Letwin: matrix and quadratic estimates - The KLS theorem and its methods

Part of the results of the literature written out here, Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels; the reading order is on the full proofs page.

Overview. This dossier gives a proof of the precise matrix and quadratic estimates imported as Theorem 4.2 and Theorem 25.1. It follows the mechanism of Letwin, 2026, pinned to arXiv:2607.24164v1, and expands the coordinate change and the analytic domains needed in that mechanism. This is an author’s dossier, not its independent certification.

On a bounded regular target, construct cutoffs in the source coordinates and justify integrating the differentiated Monge–Ampère identity before using its positive terms.
Transform the third-order tensor and the fixed matrix together. The pointwise comparison then combines with Brascamp–Lieb to give the matrix estimate.
Transfer the Stein kernel by congruence, apply the published $H^{-1}$ inequality on a generally non-isotropic law, and remove singular matrices and target regularity.

The source locations are Theorem 2.5 (PDF pp. 8–13), Theorem 1.2 (p. 3, proof on pp. 14–16), and Appendix A (pp. 16–18). Page numbers refer to the PDF, with its first page numbered one. The versioned HTML also carries the theorem and lemma numbers. The proof of the general KLS bound, Theorem 1.1 of the source, is outside this dossier.

Statements and published inputs¶

The established inputs used below are the compact-target regularity theorem Theorem 4.1, the pointwise estimate of Klartag, 2014, Theorem 1.1, the convex-boundary Brascamp–Lieb inequality Kolesnikov & Milman, 2017, Theorem 1.2(1), and Barthe & Klartag, 2019, Propositions 10 and 27. We use these as published results, not the conclusions of either recent preprint. The application of each is specified below. Gaussian convolution preserves log-concavity, and finite-dimensional log-concave probabilities have finite polynomial moments; these standard facts are used only in the last approximation step.

Regularity, symmetry and integration¶

The regularity node applies with $P=\overline K$ . Indeed, multiplying $V$ by a smooth cutoff equal to one near $P$ gives a globally smooth extension of $V|_P$ ; its exponential is positive and globally smooth, as required by that node. The extension need not be convex outside $P$ . Thus $\varphi$ is smooth, $H\succ0$ , and $\nabla\varphi$ maps $\mathbb R^n$ diffeomorphically onto $K$ . The change-of-variables equation is

\log\det H=-\varphi+V(\nabla\varphi).

(103.3)

The hypotheses of Klartag’s Theorem 1.1 are satisfied: $V$ and each of its derivatives are bounded on $K$ , by smoothness near the compact set $\overline K$ . It gives $\operatorname{Tr}H\le2R_K^2$ , where $R_K=\sup_{x\in K}|x|$ . In particular $H$ and $\nabla\varphi$ are bounded; no positive uniform lower bound on $H$ is asserted or used.

Use $H^{ab}$ for the entries of $H^{-1}$ and define

Lf=H^{ab}\partial_{ab}f-V_a(\nabla\varphi)\partial_af, \qquad \Gamma(f,g)=H^{ab}\partial_af\partial_bg.

(103.4)

Repeated indices are summed. Differentiating the inverse Hessian and using symmetry of third derivatives gives $\partial_a H^{ab}=-H^{bi}\partial_i\log\det H$ . The logarithmic equation therefore implies the divergence formula

Lf=e^{\varphi}\partial_a(e^{-\varphi}H^{ab}\partial_bf).

(103.5)

Consequently $\int (Lf)g\,d\nu=-\int\Gamma(f,g)\,d\nu$ when either smooth function has compact support. This assertion is local and requires no tail estimate on $H^{-1}$ .

Ordinary Euclidean integration by parts with a cutoff $\zeta(y/R)$ gives $\mathbb E\partial_i f=\mathbb E f\partial_i\varphi$ whenever all displayed terms and $f$ are integrable: the extra term is bounded by $C R^{-1}\mathbb E|f|$ . Applying it to $f=\partial_j\varphi$ , whose derivative and all other terms are bounded, gives

\mathbb E_\nu H_{ij}=\mathbb E_\nu(\partial_i\varphi\partial_j\varphi) =\mathbb E_\mu X_iX_j=\delta_{ij}.

