Balasubramanian–Kasiviswanathan: compatible integration

What to retain. Balasubramanian and Kasiviswanathan (BK) bound every power of an integration operator on compatible symmetric tensor fields with one common prefactor, from finitely many polynomial observations. Localization transfers these bounds back to the next Appell coefficient, which closes an induction $c_d\le10^{8d}/(d+1)^4$ (Theorem 11.1); letting the number of observations tend to infinity gives $\CP\le1+2\cdot10^{16}$ (Theorem 11.2).

This chapter works through the BK preprint Balasubramanian & Kasiviswanathan, 2026, arXiv v1 of 6 October 2026; the bibliography entry identifies the text read, and how the statements here were checked is explained on the welcome page. BK start, like the first version of Song–Zhang (SZ v1, Chapter Song–Zhang, first version: polynomial estimates and curvature), from the Appell normalization and Letwin’s quadratic variance bound. The exponential criterion Proposition 7.1 describes what they prove, but they obtain its coefficient hypothesis without any coefficient bound from the proofs of Bizeul–Klartag–Lehec (BKL) or of the second version of Song–Zhang (SZ v2), and reach the Poincaré constant through their integration estimate rather than through that criterion.

A Gaussian calculation and the proof’s design¶

For the standard Gaussian on the line, centered integration sends $x$ to $(x^2-1)/2$ , then to $(x^3-3x)/6$ , and in general $A_d/d!$ to $A_{d+1}/(d+1)!$ ; the coefficients $c_d=1/\sqrt{d!}$ are the norms of its successive powers applied to 1 (Section Balasubramanian–Kasiviswanathan: compatible integration).

For a general measure there are three separate difficulties. Weighted divergence need not preserve the tensor fields that can be integrated; the Hodge estimate controls the projection needed to restore this property. A bound for one integration repeated naively pays its loss at every step; the operator estimate instead retains one common factor for every power. Finally, polynomial norms change with the measure; covariance-normalized localization transfers the curved integration estimate back to the original law. The induction closes only if all three estimates have compatible constants. These are the successive parts of this chapter.

Appell normalization and regular approximation¶

The coefficient convention agrees exactly with that of SZ v1. The supremum over covariance at most identity is useful because localization and subsequent linear changes of variables need not produce an isotropic law at every intermediate step.

Definition 11.1 (Uniform Appell coefficient conventions — def:bk-uniform-appell-coefficients · Correction)

For a probability law $\mu$ on $\mathbb R^n$ with all moments, define the symmetric Appell tensors by the formal identity

\frac{e^{\langle z,x\rangle}}{\mathbb E_\mu e^{\langle z,X\rangle}} =\sum_{d\ge0}\frac{\langle A_d^\mu(x),z^{\otimes d}\rangle}{d!}.

(11.1)

For $T\in\operatorname{Sym}^d\mathbb R^n$ , put $P_d^\mu[T]=\langle T,A_d^\mu\rangle$ and

c_d(\mu)=\frac1{d!}\sup_{\|T\|_{\mathrm{HS}}=1}\|P_d^\mu[T]\|_{L^2(\mu)},\qquad d\ge1,

(11.2)

where the tensor norm sums squares over ordered indices. Let $c_d^*$ be the supremum of $c_d(\mu)$ over all dimensions and all centered log-concave laws with $\operatorname{Cov}(\mu)\preceq I$ , allowing proper affine supports. Positive-degree Appell polynomials have mean zero, so these $c_d$ coincide with the coefficients in Theorem 7.1.

The analytic construction starts with a smooth potential bounded above and below in Hessian. Approximation is needed both to enter this class and to return to nonsmooth log-concave laws. The following approximation result keeps covariance, curvature and fixed-degree polynomial norms together.

Lemma 11.1 (Regular approximation preserving covariance and curvature — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:bk-regular-approximation · Correction)

Let $\mu$ be a centered full-dimensional log-concave law with $\operatorname{Cov}(\mu)\preceq I$ and curvature at least $aI$ , where $0<a\le1$ . If $X\sim\mu$ and $G$ is an independent standard Gaussian, then

\mu_\varepsilon=\mathcal L\big((X+\sqrt\varepsilon G)/\sqrt{1+\varepsilon}\big)

(11.3)

is regular in the sense of Definition 11.2, has covariance at most $I$ and curvature at least $aI$ , and converges weakly and in every fixed moment to $\mu$ . For fixed dimension and degree, its Appell coefficients and the squared $L^2$ norms of every fixed-order derivative of a fixed-degree Appell polynomial with fixed leading tensor converge to those of $\mu$ . Every isotropic log-concave law is also the weak and fixed-moment limit of regular isotropic laws. A common Poincaré bound for the approximating laws passes to locally Lipschitz finite-energy functions under the limit law, including square integrability.

