves it exactly under the additional hypothesis of Proposition D12.2, and proves unconditionally the statement that the manuscript’s “gap-mode coefficient” actually uses, namely $\E_\nu[\varphi_{ij}\varphi_k]=\tfrac12\E_\mu[X_iX_jX_k]$ (Theorem D12.1(a)): in the decomposition of Theorem D12.1(b) the coefficient of $X_b$ in $(\tau a)_i$ is $\E_\mu[(\tau a)_iX_b]=\sum_ka_k\E_\nu[\varphi_{ik}\varphi_b]$ , which involves only second derivatives of $\varphi$ . The identification with the coefficient tensor of Lemma 16.1 is a matter of that lemma’s definitions and is not asserted here; nothing in this dossier depends on that lemma. On the compact-target regular class the extra hypothesis is a separate claim (bounded $H$ does not by itself bound $D^3\varphi$ ) and is left to whichever dossier certifies that class.

8. Remarks¶

Remark D12.2 (Where each hypothesis is used)

Convexity and $C^2$ regularity of $\varphi$ : the divergence-theorem computations (Proposition D12.1), the structure Lemma D12.4 (convexity is what makes the fibres of $\nabla\varphi$ segments on which $H$ degenerates), and $\tau\succeq0$ . Absolute continuity of $\mu$ : only to make $\nabla\varphi$ injective off a $\nu$ -null set (Lemma D12.4(ii)), i.e. to give (4.19) a meaning; a reader who prefers to define $\tau$ only where $\nabla\varphi$ is invertible needs nothing else. Finite third moment of $\mu$ : absolute convergence of the left side of (D12.8) for quadratic $f$ , and the definition of $T_3$ . Isotropy: orthonormality of $\{\mathbf 1,X_b\}$ and $\E_\nu\abs{\nabla\varphi}^2=n$ . $(\mathrm H3)$ : $\tau_{ij}\in L^2(\mu)$ , the domination of the interior integrands, and the shell function $\norm H_{\HS}$ . Log-concavity of $\mu$ is used only through Lemma D12.1, i.e. to supply absolute continuity and finite third moments; the theorem holds for any moment measure with those two properties. The essential uniqueness of the moment potential Cordero-Erausquin & Klartag, 2015 is not used in any proof; it is what identifies the $\varphi$ of the hypothesis with “the” potential of Conjecture 16.1 (any two differ by a translation, under which $\E_\nu[H^2]$ is invariant).

Remark D12.3 (Symmetric laws and the compact-target class)

If $\mu$ is symmetric ( $X$ and $-X$ have the same law) then $T_3(\mu)=0$ , so $\tau a=a+v_a$ and $\E_\mu\abs{\tau a}^2=\abs a^2+\E_\mu\abs{v_a}^2$ : the whole of $a^\top\E_\mu[\tau^2]a-\abs a^2$ is the high-mode term. For a product of centered exponentials the opposite holds (Example D12.1). For a member of the compact-target regular class of Theorem 4.1 which is isotropic and log-concave, $(\mathrm H)$ holds: $(\mathrm H1)$ and the diffeomorphism property are that theorem, $(\mathrm H2)$ follows from the density and the compact support, and $(\mathrm H3)$ follows from the pointwise bound $0\preceq D^2\varphi\preceq2R(P)^2\Id$ of Klartag, 2014, Thm. 1.1 ( $R(P)$ the radius of a Euclidean ball centred at the origin that contains $P$ ), as recorded in the manuscript after Lemma 16.2. This published bound is used only in this remark.

Obstructions respected. Neither node carries a bounded_by or heuristic_barriers entry. Of the ledger’s six obstruction nodes: rem:two-tail-slice-bounds, rem:projection-ceiling, rem:crude-insufficient and rem:single-coordinate-cuts concern localization-route estimates and are not touched, since nothing here is a bound on a Poincaré or Cheeger constant; rem:relative-ceiling is respected because nothing here is a statement of KLS-equivalent strength (the theorem is an exact identity on a fixed measure, and the corollary is an implication whose antecedent is the open Conjecture 16.1); rem:profile-circularity is respected because no isoperimetric profile is used. The CMH-route guardrail prop:letwin-not-gate-zero (the constant-matrix estimate does not imply gate zero) is respected: this dossier neither uses the constant-matrix estimate nor claims gate zero; it proves only that gate zero implies a directional third-moment bound. Program constraint P1 is respected: no member of the trace-upgrade cluster is opened, and no transfer between its members is asserted.

References¶

Cordero-Erausquin, D., & Klartag, B. (2015). Moment Measures. Journal of Functional Analysis, 268(12), 3834–3866. 10.1016/j.jfa.2015.04.001
Klartag, B. (2014). Logarithmically-Concave Moment Measures I. In Geometric Aspects of Functional Analysis (Vol. 2116, pp. 231–260). Springer. 10.1007/978-3-319-09477-9_16