(103.6)

Proof 103.1

An integrable exponential of a finite convex potential has bounded sublevel sets: otherwise a sublevel set with nonempty interior would be an unbounded convex set containing a ball, hence have infinite volume, contradicting integrability. Thus $\varphi$ is coercive and attains its minimum. Put $W=\varphi-\min\varphi+1$ . The formula for $L$ gives $LW=n-\nabla V(\nabla\varphi)\cdot\nabla\varphi$ , so $D=\|LW\|_\infty<\infty$ .

Choose a smooth nonincreasing $\eta$ with values in $[0,1]$ , equal to one on $(-\infty,1]$ and zero on $[2,\infty)$ , and set $\chi_R=\eta(W/R)$ . These cutoffs are compactly supported. Choose smooth compactly supported $w\ge|\eta''|$ , $w\ge0$ , and put $q_R(t)=R^{-1}\int_{t/R}^{\infty}w(s)\,ds$ . Then $q_R(W)$ has compact support, $\|q_R\|_\infty\le R^{-1}\|w\|_1$ , and $-q_R'(t)=R^{-2}w(t/R)$ . Compactly supported integration by parts yields

R^{-2}\int w(W/R)\Gamma(W)\,d\nu =\int q_R(W)LW\,d\nu\le D\|w\|_1/R.

(103.7)

The chain rule $L\chi_R=R^{-1}\eta'(W/R)LW+R^{-2}\eta''(W/R)\Gamma(W)$ now gives $\|L\chi_R\|_1\le D(\|\eta'\|_\infty+\|w\|_1)/R$ .

For the last assertion, compact support and symmetry give

\int\chi_R(Lf+cf)\,d\nu=c\int\chi_R f\,d\nu+\int fL\chi_R\,d\nu.

(103.8)

The left integrand is nonnegative and the right side is uniformly bounded. Fatou’s lemma gives $Lf+cf\in L^1$ , hence $Lf\in L^1$ . Dominated convergence and $\|L\chi_R\|_1\to0$ then give $\int Lf\,d\nu=0$ .

For later use, every bounded smooth $u$ satisfies $\operatorname{Var}_\nu u\le\int\Gamma(u)\,d\nu$ , allowing infinity on the right. Here the precise boundary input is Kolesnikov–Milman, Theorem 1.2(1) (DOI). Set $M_R=\{\varphi\le R\}$ for $R>\min\varphi$ , with its Euclidean metric and conditional probability $\nu_R=\nu|_{M_R}/\nu(M_R)$ . These closed sublevels are compact, connected and convex. Since $H\succ0$ , their boundaries are smooth and $\nabla\varphi$ does not vanish there. With outward normal $\nabla\varphi/|\nabla\varphi|$ , their second fundamental form on tangent vectors is $H/|\nabla\varphi|\succeq0$ (vacuously on the zero-dimensional boundary when $n=1$ ). The conditional potential is $\varphi+\log\nu(M_R)$ , so the Euclidean $N=\infty$ curvature tensor in the cited theorem is exactly $D^2\varphi=H\succ0$ ; its prefactor $N/(N-1)$ equals one. The theorem applies to arbitrary $C^1(M_R)$ tests, with no Neumann condition on $u$ , and yields

\operatorname{Var}_{\nu_R}u \le\frac{1}{\nu(M_R)}\int_{M_R}\nabla u^TH^{-1}\nabla u\,d\nu.

(103.9)

The domains exhaust $\mathbb R^n$ and $\nu(M_R)\to1$ . Boundedness of $u$ passes its conditional first and second moments; monotone convergence passes the unnormalized nonnegative energy integrals. This proves the claimed inequality, including the case of infinite energy.

The full tensor comparison¶

Differentiating the logarithmic Monge–Ampère identity twice gives

LH+H=A+Q,\qquad A=H(D^2V\circ\nabla\varphi)H,\qquad Q_{ij}=H^{ac}H^{bd}\varphi_{abi}\varphi_{cdj}.

(103.10)

For clarity, its first derivative is $H^{ab}\varphi_{abi}=-\varphi_i+V_a\varphi_{ai}$ , and differentiating it uses $\partial_jH^{ab}=-H^{ac}\varphi_{cdj}H^{db}$ . The drift term $V_a\varphi_{aij}$ moves to the left and gives $LH_{ij}$ . Both $A$ and $Q$ are positive semidefinite: convexity gives the first, and the second is the Gram matrix of $H^{-1/2}(\partial_iH)H^{-1/2}$ .

