BKL: analytic foundations and tensor symmetrization

Part of the Bizeul–Klartag–Lehec proof, Chapter Bizeul–Klartag–Lehec: cumulants and suspension; the reading order is on the full proofs page.

Overview. This reconstructs the analytic interfaces of Section 2 of Bizeul et al., 2026, version arXiv:2610.05474v1. We establish closure of the polynomial-growth class under the inverse Laplacian, justify the eigenfunction used by the criterion, and prove the finite-dimensional tensor recovery inequality by an incidence calculation. Regular approximation is arranged through compactly supported intermediate measures; this makes the extra derivative-growth requirement explicit.

Dependencies. The normalization is Definition 8.1. The spectral construction, graph core, Bochner identity and scalar Poincaré stability for smooth potentials with two-sided positive Hessian bounds are supplied by Lemma 7.1. We prove below the additional polynomial-growth assertions needed here. The tensor argument has no analytic or probabilistic input. In particular, neither KLS nor the uniform Appell bound is used.

The analytic class and its operators¶

Theorem 130.1 (Analytic foundations in the BKL regular class)

Let $\mu(dx)=e^{-V(x)}dx$ be a probability on $\mathbb R^n$ , where $V\in C^\infty$ , $aI\preceq D^2V\preceq bI$ for some $0<a\le b<\infty$ , and every derivative of $V$ has at most polynomial growth. Let $\mathcal A$ consist of the smooth functions all of whose derivatives have at most polynomial growth, and put $\mathcal A_0=\{g\in\mathcal A:\int g\,d\mu=0\}$ . Set $L=\Delta-\nabla V\cdot\nabla$ .

The operator $L$ preserves $\mathcal A$ , and for $u,v\in\mathcal A$ ,

\int (Lu)v\,d\mu=-\int\nabla u\cdot\nabla v\,d\mu, \qquad \|Lu\|_2^2=\|D^2u\|_2^2+ \int\langle D^2V\nabla u,\nabla u\rangle\,d\mu.

(130.1)

Writing $\lambda=C_P(\mu)^{-1}$ , one has $\lambda>0$ and a real $f\in\mathcal A_0$ with $\|f\|_2=1$ and $-Lf=\lambda f$ . For every $g\in\mathcal A_0$ there is exactly one $u\in\mathcal A_0$ satisfying $-Lu=g$ , and

\|u\|_2\le\lambda^{-1}\|g\|_2,\qquad \|\nabla u\|_2^2=\langle g,u\rangle \le\lambda^{-1}\|g\|_2^2,\qquad \|\nabla_0u\|_2\le\lambda^{-1/2}\|g\|_2, \tag{A1}

(130.2)

where $\nabla_0u=\nabla u-\int\nabla u\,d\mu$ . All identities and inequalities extend componentwise to tensors, using the sum over ordered indices. Every isotropic log-concave probability is a weak limit of isotropic probabilities in this regular class, with convergence of every polynomial moment. A uniform scalar Poincaré inequality on that class passes to the limit. These are the assertions of Lemma 8.1.

Proof 130.1

Strong convexity gives $V(x)\ge V(0)+\nabla V(0)\cdot x+a|x|^2/2$ . Thus polynomial-growth functions and their derivatives belong to every finite $L^p(\mu)$ . Products and derivatives preserve $\mathcal A$ , as does $L$ . Cut off at radius $R$ with a smooth function $\chi_R$ whose first two derivatives are bounded by $C/R$ and $C/R^2$ . In integration by parts the error is an integral of a polynomial-growth function times a derivative of $\chi_R$ , supported where $|x|\ge R$ ; Gaussian decay makes it tend to zero. This proves the first identity. In particular $u,Lu\in L^2(\mu)$ and $u$ is in the weak domain of $-L$ . The graph-core and Bochner assertions of Lemma 7.1 apply and give the second identity. They also give compact resolvent, constants as the kernel, and a normalized first nonconstant eigenfunction in the operator domain, with eigenvalue $\lambda=C_P^{-1}>0$ .

We next show that this eigenfunction lies in $\mathcal A$ ; the argument also specifies the growth issue behind this regularity statement. Interior elliptic regularity first makes $f$ smooth. Choose an integer $m$ with $2ma>\lambda+1$ and put $w(x)=(1+|x|^2)^m$ . The Hessian bounds imply

\langle x,\nabla V(x)\rangle\ge a|x|^2-O(|x|), \qquad \frac{Lw}{w}\le-2ma+O(|x|^{-1})\quad (|x|\longrightarrow\infty). \tag{A2}

(130.3)

Indeed, $\nabla w/w=2mx/(1+|x|^2)$ and $\Delta w/w=O(|x|^{-2})$ ; insert these in $Lw/w$ . Fix a radius outside which $c=-Lw/w-\lambda\ge c_0>0$ . Let $v=f/w$ and let $d\rho=w^2\,d\mu$ . The equation for $f$ becomes, in weak form,

