A small-gap fourth-moment bound - The KLS theorem and its methods

Part of the fixed-eigenfunction mechanism, Chapter The fixed eigenfunction: following one eigenfunction through localization; the reading order is on the full proofs page.

Overview. This dossier proves Lemma 21.5 as Theorem D16.1. The result is an implication from a uniform Poincaré constant $K_n$ for isotropic log-concave laws (Assumption D16.1): a normalized first eigenfunction of a regular isotropic approximant with $\lambda\le3/(8K_n)$ satisfies $\E_\mu f^4\le2$ . The intended instantiation $K_n=C_K\log n$ is the published import Theorem 26.2, which is cited, not proved. The dossier notes that ledger acceptance of this import is pending, and it does not use Theorem 25.1. The proof bootstraps an eigenvalue–energy identity through the Poincaré inequality applied to $f^2$ .

Hypercontractivity under strong log-concavity gives $\E_\mu f^4<\infty$ . This bound depends on $\varepsilon$ and is used only for finiteness.
Pairing the eigenequation with truncated cubes, and passing to the limit by monotone convergence, gives $\lambda\E_\mu f^4=3\E_\mu[f^2\abs{\nabla f}^2]$ ((D16.9)).
Assumption D16.1 applied to $h=f^2$ gives (D16.10).
Substituting step 2 into step 3 under $\lambda\le3/(8K_n)$ gives $\E f^4-1\le\tfrac12\E f^4$ . Finiteness from step 1 then yields (D16.5).
Remarks: the self-bootstrap with $K=1/\lambda$ never closes, so the external input is needed (Remark D16.1). The large-gap branch is left to the consuming assembly (Remark D16.2).

Refined statement and standing. The lemma is an unconditional implication: it assumes a uniform Poincaré constant $K_n$ for isotropic log-concave laws in dimension $n$ and bootstraps a dimension-free fourth moment for first eigenfunctions whose gap is small on the scale $1/K_n$ . Its intended instantiation is the published frontier $K_n=C_K\log n$ ( $n\ge2$ ): Klartag Klartag, 2023 proves the Cheeger constant bound $\psi_n\le C\sqrt{\log n}$ , and Cheeger’s inequality converts it into $\CP\le C_K\log n$ for every isotropic log-concave probability on $\R^n$ . This instantiation is the import node thm:klartag-logn (published import; ledger acceptance pending as of this dossier’s date); it is cited, not proved, here. No unreviewed preprint input occurs anywhere in this dossier; in particular Theorem 25.1 is not used.

Theorem D16.1 (Small-gap fourth moment)

Assume Assumption D16.1. Let

\dd\mu(x)=Z^{-1}e^{-V(x)}\dd x, \qquad V\in C^\infty(\R^n),\qquad\nabla^2V\succeq\varepsilon I_n,\ \varepsilon>0,

(D16.2)

be a smooth, strongly log-concave, centered isotropic probability measure (a regular isotropic approximant in the sense of Section Setup and the exact fixed-function SDE), let $L=\Delta-\nabla V\cdot\nabla$ be its Friedrichs realization, and let $f$ be a normalized first nonconstant eigenfunction:

-Lf=\lambda f,\qquad\E_\mu f=0,\qquad\E_\mu f^2=1 .

(D16.3)

\lambda\ \le\ \frac{3}{8K_n},

(D16.4)

then

\E_\mu f^4\ \le\ 2 .

(D16.5)

The bound 2 is universal: it depends on neither $n$ , $\varepsilon$ , $\mu$ , nor $\lambda$ . The regularization enters only through the qualitative finiteness $\E_\mu f^4<\infty$ .