Fix $B\succeq0$ and introduce

S=\operatorname{Tr}(BHBH),\quad D_B=H^{ab}\operatorname{Tr}(B\partial_aH B\partial_bH),\quad C_B=\operatorname{Tr}(BHBQ),\quad A_B=\operatorname{Tr}(BHBA).

(103.11)

All four are nonnegative. For $D_B$ this follows by contracting the Gram matrix of $B^{1/2}(\partial_aH)B^{1/2}$ with $H^{-1}$ ; for $C_B,A_B$ it follows from the trace of a product of positive semidefinite matrices. Product differentiation gives

LS=-2S+2(D_B+A_B+C_B).

(103.12)

$S$ is bounded. The cutoff lemma with $c=2$ first gives $LS\in L^1$ and $\mathbb E LS=0$ . The nonnegative sum is therefore integrable, and each summand is integrable separately. Only at this point do we obtain

\mathbb E S=\mathbb E D_B+\mathbb E A_B+\mathbb E C_B.

(103.13)

The pointwise comparison is $C_B\ge D_B$ . Here is the coordinate calculation including the invariance that permits normalization. At the point in question take an invertible constant matrix $P$ , put $y=Pz$ , and transform

\widetilde H=P^THP,\quad \widetilde B=P^{-1}BP^{-T},\quad \widetilde T_{abc}=P_{ia}P_{jb}P_{kc}T_{ijk},\qquad T_{ijk}=\varphi_{ijk}.

(103.14)

The inverse Hessian transforms as $P^{-1}H^{-1}P^{-T}$ . Contracting two copies of $\widetilde T$ with two inverse Hessians gives $\widetilde Q=P^TQP$ ; hence $\operatorname{Tr}(\widetilde B\widetilde H\widetilde B\widetilde Q)=C_B$ . Moreover

\partial_{z_a}\widetilde H =\sum_i P_{ia}P^T(\partial_{y_i}H)P,

(103.15)

so cyclicity of the trace and $\sum_{ab}(\widetilde H^{-1})_{ab}P_{ia}P_{jb}=(H^{-1})_{ij}$ show $\widetilde D_{\widetilde B}=D_B$ . Thus each scalar in the comparison is invariant. Choose $P$ so that $P^THP=I$ , followed by an orthogonal change that diagonalizes $\widetilde B=\operatorname{diag}(b_1,\ldots,b_n)$ . It remains symmetric and positive semidefinite by congruence. In these coordinates

D_B=\sum_{ijk}b_i b_j T_{ijk}^2,\qquad C_B=\sum_{ijk}b_i^2T_{ijk}^2,

(103.16)

and full symmetry of $T$ implies

C_B-D_B=\frac12\sum_{ijk}(b_i-b_j)^2T_{ijk}^2\ge0.

(103.17)

This is a change at a single point with a constant $P$ ; no derivative of a moving normalizing frame is taken. The global mean identity and Brascamp–Lieb step remain in the original coordinates. We conclude that $\mathbb E D_B\le\tfrac12\mathbb E S$ .

Proof of the matrix assertion¶

Proof 103.2

For $B\succeq0$ , put $F=B^{1/2}HB^{1/2}$ . Its entries are bounded smooth functions, its mean is $B$ , and summing their Brascamp–Lieb inequalities gives

\mathbb E S-\operatorname{Tr}(B^2) =\sum_{ij}\operatorname{Var}(F_{ij}) \le\mathbb E D_B\le\tfrac12\mathbb E S.

(103.18)

This proves the matrix estimate for positive semidefinite $B$ , including singular $B$ . For any symmetric $B$ , in its orthonormal eigenbasis $\operatorname{Tr}(BHBH)=\sum_{ij}b_i b_j H_{ij}^2 \le\sum_{ij}|b_i||b_j|H_{ij}^2=\operatorname{Tr}(|B|H|B|H)$ . Apply the established positive case to $|B|$ , whose squared trace norm is unchanged. This proves the first assertion of Theorem 103.1.

Quadratic functions, congruence, and singular matrices¶

For a centered log-concave probability $\lambda$ , use the homogeneous dual norm

\|h\|_{H^{-1}(\lambda)} =\sup\left\{\int hg\,d\lambda:\ g\text{ locally Lipschitz},\quad g\in L^2(\lambda),\quad\int|\nabla g|^2\,d\lambda\le1\right\}, \qquad \int h\,d\lambda=0.