Compatible tensors and the Hodge projection¶

A vector field can be integrated to a scalar only if it is curl-free. At higher rank, the corresponding condition says that differentiating in any new slot produces a fully symmetric tensor. Subtracting a constant preserves this condition, so a centered primitive can be chosen at each stage. The derivative itself is left uncentered: its constant part carries the polynomial information that must not be discarded.

Definition 11.2 (Compatible symmetric tensor calculus — def:bk-compatible-calculus · Correction)

For a centered law $\mu(dx)=Z^{-1}e^{-V(x)}dx$ on $\mathbb R^n$ with $V\in C^\infty$ and $0<aI\preceq D^2V\preceq a_+I$ , let $E_r=\operatorname{Sym}^r\mathbb R^n$ , with the norm summing squares over all ordered indices. Put $C_0=L^2(\mu)$ ; for $r\ge1$ , let $C_r$ be the closed subspace of $L^2(\mu;E_r)$ satisfying $\partial_iF_{jI}=\partial_jF_{iI}$ in distributions for every $(r-1)$ -tuple $I$ . Let $G_r=\{F\in C_r:\mathbb EF=0\}$ , so $C_r=E_r\oplus G_r$ . The raw derivative is $D_r:G_r\supset G_r\cap W^{1,2}(\mu;E_r)\to C_{r+1}$ , $D_rU=\nabla U$ . Its Hilbert adjoint is $D_r^*$ , and $H_r=D_r^*D_r$ is the operator associated with the gradient form on $G_r$ . When the inverse exists, write $\widetilde J_r=D_r^{-1}$ and $J_r=\widetilde J_r|_{G_{r+1}}$ . Define $L_r:E_{r+1}\to G_r$ by $L_rT(x)=T\mathbin{\lrcorner}x$ . Set $\mathcal H=\bigoplus_{r\ge0}G_r$ , $\mathcal E=\bigoplus_{r\ge0}E_{r+1}$ , $(JF)_r=J_rF_{r+1}$ and $(LT)_r=L_rT_r$ whenever these formulas define bounded operators.

The smallest matrix example¶

Take $F=D^2\psi$ on $\mathbb R^2$ , with $\psi$ smooth and compactly supported, and let $Y_i=\sum_j(-\partial_j+\partial_jV)F_{ij}$ be its weighted divergence. At a point where $D^2V$ is diagonal with eigenvalues $\kappa_1,\kappa_2$ , direct differentiation gives

\partial_1Y_2-\partial_2Y_1=(\kappa_1-\kappa_2)F_{12}.

(11.4)

Thus even a Hessian can acquire curl under weighted divergence. The Hodge projection estimate of the preprint bounds the squared norm of the discarded part by

\mathbb E\frac{(\kappa_1-\kappa_2)^2}{\kappa_1+\kappa_2}F_{12}^2.

(11.5)

The mixed entry initially contributes $(\kappa_1+\kappa_2)F_{12}^2$ to the curvature energy. After subtracting this loss, its remaining contribution is

\left(\kappa_1+\kappa_2- \frac{(\kappa_1-\kappa_2)^2}{\kappa_1+\kappa_2}\right)F_{12}^2 =\frac{2\kappa_1\kappa_2}{\kappa_1+\kappa_2}\,(2F_{12}^2).

(11.6)

The factor $2F_{12}^2$ is exactly the ordered-index norm of the mixed symmetric component. Its remaining weight is the harmonic mean, at least $a$ when both curvatures are at least $a$ . Equal curvatures produce no projection loss. The higher-rank estimate below retains this same lower bound without deterioration as more tensor slots are added.

This Hodge estimate concerns the potential $V$ of the original measure; it is not the moment-Hessian comparison Corollary 16.1 (Section A deterministic inequality, sharp on products and Dirichlet laws).