-\operatorname{div}(w^2e^{-V}\nabla v) +c\,w^2e^{-V}v=0. \tag{A3}

(130.4)

Here $v\in L^2(\rho)$ and $\nabla v\in L^2(\rho)$ : the latter follows from $\nabla f\in L^2(\mu)$ and boundedness of $\nabla\log w$ . The function $c$ is bounded, since $\nabla V$ has linear growth. Choose $C>0$ so that $v<C$ on a slightly larger closed ball and set $h=(v-C)_+$ . Its support lies in the exterior where $c\ge c_0$ . Testing (A3) against $\chi_R^2h$ gives

\int\chi_R^2|\nabla h|^2\,d\rho+ \int c\,v h\,\chi_R^2\,d\rho =-2\int\chi_Rh\,\nabla h\cdot\nabla\chi_R\,d\rho.

(130.5)

On $\{h>0\}$ , $v=h+C$ , so the second integral is nonnegative. Young’s inequality therefore bounds half the gradient integral and the entire second integral by $2\int h^2|\nabla\chi_R|^2\,d\rho\le CR^{-2}\|v\|_{L^2(\rho)}^2$ . Letting $R\to\infty$ gives $h=0$ , since $c\,vh\ge c_0h^2$ . Apply the same argument to $-f$ . Thus $|f|\le Cw$ . This weak barrier argument does not require pointwise decay of a Schrödinger transform of $f$ at infinity.

For completeness, polynomial growth of all derivatives follows from local elliptic estimates with a shrinking radius. At $x_0$ use $r=(1+|x_0|)^{-1}$ and rescale $x=x_0+ry$ to a fixed unit ball. The rescaled equation has Laplacian principal part and drift $r\nabla V(x_0+ry)$ bounded uniformly on that ball. Its first derivatives are bounded uniformly because $D^2V$ is bounded. Interior gradient estimates bound $|\nabla f(x_0)|$ by a fixed power of $1+|x_0|$ times the supremum of $|f|$ on the ball. For a multi-index $\alpha$ , differentiating $(L+\lambda)f=0$ yields

(L+\lambda)\partial^\alpha f =\sum_{0<\beta\le\alpha}\binom{\alpha}{\beta} (\partial^\beta\nabla V)\cdot \nabla\partial^{\alpha-\beta}f. \tag{A4}

(130.6)

If the derivatives of $f$ through order $|\alpha|$ have polynomial growth, the right side does too. Apply the same interior gradient estimate to $\partial^\alpha f$ , now with this right side. Rescaling contributes only powers of $r^{-1}$ ; the coefficient derivatives are polynomially bounded by the hypotheses on $V$ . Induction gives polynomial growth at every order. Hence $f\in\mathcal A$ . The estimates invoked here are the classical interior estimates for a uniformly elliptic equation with smooth coefficients on a fixed ball; no estimate uniform in derivative order, dimension or $\mu$ is needed.

For $g\in\mathcal A_0$ , spectral calculus on the centered subspace gives a unique $u\in\operatorname{Dom}(-L)\cap L^2_0(\mu)$ with $-Lu=g$ and $\|u\|_2\le\lambda^{-1}\|g\|_2$ . The form identity gives the middle statement of (A1). Interior regularity again makes $u$ smooth. Choose $m$ large enough that the same polynomial $w$ satisfies $-Lw\ge|g|$ outside a ball; (A2) and polynomial growth of $g$ permit this choice. Choose $C\ge1$ with $u<Cw$ on a slightly larger ball. Now $h=(u-Cw)_+$ vanishes near that ball, lies in the form domain, and satisfies $0\le h\le |u|$ . The weak equation for $u-Cw$ gives

\int\chi_R^2|\nabla h|^2\,d\mu +2\int\chi_Rh\,\nabla h\cdot\nabla\chi_R\,d\mu =\int(g+CLw)\chi_R^2h\,d\mu\le0.

(130.7)

Consequently $\int\chi_R^2|\nabla h|^2\,d\mu \le4\int h^2|\nabla\chi_R|^2\,d\mu \le CR^{-2}\|u\|_2^2$ . Exhaustion makes $\nabla h=0$ , and $h$ vanishes on a ball, so $h=0$ . Repeating with $-u$ gives $|u|\le Cw$ . Differentiate $Lu=-g$ and repeat the rescaled interior estimate: the new right sides involve derivatives of $g,V$ and already controlled derivatives of $u$ . Thus $u\in\mathcal A_0$ . Uniqueness in this class follows from the form identity: a centered solution of $Lv=0$ has zero gradient and is zero. Subtracting the mean from $\nabla u$ is an orthogonal projection in $L^2$ , proving the last inequality of (A1). Summing the scalar arguments over tensor components proves all componentwise claims.