(103.19)

The published Barthe–Klartag Proposition 10 states that $\operatorname{Var}_\lambda f\le\sum_i\|\partial_i f\|_{H^{-1}(\lambda)}^2$ provided $f,\partial_i f\in L^2$ and $\int\partial_i f\,d\lambda=0$ for every $i$ . It requires log-concavity, not isotropy. Their Proposition 27 supplies density of $C_c^\infty(\mathbb R^n)$ in the weighted $H^1$ norm.

In target coordinates let $\tau(x)=H((\nabla\varphi)^{-1}(x))$ . For $g\in C_c^\infty(\mathbb R^n)$ the bounded functions $g(\nabla\varphi)$ and their derivatives permit ordinary source integration by parts:

\mathbb E_\mu X_i g(X) =\mathbb E_\nu\varphi_i g(\nabla\varphi) =\mathbb E_\nu\sum_jH_{ij}(\partial_jg)(\nabla\varphi) =\mathbb E_\mu\sum_j\tau_{ij}(X)\partial_jg(X).

(103.20)

If a centered full-dimensional log-concave $\lambda$ has a symmetric Stein kernel $\tau_\lambda\in L^2(\lambda)$ , Cauchy–Schwarz gives for such $g$

\left|\mathbb E (v\cdot Z)g(Z)\right| \le\left(\mathbb E|\tau_\lambda(Z)v|^2\right)^{1/2} \left(\mathbb E|\nabla g(Z)|^2\right)^{1/2}.

(103.21)

Proposition 27 extends this to every test in the dual norm: both sides of the Stein identity converge in an $H^1$ approximation by Cauchy–Schwarz, since $v\cdot Z$ and $\tau_\lambda v$ are square integrable. Therefore $\|v\cdot z\|_{H^{-1}(\lambda)}^2\le\mathbb E|\tau_\lambda(Z)v|^2$ .

Proof 103.3 (Proof of the quadratic assertion on regular targets)

First suppose $M$ is symmetric and invertible. Write $P=|M|^{1/2}$ , $J=M|M|^{-1}$ , $Z=PX$ , and $\lambda=P_\#\mu$ . Then $J^TJ=I$ , $\mathbb EZ=0$ , and $\operatorname{Cov}Z=|M|$ . Changing variables in the Stein identity gives the symmetric Stein kernel $\tau_\lambda(Z)=P\tau(X)P$ . In particular,

\mathbb E\|\tau_\lambda(Z)\|_{\mathrm{HS}}^2 =\mathbb E_\nu\operatorname{Tr}(|M|H|M|H) \le2\operatorname{Tr}(M^2).

(103.22)

The function $f(z)=z^TJz-\operatorname{Tr}M$ has zero mean and derivatives $\partial_i f(z)=2(Je_i)\cdot z$ with zero means. Since $\lambda$ has bounded support, $f$ and its derivatives are in $L^2$ . The published inequality and the dual Stein bound therefore give

\begin{aligned} \operatorname{Var}(X^TMX) &\le4\sum_i\|(Je_i)\cdot z\|_{H^{-1}(\lambda)}^2\\ &\le4\mathbb E\sum_i|\tau_\lambda(Z)Je_i|^2 =4\mathbb E\|\tau_\lambda(Z)\|_{\mathrm{HS}}^2 \le8\operatorname{Tr}(M^2). \end{aligned}

(103.23)

For a singular symmetric $M$ , let $P_0$ be the orthogonal projection onto its kernel and set $M_\delta=M+\delta P_0$ , $\delta>0$ . This matrix is invertible, $\operatorname{Tr}(M_\delta^2)=\operatorname{Tr}(M^2)+\delta^2\dim\ker M$ , and $\|X^T(M_\delta-M)X\|_2\le\delta(\mathbb E|X|^4)^{1/2}$ . Hence the estimate passes to $M$ in $L^2$ . This order avoids using an orthogonal sign matrix or an invertible congruence for a singular $M$ .