Lemma 11.2 (Compatible-tensor Hodge estimate and operator domains — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:bk-compatible-hodge · Correction)

For every law in Definition 11.2 and every integer $r\ge0$ , $D_r$ is closed, densely defined and bijective, and $\|D_r^{-1}\|\le\sqrt{\CP(\mu)}$ . The fields $\{\nabla^{r+1}\psi:\psi\in C_c^\infty(\mathbb R^n)\}$ form a graph core for $D_r^*$ . Every $F\in\operatorname{Dom}(D_r^*)$ belongs to $W^{1,2}(\mu;E_{r+1})$ and satisfies

\|D_r^*F\|_2^2\ge\|\nabla F\|_2^2+a\|F\|_2^2.

(11.7)

The coefficient $a$ is independent of $r$ and $n$ .

The proof begins with weighted integration by parts, then subtracts the energy lost by projecting divergence onto compatible fields. The preceding harmonic-mean calculation illustrates the curvature left after subtraction. At general rank, symmetry must prevent a loss growing with the number of indices. Extending the estimate from compactly supported potentials to the adjoint domain is part of the assertion, not a formal consequence of the calculation on smooth fields. The inverse derivative then provides the integration operators used below.

From polynomial tests to every integration power¶

The decomposition into constant and centered tensors separates the inverse derivative into $L$ and $J$ . Constants produce linear functions through $L$ ; repeated application of $J$ produces normalized Appell polynomials. This is the abstract version of the Gaussian calculation above.

Proposition 11.1 (Compatible integration and Appell coefficients — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · prop:bk-integration-calculus · Correction)

For every law in Definition 11.2 with $\operatorname{Cov}(\mu)\preceq I$ , the operators $J$ and $L$ are bounded, and

\|J\|^2\le \CP(\mu)\le1+\|J\|^2,\qquad \|L\|\le1.

(11.8)

For every $r\ge0$ ,

H_r^{-1}=J_rJ_r^*+L_rL_r^*,

(11.9)

and, with $g_a(t)=t/(1+at)$ defined on positive operators by spectral calculus,

J^*J\preceq g_a(JJ^*+LL^*).

(11.10)

For every $j\ge1$ , with the Appell normalization of Definition 11.1,

\|J^{j-1}L\|\le c_j(\mu).

(11.11)

More precisely, for every $r,k\ge0$ and $T\in E_{r+k+1}$ ,

J_r\cdots J_{r+k-1}L_{r+k}T=\frac1{(k+1)!}T\mathbin{\lrcorner}A_{k+1}^\mu,

(11.12)

where the product is $L_r$ if $k=0$ and the contraction leaves $r$ free indices.

The identity for $H_r^{-1}$ comes from this orthogonal decomposition. The Hodge lower bound, followed by inversion of positive operators, gives the inequality with $g_a$ . The Appell identity follows by differentiating their generating series and selecting the mean-zero primitive at every step. The remaining issue is operator theoretic: estimates on the special inputs $J^{j-1}L$ must control $J^k$ on arbitrary compatible fields.

The following operator lemma isolates that issue on any Hilbert space. Its crucial feature is that $Q_D(B)$ does not depend on the power $k$ .

Lemma 11.3 (Uniform bound for powers from finitely many observations — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:bk-uniform-power-bound · Correction)

Let $R:X\to X$ and $L:Y\to X$ be bounded operators on real or complex Hilbert spaces, let $a>0$ , and suppose

R^*R\preceq g(RR^*+LL^*),\qquad g(u)=\frac{u}{1+au}.

(11.13)

For an integer $D\ge2$ , let $\gamma_2,\ldots,\gamma_D\ge0$ be finite and satisfy $\|R^{j-1}L\|\le\gamma_j$ for $2\le j\le D$ . Then for every $B>0$ with $aB^{D+1}>\gamma_D^2$ , simultaneously for all integers $k\ge0$ ,

\|R^k\|^2\le B^kQ_D(B),\qquad Q_D(B)=1+\sum_{j=1}^{D-2}\frac{j\gamma_{j+1}^2}{B^{j+1}}.

(11.14)

The sum is empty for $D=2$ .

The preprint uses the concavity and monotonicity of $g(u)=u/(1+au)$ to compare successive operator powers. The observed terms $R^{j-1}L$ control the boundary contributions, while the strict inequality $aB^{D+1}>\gamma_D^2$ absorbs the last one. Retaining the finite sum $Q_D(B)$ avoids multiplying a separate prefactor for every integration. At $D=2$ the sum is empty and the prefactor is exactly one; this is also what makes a degree-two estimate sufficient to start the argument.

