BK: integration and Appell observations - The KLS theorem and its methods

Part of the Balasubramanian–Kasiviswanathan proof, Chapter Balasubramanian–Kasiviswanathan: compatible integration; the reading order is on the full proofs page.

Overview. We reconstruct Section 4 of Balasubramanian & Kasiviswanathan, 2026, at Git commit 4837c33649ba2271f43c9684e9350ecbdd725f95. The bounded inverse of the raw compatible derivative splits into centered and constant input blocks. The Hodge estimate gives a form comparison whose inverse controls the centered integration operators. Taking a Hilbert direct sum preserves the rank-uniform inequalities; Appell differentiation identifies the constant-input products exactly.

Dependencies. The substantive analytic input is Lemma 11.2. Notation and coefficient normalization are Definition 11.2 and Definition 11.1. We use only the formal Appell definition, not any uniform bound for its coefficients. No KLS, BKL, Song–Zhang v2 or later BK theorem is an input.

Theorem 152.1 (Integration operators on all tensor ranks)

Let $\mu(dx)=Z^{-1}e^{-V(x)}dx$ be centered, $V\in C^\infty$ , with $0<aI\preceq D^2V\preceq a_+I$ and $\operatorname{Cov}(\mu)\preceq I$ . For the notation of Definition 11.2, all $\widetilde J_r$ exist, and $J,L$ are bounded on their indicated direct-sum spaces. They obey

\|J\|^2\le C_P(\mu)\le1+\|J\|^2,\qquad \|L\|\le1, \qquad H_r^{-1}=J_rJ_r^*+L_rL_r^* \quad(r\ge0).

(152.1)

For $g_a(t)=t/(1+at)$ , spectral calculus gives

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

(152.2)

For every $j\ge1$ ,

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

(152.3)

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 =\frac{1}{(k+1)!}T\mathbin{\lrcorner}A_{k+1}^{\mu},

(152.4)

where the contraction leaves $r$ free indices and the left side is $L_rT$ when $k=0$ . This is Proposition 11.1.

Proof 152.1

Set $p=C_P(\mu)$ . By Lemma 11.2, $D_r$ is closed, densely defined and bijective, with bounded inverse $\widetilde J_r:C_{r+1}\to G_r$ , satisfying $\nabla\widetilde J_rF=F$ and $\mathbb E\widetilde J_rF=0$ . Its operator norm is at most $\sqrt p$ . Since $C_{r+1}=E_{r+1}\oplus G_{r+1}$ , the inverse is the row operator $(L_r,J_r)$ : on constants the centered primitive is $L_rT=T\mathbin{\lrcorner}x$ . This uses $\mathbb EX=0$ . For each ordered $r$ -tuple $I$ , the covariance bound gives

\|L_rT\|_2^2 =\sum_I\mathbb E\Big(\sum_jT_{Ij}X_j\Big)^2 \le\sum_{I,j}T_{Ij}^2=\|T\|^2. \tag{1}

(152.5)

Thus $\|L_r\|\le1$ independently of $r$ . Similarly, scalar Poincaré on every centered component of $J_rF$ gives $\|J_rF\|_2^2\le p\|F\|_2^2$ ; equivalently this follows from the inverse bound in the Hodge lemma.

Inverse identities, with domains. A closed densely defined bijective operator with everywhere bounded inverse has bijective adjoint and $(D_r^*)^{-1}=(D_r^{-1})^*=\widetilde J_r^*$ . Indeed, the equations defining the Hilbert adjoint of the bounded inverse are exactly those defining the inverse of the unbounded adjoint; injectivity uses density of $\operatorname{Dom}(D_r)$ and surjectivity of $D_r$ . In particular $\widetilde J_r\widetilde J_r^*f$ lies in $\operatorname{Dom}(D_r^*D_r)$ : applying $D_r$ gives $\widetilde J_r^*f\in\operatorname{Dom}(D_r^*)$ , whose adjoint image is $f$ . Conversely applying the product inverse to $D_r^*D_ru$ recovers $u$ . Therefore

H_r^{-1}=\widetilde J_r\widetilde J_r^* =J_rJ_r^*+L_rL_r^*. \tag{2}

(152.6)

The same argument in the other order gives $(D_rD_r^*)^{-1}=\widetilde J_r^*\widetilde J_r$ . The operators $D_r^*D_r$ and $D_rD_r^*$ are the positive self-adjoint operators associated with their standard closed forms.

Inverting the Hodge form inequality. Let $\widetilde H_{r+1}$ be the operator on $C_{r+1}$ associated with the full gradient form, whose domain is $C_{r+1}\cap W^{1,2}$ . The potential cores from the Hodge lemma’s proof ensure density. Splitting off constants shows $\widetilde H_{r+1}=0\oplus H_{r+1}$ relative to $C_{r+1}=E_{r+1}\oplus G_{r+1}$ . The Hodge lemma gives both

\operatorname{Dom}(D_r^*)\subset \operatorname{Dom}(\widetilde H_{r+1}^{1/2}),\qquad \|D_r^*F\|_2^2\ge \|\widetilde H_{r+1}^{1/2}F\|_2^2+a\|F\|_2^2.

(152.7)

This is the form order $D_rD_r^*\succeq\widetilde H_{r+1}+aI$ . The order includes a domain inclusion; it is not merely an identity on a smooth core. For a positive self-adjoint $T$ with positive lower bound,

\langle f,T^{-1}f\rangle= \sup_{u\in\operatorname{Dom}(T^{1/2})} \{2\operatorname{Re}\langle f,u\rangle-\|T^{1/2}u\|^2\}.

