Family 3: Bochner, heat flow, and the H^{-1} calculus

KLS is now proved, by three different arguments compared in Chapter The proofs of KLS compared. This chapter describes what the Bochner and heat-flow calculus controls on its own and what it does not give without them.

Object followed. The first eigenspace of the diffusion generator, and the energy of derivatives of test functions, measured in the negative Sobolev norm attached to that generator.

What it buys. Two things that no other family supplies. First, the conversion device: an inequality that turns “curvature $t$ plus covariance $\norm{\Cov\mu}_\op$” into a spectral gap, sharper than Bakry–Émery. This is the box marked Improved Lichnerowicz in Figure Figure 0.1 and it is the reason short-time localization is usable at all. Second, a calculus in which coordinate functions and quadratic functions have computable spectral mass, which is what makes the moment-map input of Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels usable.

The improved Lichnerowicz inequality¶

Let $\mu=e^{-U}\dd x$ with $\Hess U\succeq tI$. Bakry–Émery, equivalently Brascamp–Lieb, gives $\CP(\mu)\le1/t$. Klartag’s sharper geometric-mean estimate is the following Klartag, 2023.

This is Klartag’s theorem, quoted from Klartag, 2023; the argument is short enough to record, because the shape of it explains where each factor in the bridge (0.14) comes from.

Note the two places the argument spends: (3.2) is where a gradient energy is converted into information about $\int\nabla f\dd\mu$, a single vector, and (3.3) is where that vector is charged to the operator norm of the covariance. Both steps are sharp for the Gaussian; neither knows anything about $f$ beyond its first moments. That is the sense in which this family “averages away” the test function, and it is the source of the limitation recorded at the end of this chapter.

The $H^{-1}$ norm and the Barthe–Klartag inequality¶

Let $L=\Delta-\nabla V\cdot\nabla$ be the reversible generator of $\mu=e^{-V}\dd x$, so that $\CP(\mu)=\lambda_1(-L)^{-1}$. For centered $g$ define

where $\nu_g$ is the spectral measure of $g$. The last expression is the useful one: the $H^{-1}$ norm is the spectral mass of $g$ weighted by $1/\lambda$, so it is large exactly when $g$ loads the bottom of the spectrum.

Barthe and Klartag proved that, when the derivatives of $f$ are centered Barthe & Klartag, 2019,

Applied to $f(x)=|x|^2$, whose derivatives are the coordinate functions up to a factor 2, this gives

so the thin-shell problem becomes a statement about the low spectral mass of coordinate functions. This is the entry point through which the moment-map estimates of Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels are used.

Heat flow and spectral monotonicity¶

Smooth the measure by Gaussian convolution, $\mu_s=\mu*\gamma_s$, and define the conditional expectation operator

with $\rho$ the density of $\mu$ and $P_s$ the heat semigroup. The operator $Q_s$ is the deterministic, time-reversed counterpart of stochastic localization; concretely,

the second identity being (2.4) in expectation. So the two families are two readings of the same object.

Klartag and Putterman proved that the Rayleigh quotient of $Q_sg$ decreases under the heat flow Klartag & Putterman, 2021; consequently low-frequency modes cannot disappear too fast,

with $E(g)$ the Rayleigh quotient of $g$. This converts covariance estimates into bounds on the low spectral mass of coordinate functions; integrating that mass against $1/\lambda$ as in (3.4) is how Klartag–Lehec obtained their polylogarithmic thin-shell and KLS bounds Klartag & Lehec, 2022.

What it does not reach alone. Uniform control of $\norm{\partial_if}_{H^{-1}}$ for the derivatives of an arbitrary test function, rather than for coordinates and for derivatives of quadratics. Every step above is an averaging step. Bochner in (3.2) keeps only $\int\nabla f\dd\mu$; (3.5) keeps only the spectral masses of the $\partial_if$; (3.9) is a statement about a single Rayleigh quotient. Trace, coordinate or averaged spectral information does not bound every slow mode, and a near-extremizing eigenfunction of a general log-concave measure is exactly the object about which these averages say least.

Where this family meets the alternative mechanisms. The fixed eigenfunction (Chapter The fixed eigenfunction: following one eigenfunction through localization) is exactly the attempt to keep the eigenfunction itself in the argument rather than averaging it away, and it is the mechanism this family points at most directly: it carries the first spectral mode through localization instead of replacing it by a spectral mass. The $H^{-1}$ residual bound of Section The $H^{-1}$ endpoint is where the two meet; it is an endpoint in the sense that it is as strong as KLS itself, usable as a final target but not as a first step.