Corollary 11.1 (Uniform integration powers — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · cor:bk-integration-powers · Correction)

Let $\nu$ be a centered regular log-concave law with $\operatorname{Cov}(\nu)\preceq I$ and $D^2V\succeq aI$ for $a>0$ , with coefficients and integration operators as in Definition 11.1 and Definition 11.2. If $D\ge2$ is an integer, $\rho\ge1$ , and $c_j(\nu)\le\rho^j/(j+1)^4$ for $2\le j\le D$ , then with $B=\rho^2\max\{1,a^{-1/(D+1)}\}$ one has, simultaneously for all integers $q\ge1$ ,

\|J^q\|^2\le2B^q,\qquad\|J_0\cdots J_{q-1}\|^2\le2B^q.

(11.15)

In particular $\CP(\nu)\le1+2B$ .

Substituting the coefficient majorant in $Q_D$ gives a convergent sum bounded by two. This yields all powers with one factor two, rather than a factor $2^q$ . The seed below uses only Letwin’s quadratic variance estimate Theorem 25.1: it gives $c_2\le\sqrt2$ , hence $\|JL\|\le\sqrt2$ . The $D=2$ operator estimate then gives the stated bound by taking $B$ down to $(2/a)^{1/3}$ . No higher-degree coefficient estimate is needed to start.

Returning from curved laws to polynomial norms¶

Localization introduces curvature, but the polynomial and its law both move. BK use noise normalized by the inverse covariance. Letwin’s quadratic estimate controls the third moments in the covariance equation; tensor covariance bounds keep that control when several derivatives are contracted together. These estimates hold on the full ordered tensor spaces, so their use does not hide a dimension-dependent trace.

Lemma 11.4 (Global covariance-normalized localization — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:bk-localization-covariance · Correction)

Let $\mu_0$ be any compactly supported isotropic log-concave law on $\mathbb R^n$ . Write

\mu_{\theta,\Lambda}(dx)=Z(\theta,\Lambda)^{-1} e^{\theta\cdot x-x^T\Lambda x/2}\mu_0(dx),

(11.17)

and let $m(\theta,\Lambda)$ and $A(\theta,\Lambda)$ be its mean and covariance. The SDE, started at $(0,0)$ ,

d\theta_t=A_t^{-1/2}dB_t+A_t^{-1}m_tdt, \qquad d\Lambda_t=A_t^{-1}dt, \qquad \mu_t=\mu_{\theta_t,\Lambda_t},

(11.18)

has a global solution with $A_t\succ0$ . Every bounded Borel test has $\int f\,d\mu_t$ a true martingale. On the full ordered tensor spaces, for every integer $k\ge1$ , $0\le s\le t$ , every integer $d\ge1$ , and $s_1,\ldots,s_d\in[0,t]$ ,

\mathbb E[A_t^{\otimes k}\mid\mathcal F_s] \preceq e^{4k^2(t-s)}A_s^{\otimes k},\qquad \mathbb E[A_{s_1}\otimes\cdots\otimes A_{s_d}] \preceq e^{4d^2t}I.

(11.19)

For $t>0$ , $0\le q\le d$ and $\delta\ge0$ ,

\mathbb E[(A_t+\delta\Lambda_t^{-1})^{\otimes q} \otimes A_t^{\otimes(d-q)}] \preceq (1+\delta/t)^q e^{4d^2t}I.

(11.20)

The covariance proof first stops the process where the covariance and its inverse are bounded, derives the tensor inequalities there, and then removes the stopping. The mixed-time bound is needed because the accumulated curvature integrates inverse covariances over earlier times. The estimate with $\Lambda_t^{-1}$ is what allows that curvature to enter the later change of coordinates. Global existence and true martingales are included explicitly to justify the expectation identities.

The next assertion follows the polynomial adapted to the current law. Its lower-degree errors involve a convolution of previously bounded Appell coefficients. This is the induction’s error term.