(152.8)

Thus inversion reverses this form order, and

\widetilde J_r^*\widetilde J_r \preceq(\widetilde H_{r+1}+aI)^{-1}.

(152.9)

Compress to $G_{r+1}$ to obtain

J_r^*J_r\preceq(H_{r+1}+aI)^{-1} =g_a(H_{r+1}^{-1}) =g_a(J_{r+1}J_{r+1}^*+L_{r+1}L_{r+1}^*). \tag{3}

(152.10)

The middle equality follows on each spectral value $\lambda>0$ from $(\lambda+a)^{-1}=g_a(\lambda^{-1})$ ; continuity of $g_a$ at zero also covers spectral accumulation of $H_{r+1}^{-1}$ there. No operator monotonicity of an arbitrary scalar concave function is asserted or needed.

The simultaneous direct sum. For square-summable $F=(F_r)$ and $T=(T_r)$ , (1) and the uniform bound $\|J_r\|\le\sqrt p$ imply

\sum_{r\ge0}\|J_rF_{r+1}\|_2^2\le p\sum_{r\ge0}\|F_r\|_2^2, \qquad \sum_{r\ge0}\|L_rT_r\|_2^2\le\sum_{r\ge0}\|T_r\|^2.

(152.11)

So $J$ and $L$ extend to bounded operators with $\|J\|^2\le p$ and $\|L\|\le1$ . Their norms are the suprema of the block norms. At rank zero $H_0$ is the scalar Poincaré form operator on mean-zero $L^2$ , so its variational definition gives $p=\|H_0^{-1}\|$ . Equation (2) and (1) yield

p\le\|J_0\|^2+\|L_0\|^2\le\|J\|^2+1.

(152.12)

Both $JJ^*$ and $LL^*$ preserve each $G_r$ and have respective blocks $J_rJ_r^*$ and $L_rL_r^*$ . Functional calculus of their sum preserves the same decomposition (first for polynomials, then by uniform approximation on its bounded spectrum). On $G_r$ for $r\ge1$ , $J^*J$ has block $J_{r-1}^*J_{r-1}$ , so (3) proves the desired comparison there. On $G_0$ its block is zero and the right side is positive. Summing the quadratic forms proves $J^*J\preceq g_a(JJ^*+LL^*)$ on all of $\mathcal H$ .

Appell normalization and all constant-input products. The lower curvature bound gives $V(x)\ge V(0)+\langle\nabla V(0),x\rangle+a|x|^2/2$ , so all polynomial moments exist. The formal generating identity defining $A_j^\mu$ gives $A_1^\mu(x)=x$ , $\mathbb EA_j^\mu=0$ for $j\ge1$ , and

\nabla\left[\frac1{j!}T\mathbin{\lrcorner}A_j^\mu\right] =\frac1{(j-1)!}T\mathbin{\lrcorner}A_{j-1}^\mu, \tag{4}

(152.13)

where the right contraction has one additional free index. To derive these facts, take expectations in the formal generating series, yielding the constant series one, and differentiate it in $x$ , which multiplies it by the formal variable. Coefficient comparison proves (4). The polynomial field on the left is compatible since its derivative is fully symmetric; it is centered for $j\ge1$ and belongs to all required finite Sobolev orders.

For $k=0$ the claimed identity is $L_rT=T\mathbin{\lrcorner}x$ . For $k\ge1$ , the derivative of its proposed right side equals $T\mathbin{\lrcorner}A_k^\mu/k!$ , which by induction is $J_{r+1}\cdots J_{r+k-1}L_{r+k}T$ and is centered. Uniqueness of the centered primitive therefore proves the identity at rank $r$ .

For an ordered free-index tuple $I$ of length $r$ , let $T_I$ be the symmetric rank- $k+1$ slice of $T$ . The definition of $c_{k+1}(\mu)$ gives

\left\|\frac{\langle T_I,A_{k+1}^\mu\rangle}{(k+1)!}\right\|_2^2 \le c_{k+1}(\mu)^2\|T_I\|^2.

(152.14)

Summing over $I$ and using $\sum_I\|T_I\|^2=\|T\|^2$ proves

\|J_r\cdots J_{r+k-1}L_{r+k}\|\le c_{k+1}(\mu).

(152.15)

For fixed $j\ge1$ , the nonzero blocks of $J^{j-1}L$ are precisely these products with $k=j-1$ . Their source summands $E_{r+j}$ and target summands $G_r$ are mutually orthogonal as $r$ varies; the lower source summands are annihilated by the shift. Taking the supremum of block norms proves $\|J^{j-1}L\|\le c_j(\mu)$ .

Hypotheses and limits. Centering identifies $L_r$ with contraction against $x$ ; the covariance bound supplies its norm at most one. Positive curvature and regularity are used only through the Hodge/domain result and existence of moments. The common bound $\sqrt{C_P(\mu)}$ justifies the infinite direct sum. Every $c_j(\mu)$ here is for this fixed law; no dimension-uniform or degree-uniform coefficient estimate has been inferred.

Fences respected. No bounded_by nodes are assigned. This calculus supplies operator inputs for a later argument. It makes no CMH, occupation, trace-upgrade, or curvature-free spectral conclusion and takes none as an input.

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