Proof D16.1

Step 1: qualitative $L^4$ finiteness ( $\varepsilon$ -dependent, used only for finiteness). The Bakry–Émery criterion Bakry & Émery, 1985 for the $\varepsilon$ -uniformly convex potential gives a logarithmic Sobolev inequality for $\mu$ with constant depending only on $\varepsilon$ , and hence, by Gross’s theorem, hypercontractivity of the semigroup $P_t=e^{tL}$ : there are $t_0=t_0(\varepsilon)>0$ and an exponent $q(t_0)\ge4$ with $\norm{P_{t_0}g}_{L^4(\mu)}\le\norm g_{L^2(\mu)}$ (see Bakry et al., 2014, Ch. 5). Since $P_{t_0}f=e^{-\lambda t_0}f$ ,

\norm f_{L^4(\mu)}=e^{\lambda t_0}\norm{P_{t_0}f}_{L^4(\mu)} \le e^{\lambda t_0}\norm f_{L^2(\mu)}<\infty .

(D16.6)

Thus $\E_\mu f^4<\infty$ . The bound is $\varepsilon$ -dependent and is used only to license the rearrangement in Step 4; no quantitative use is made of it. We also record that $f\in C^\infty(\R^n)$ by elliptic regularity for the equation $\Delta f-\nabla V\cdot\nabla f=-\lambda f$ with smooth coefficients, and that $f$ lies in the form domain $W^{1,2}(\mu)$ with $\E_\mu\abs{\nabla f}^2=\langle f,-Lf\rangle=\lambda$ .

Step 2: the eigenvalue–energy identity $\lambda\,\E f^4=3\,\E f^2\abs{\nabla f}^2$ . For $k\in\mathbb N$ let $\varphi_k:\R\to\R$ be the truncated cubic

\varphi_k(s)= \begin{cases} s^3, & \abs s\le k,\\ k^3+3k^2(s-k), & s>k,\\ -k^3+3k^2(s+k), & s<-k, \end{cases}

(D16.7)

which is $C^1$ , odd, nondecreasing, globally Lipschitz with constant $3k^2$ , and satisfies $\varphi_k(0)=0$ , $\varphi_k'(s)=3(\abs s\wedge k)^2$ . Since the Dirichlet form of the Friedrichs realization is a Dirichlet form, composition of the form-domain element $f$ with a Lipschitz function vanishing at 0 stays in the form domain, with the chain rule $\nabla\varphi_k(f)=\varphi_k'(f)\nabla f$ ; concretely, $\varphi_k(f)\in L^2(\mu)$ because $\abs{\varphi_k(f)}\le3k^2\abs f$ , and $\abs{\nabla\varphi_k(f)}\le3k^2\abs{\nabla f}\in L^2(\mu)$ . Pairing the eigenequation with $\varphi_k(f)$ in the form,

\lambda\,\E_\mu\bigl[f\,\varphi_k(f)\bigr] =\E_\mu\bigl[\nabla f\cdot\nabla\varphi_k(f)\bigr] =\E_\mu\bigl[\varphi_k'(f)\abs{\nabla f}^2\bigr] =3\,\E_\mu\bigl[(\abs f\wedge k)^2\abs{\nabla f}^2\bigr].

(D16.8)

Both extreme members are monotone nondecreasing in $k$ : $s\,\varphi_k(s)\ge0$ increases pointwise to $s^4$ (for $\abs s>k$ , $s\varphi_k(s)=\abs s\,(k^3+3k^2(\abs s-k))\le s^4$ and is nondecreasing in $k$ ), and $(\abs f\wedge k)^2\abs{\nabla f}^2$ increases pointwise to $f^2\abs{\nabla f}^2$ . Monotone convergence on both sides of (D16.8) yields the identity in $[0,\infty]$ :

\lambda\,\E_\mu f^4=3\,\E_\mu\bigl[f^2\abs{\nabla f}^2\bigr].

(D16.9)

By Step 1 the left side is finite, hence so is $\E_\mu[f^2\abs{\nabla f}^2]=\lambda\,\E_\mu f^4/3$ (recall $\lambda>0$ : a first nonconstant eigenfunction has $\lambda=\E\abs{\nabla f}^2>0$ ).