Finally consider an arbitrary isotropic log-concave $\mu$ . Condition on growing balls, then center and whiten. The resulting compactly supported isotropic laws converge weakly to $\mu$ , and their moments converge at each fixed order, by integrability of all polynomials and convergence of the affine normalizations to the identity. For a fixed one of these compactly supported laws, write it as the law of $X$ , and convolve with $N(0,\delta I)$ . Apart from a constant its density is

p_\delta(y)= e^{-|y|^2/(2\delta)} \int e^{y\cdot x/\delta-|x|^2/(2\delta)}\,d\mu_X(x). \tag{A5}

(130.8)

The logarithmic derivatives of the last integral are cumulants of a probability supported in the same fixed compact set. Every such derivative, of each fixed order, is bounded uniformly in $y$ : it is a finite polynomial in moments of bounded coordinates. Thus derivatives of $-\log p_\delta$ of order at least two are bounded, while its first derivative has at most linear growth. The Hessian formula and convolution log-concavity in Lemma 7.1 give $0\preceq D^2(-\log p_\delta)\preceq\delta^{-1}I$ . Multiplication of $p_\delta$ by $e^{-\epsilon|y|^2/2}$ , followed by normalization, gives a potential with $\epsilon I\preceq D^2V\preceq(\epsilon+\delta^{-1})I$ and all derivatives polynomially bounded. Centering and whitening preserve this class. For each compact intermediate law, letting $\delta\to0$ preserves every fixed polynomial moment, by the binomial expansion of $X+\sqrt \delta Z$ and finiteness of Gaussian moments. For fixed $\delta$ , letting $\epsilon\to0$ preserves those moments by dominated convergence. For the $j$ th intermediate law choose $\delta_j,\epsilon_j$ so that the errors of all moments of degree at most $j$ are at most $1/j$ . Include convergence against a countable determining family of bounded continuous functions in this diagonal choice. Its means and covariances tend to $0,I$ ; centering and whitening therefore preserve weak convergence and convergence of every fixed polynomial moment. The individual Hessian bounds need not be uniform. The scalar stability assertion of Lemma 7.1 passes any common Poincaré bound to $\mu$ , including locally Lipschitz tests of finite energy. This completes the analytic assertions.

Recovering a tensor from partial symmetrization¶

Proof 130.2

Put $\Omega=\{1,\ldots,3d\}$ . For each $d$ -element subset $F$ of $\Omega$ , let $T_F$ be the copy of $T$ obtained by moving its first block into the slots $F$ and the second block into $\Omega\setminus F$ . Leave all other slots fixed. The assumed symmetries make $T_F$ independent of the order chosen within either block. Expand the following sums using the Hilbert-space scalar product:

\sum_{\substack{E\subset\Omega\\|E|=2d}} \left|\sum_{\substack{F\subset E\\|F|=d}}T_F\right|^2 = \sum_{s=0}^d\binom ds \sum_{\substack{S\subset\Omega\\|S|=s}} \left|\sum_{\substack{F\supset S\\|F|=d}}T_F\right|^2. \tag{A7}

(130.10)

To verify equality, fix $F,F'$ and write $r=|F\cap F'|$ . The coefficient of $\langle T_F,T_{F'}\rangle$ on the left is the number of $2d$ -sets containing $F\cup F'$ : $\binom{d+r}{r}$ . On the right it is $\sum_{s=0}^d\binom ds\binom rs=\binom{d+r}{d}$ by the Vandermonde identity. The two coefficients agree, proving (A7) term by term.

Every summand on the right is nonnegative. Keeping only $s=d$ bounds that side below by $\sum_{|F|=d}|T_F|^2=\binom{3d}{d}|T|^2$ . For $E=\{1,\ldots,2d\}$ , averaging permutations of $E$ sends the first block of $T$ uniformly onto its $d$ -subsets. Therefore the inner sum on the left is $\binom{2d}{d}\mathcal S_{2d}T$ . For any other $E$ it is a permutation of this same tensor, so it has the same norm. The left side is consequently $\binom{3d}{2d}\binom{2d}{d}^2|\mathcal S_{2d}T|^2$ . Since $\binom{3d}{2d}=\binom{3d}{d}$ , cancellation proves the first inequality of (A6). The second follows from $\binom{2d}{d}\le\sum_{j=0}^{2d}\binom{2d}{j}=2^{2d}$ .

Source concordance. The operator assertions and the inverse on $\mathcal A_0$ reconstruct BKL Section 2 and Lemma 2.1. The incidence identity reconstructs Lemma 2.2. The polynomial barrier for the eigenfunction above is a weak-form justification of the growth assertion; the compact-support first approximation supplies an explicit route to the stronger regular class.

Fences respected. No additional bounded-by edge is proposed. Remark 25.4 is respected by recovering full tensors. Remark 33.6, Remark 33.5, and Remark 32.6 concern stochastic occupation or profiles; no such estimate is asserted here. The inverse operator is taken only at one fixed regular measure; no inverse or eigenfunction is passed through an approximation limit. None of these foundations establishes the premise of the tilt criterion.

References¶

Bizeul, P., Klartag, B., & Lehec, J. (2026). Presenting a Proof of the Kannan–Lovasz–Simonovits Conjecture. https://arxiv.org/abs/2610.05474v1