Lemma 11.5 (Moving Appell variance — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · lem:bk-moving-appell-variance · Correction)

Fix $d\ge2$ and finite constants $C_k\ge0$ such that $c_k(\nu)\le C_k$ for every isotropic log-concave law in every dimension and $1\le k<d$ . Set $\Sigma_d=\sum_{k=2}^{d-1}(d-k+1)C_kC_{d-k+1}$ , with an empty sum zero. For the localization of Lemma 11.4, fixed $T\in\operatorname{Sym}^d\mathbb R^n$ , $p_t=P_d^{\mu_t}[T]$ and $V(t)=\mathbb E E_tp_t^2$ , one has, for all $t\ge0$ ,

\sqrt{V(0)}\le e^{(d+1)t}\sqrt{V(t)} +d!\Sigma_dt\,e^{(2d^2+d+1)t}\|T\|.

(11.21)

Differentiating the normalized Appell generating function along the localization produces the evolution of its variance. Taking a square root allows the error terms to be bounded by products of lower-degree norms. Integrating the resulting differential inequality compares the original polynomial norm with its later norm. The latter is estimated by integrating $q$ times from degree $d-q$ under the now curved law.

Proposition 11.2 (Reverse coefficient transfer — Proved (from a preprint) · agent review (gpt-6-astra, 2026-10-06) · prop:bk-reverse-transfer · Correction)

Fix integers $d\ge2$ and $1\le q\le d-1$ , a real $0<\eta\le1$ , and put $\delta=\eta^2/d^2$ . Suppose $M_q<\infty$ is nonnegative and every centered regular log-concave law $\nu$ in every dimension with $\operatorname{Cov}(\nu)\preceq I$ and curvature at least $\delta I$ satisfies $\|J_0^\nu J_1^\nu\cdots J_{q-1}^\nu\|_{\mathrm{op}}\le M_q$ . Suppose finite $C_1,\ldots,C_{d-1}\ge0$ satisfy $c_k(\nu)\le C_k$ for every isotropic log-concave law in every dimension and $1\le k<d$ . Set $\Sigma_d=\sum_{k=2}^{d-1}(d-k+1)C_kC_{d-k+1}$ , with empty sum zero. Then the supremum over all centered log-concave laws in every dimension with covariance at most identity satisfies

c_d^*\le e^{3\eta}\left[(1+\eta)^{q/2}M_qC_{d-q}+\frac{\eta}{d^2}\Sigma_d\right].

(11.22)

The transfer uses time $t=\eta/d^2$ . Its curvature parameter $\delta=\eta^2/d^2$ and the covariance tensor bound together account for the factor $(1+\eta)^{q/2}$ . The principal term contains an actual chain of $q$ integration operators; replacing its norm by a product of separate one-step estimates would lose the common-factor advantage. The second term uses only degrees below $d$ , so the estimate can close an induction.

Closing the induction and extracting the constant¶

The preprint chooses the summable majorant $\beta_d=R_*^d/(d+1)^4$ . Its convolution has enough decay to absorb the moving-polynomial error. A finite initial range is supplied by the quadratic seed; for larger degrees the integration length is $q=\lfloor d/2\rfloor$ , of order $d$ , and the observation degree is $D=\lfloor\sqrt d\rfloor$ , of order $\sqrt d$ . Both are below $d$ . The strict margin in the operator bound leaves room for the transfer factors. The explicit choices in Appendix E of the preprint lead to the following value.

To extract a spectral gap, fix one regular law and its positive lower curvature $a$ . Let the observation degree $D$ tend to infinity in Corollary 11.1, with $\rho=R_*$ . Then $a^{-1/(D+1)}\to1$ , and the one-step estimate gives $\CP\le1+2R_*^2$ . Only after this degree limit is taken does approximation pass the common scalar inequality to arbitrary log-concave laws. This order avoids requiring a common positive curvature for all laws.

The extension to a proper affine support is made within that support. For nonsmooth laws, the approximation statement must transfer the inequality to finite-energy locally Lipschitz functions as well as smooth tests; square integrability is part of the conclusion. The Cheeger consequence then uses the normalization already fixed in the introduction, with the factor $\pi$ coming from its reverse comparison.

The new mechanism of BK is the combination of a rank-independent Hodge estimate, a common bound for every integration power, and reverse transfer. It is compared with BKL and SZ v2 in Chapter The proofs of KLS compared.

References¶

Balasubramanian, K., & Kasiviswanathan, S. (2026). A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures. https://arxiv.org/abs/2610.07728v1