Removing target regularity and matching the normalization¶

Proof 103.4 (Proof of the quadratic assertion for every isotropic log-concave law)

Fix the dimension and an isotropic log-concave $X$ . Let $G$ be an independent standard Gaussian and $Y_\varepsilon=X+\sqrt\varepsilon G$ , for $0<\varepsilon\le1$ . Condition its law on $|Y_\varepsilon|<\varepsilon^{-1/2}$ and denote the conditional mean and covariance by $m_\varepsilon,A_\varepsilon$ . The convolved density is smooth, positive and log-concave. Conditioning on the ball preserves log-concavity and gives positive definite covariance. After the affine map $y\mapsto A_\varepsilon^{-1/2}(y-m_\varepsilon)$ its law $\mu_\varepsilon$ is an isotropic regular target: the support is a bounded open ellipsoid and its potential is smooth convex on a neighborhood of the closed ellipsoid, with the normalization constant absorbed into the potential.

All moments of $X$ are finite, so $\sup_{0<\varepsilon\le1}\mathbb E|Y_\varepsilon|^8$ is finite. Moreover

\mathbb E\left[|Y_\varepsilon|^4 \mathbf1_{\{|Y_\varepsilon|\ge\varepsilon^{-1/2}\}}\right] \le\varepsilon^2\mathbb E|Y_\varepsilon|^8\longrightarrow0.

(103.24)

The probability of the conditioning event tends to one, and $Y_\varepsilon\to X$ in $L^4$ . Thus conditional moments through degree four converge to those of $X$ ; in particular $m_\varepsilon\to0$ and $A_\varepsilon\to I$ . Continuity of the inverse square root near $I$ shows that affine normalization preserves all these moment limits. For each fixed symmetric $M$ , apply the regular-target quadratic bound to $\mu_\varepsilon$ . Variance involves only moments through degree four, so its limit gives $\operatorname{Var}(X^TMX)\le8\operatorname{Tr}(M^2)$ .

Finally $\nabla(x^TMx)=2Mx$ and isotropy gives $\mathbb E|2MX|^2=4\operatorname{Tr}(M^2)$ , yielding the gradient formulation. The supremum in Definition 25.1 is also harmless if read over all real matrices: $x^TMx=x^T(M+M^T)x/2$ and symmetrization does not increase the Hilbert–Schmidt norm. Taking the supremum proves $\mathcal Q(\mu)\le8$ .

Transfer, hypotheses, and exclusions¶

The matrix statement matches source Theorem 2.5 on exactly the class of its Lemma A.1; the quadratic statement matches source Theorem 1.2 on the whole isotropic log-concave class. The preceding argument uses no regular moment potential for the limiting law. It does not assert convergence of Hessians under the approximation, only convergence of the polynomial moments needed for the quadratic variance.

Hypothesis	Where it is used
Centering and covariance $I$	$\mathbb EH=I$ , centered derivatives, and the final gradient normalization
Convex target and smooth convex $V$ near its closure	Regular moment map, bounded coefficients, and positivity of $A$
Bounded target	Bounded $\nabla\varphi,H,LW$ and bounded regular quadratic tests
Constant symmetric $B$	Product differentiation and the tensor comparison; no derivatives of $B$ occur
Invertible $M$ at the intermediate step	Full-dimensional congruence and orthogonality of $J$ ; removed by an explicit limit
Log-concavity of the transported law	Published $H^{-1}$ inequality and Sobolev density; isotropy is not needed there
Finite polynomial moments of the original law	Gaussian conditioning and the fourth-moment passage

Fences respected. Neither imported node has a bounded_by edge. The program’s relevant boundaries are nevertheless explicit. Proposition 16.3 is respected because the conclusion is a fixed-matrix trace inequality, not $\mathbb E H^2\preceq cI$ . Remark 25.4 is respected because the proof uses a full tensor comparison, not projection-only data. The two-tail scaling boundary Remark 25.3 is untouched: no unwhitened cut source is bounded here. The occupation, bootstrap and moving-competitor boundaries Remark 33.6, Remark 33.5, Remark 32.6 and Remark 31.6 are untouched because no stochastic-time or cut-dependent assertion is proved. No comparison between the trace-upgrade cluster members is asserted, and no open antecedent is discharged by assumption.

References¶

Letwin, B. (2026). The KLS Constant is O(\log1/4 n).
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
Kolesnikov, A. V., & Milman, E. (2017). Brascamp–Lieb-Type Inequalities on Weighted Riemannian Manifolds with Boundary. The Journal of Geometric Analysis, 27(2), 1680–1702. 10.1007/s12220-016-9736-5
Barthe, F., & Klartag, B. (2019). Spectral Gaps, Symmetries and Log-Concave Perturbations.