Step 3: Poincaré applied to $f^2$ . The measure $\mu$ is isotropic and log-concave, so Assumption D16.1 applies to it. The function $h=f^2$ is $C^1$ (hence locally Lipschitz), lies in $L^2(\mu)$ by Step 1, and has $\E\abs{\nabla h}^2=4\,\E[f^2\abs{\nabla f}^2]<\infty$ by Step 2. Therefore

\E_\mu f^4-1=\E_\mu f^4-\bigl(\E_\mu f^2\bigr)^2=\Var_\mu\bigl(f^2\bigr) \le K_n\,\E_\mu\abs{\nabla f^2}^2 =4K_n\,\E_\mu\bigl[f^2\abs{\nabla f}^2\bigr].

(D16.10)

Step 4: closure of the bootstrap at $\lambda\le3/(8K_n)$ . Substituting (D16.9) into (D16.10) and using the small-gap hypothesis,

\E_\mu f^4-1 \le\frac{4K_n\lambda}{3}\,\E_\mu f^4 \le\frac{4K_n}{3}\cdot\frac{3}{8K_n}\,\E_\mu f^4 =\frac12\,\E_\mu f^4 .

(D16.11)

Since $\E_\mu f^4<\infty$ (Step 1), the term $\tfrac12\E_\mu f^4$ may be subtracted, giving $\tfrac12\E_\mu f^4\le1$ , i.e. (D16.5).

Remark D16.1 (The external frontier input is genuinely needed)

The trivially available Poincaré constant for $\mu$ itself is $K=\CP(\mu)=1/\lambda$ (sharp for the first eigenfunction). With that choice the coefficient in Step 4 is $4K\lambda/3=4/3>1$ , and the bootstrap closes for no $L^p$ , $p>2$ : the analogous computation (pairing the eigenequation with truncations of $\abs s^{p-2}s$ , which gives $\lambda\,\E\abs f^p=(p-1)\,\E[\abs f^{p-2}\abs{\nabla f}^2]$ , then Poincaré for $\abs f^{p/2}$ ) produces the coefficient $Kp^2\lambda/(4(p-1))$ , which at $K=1/\lambda$ equals $p^2/(4(p-1))=1+(p-2)^2/(4(p-1))>1$ for every $p>2$ , independently of the dimension. An external bound $K_n$ with $K_n\lambda$ small is therefore what makes the argument close, and only on the small-gap branch $\lambda\le3/(8K_n)$ . This is the probe’s structural observation, recorded here as a remark; it is not a claim and no statement is derived from it. This use of the frontier is not circular for the downstream assembly: the pipeline consuming this lemma outputs $\CP=O(\log^2n)$ , strictly weaker than its input $O(\log n)$ ; the import is a calibrated external ingredient, not a hidden assumption of the conclusion.

Obstructions respected. The candidate node carries no bounded_by edge; the registered route fences were checked individually. The argument is static and cut-free: no slice, excess, or localized isoperimetric estimate occurs (rem:two-tail-slice-bounds, rem:profile-circularity, rem:single-coordinate-cuts); no projection or radial test is promoted to a tensor bound (rem:projection-ceiling); no covariance occupation functional appears (rem:crude-insufficient, rem:relative-ceiling). Regarding rem:relative-ceiling specifically: the a priori input here is the published frontier constant $K_n$ , used to produce a strictly weaker downstream output, not an all-measure relative occupation premise. The covariance-spike fence is untouched (no localization occurs).

References¶

Klartag, B. (2023). Logarithmic Bounds for Isoperimetry and Slices of Convex Sets. Ars Inveniendi Analytica, 2023(4), 1–17. 10.15781/jsjy-0b06
Bakry, D., & Émery, M. (1985). Diffusions hypercontractives. Séminaire de Probabilités de Strasbourg, 19, 177–206.
Bakry, D., Gentil, I., & Ledoux, M. (2014). Analysis and Geometry of Markov Diffusion Operators (Vol. 348). Springer. 10.1007/978-3-319-00227-9