Modified Logarithmic Sobolev Inequality

• Modified Logarithmic Sobolev Inequality is a functional inequality that interpolates between classical Poincaré and log-Sobolev inequalities by using a tailored Young-type function.
• It employs U-bound inequalities to manage non-log-concave measures, oscillatory potentials, and sub-quadratic growth on both continuous and discrete metric spaces.
• The criterion leverages spectral-gap conditions and defect-removal methods to extend MLSI applications to Markov processes, geometric analysis, and quantum contexts.

The modified logarithmic Sobolev inequality (MLSI) is a general functional inequality that interpolates between the classical Poincaré and log-Sobolev inequalities, with flexibility to encode tail behaviors and regularity (e.g., superquadratic, subquadratic, or non-Euclidean gradient structures) beyond classical settings. The MLSI can be formulated on both continuous and discrete metric measure spaces, including probability spaces, Markov chains, and quantum (matrix-valued) contexts. In the framework developed by Papageorgiou (Papageorgiou, 2010), an explicit criterion is presented using U-bound inequalities to extend MLSI well outside the log-concave regime, particularly for probability measures on noncompact metric spaces with sub-quadratic potentials and oscillatory perturbations.

1. Precise Formulation of the Modified Logarithmic Sobolev Inequality

For a probability measure $\mu$ on a metric space $(X, d)$ equipped with a sub-gradient operator $\nabla$, and a fixed exponent $q > 2$, define the Young-type function

$$H_q(t) = \begin{cases} t^2, & |t| \leq 1, \ |t|^q, & |t| > 1. \end{cases}$$

The measure $\mu$ is said to satisfy the MLSI with respect to $H_q$ (denoted MLS$(H_q)$) if there exists a constant $C_{\rm MLS} > 0$ such that for all compactly supported smooth $f > 0$,

$$\int_X f^2 \log\left(\frac{f^2}{\int_X f^2\,d\mu}\right)\,d\mu \leq C_{\rm MLS} \int_X H_q\left(\frac{|\nabla f|}{f}\right) f^2\,d\mu.$$

This interpolates between the classical quadratic log-Sobolev ($q=2$) and higher moments, thereby controlling both small and large gradients.

2. U-Bound Inequalities and Their Analytical Role

Central to Papageorgiou’s MLSI criterion is the use of U-bound inequalities, first introduced by Hebisch and Zegarlinski. These inequalities take the form: $$\int_X U(x)\,f(x)^2\,d\mu(x) \le C \int_X H_q\left(\frac{|\nabla f|}{f}\right) f^2\,d\mu + D \int_X f^2\,d\mu,$$ where $U(x) \ge 0$ is a weight tailored to the reference measure. In particular, Papageorgiou considers

$$U(x) = |\nabla \Phi(x)|^2 + \Phi(x), \quad \Phi(x) = d(x)^p + W(x), \;\; p \in (1,2],$$

with $W$ a differentiable perturbation. The U-bound is established first for such non-convex, oscillatory potentials and is the technical anchor for propagating defective MLSI estimates (with additional lower order terms) to full MLSI via a spectral-gap condition and abstract defect-removal arguments.

3. Structural Assumptions on the Metric-Measure Space and Potential

The criterion is set on a noncompact, complete metric measure space $(X,d,\lambda)$:

• There is a sub-gradient operator $\nabla$ so that $0 < |\nabla d| \leq 1$ everywhere and $\Delta d \leq K$ outside a unit ball.
• The reference measure $\lambda$ satisfies the classical Sobolev inequality for dimension $n \geq 3$:

$$\Big(\int_X |f|^{2^*}\,d\lambda\Big)^{2/2^*} \leq a \int_X |\nabla f|^2\,d\lambda + b \int_X |f|^2\,d\lambda,$$

with $2^* = 2n/(n-2)$, and admits a local Poincaré inequality.

• Probability measures of interest have density $d\mu = Z^{-1}e^{-\Phi(x)} d\lambda(x)$, with $\Phi(x) = d(x)^p + W(x)$, $1 < p < 2$, and the perturbation $W$ satisfies $|\nabla W(x)| \leq \delta d(x)^{p-1} + \gamma$ with $0 < \delta < 1$, $\gamma > 0$.

These conditions allow for non-log-concave measures and oscillatory or sub-quadratic growth in the potential, far exceeding the classical strictly convex setting.

4. The Main Criterion and Its Implementation

Fix $q > 2$ and let $q' = q/(q-1)$. If $p \ge q'$ and $\Phi = d^p + W$ with the above $W$, there is $c > 0$ (depending on parameters) such that for all compactly supported, smooth $f > 0$,

$$\int_X f^2\log\left(\frac{f^2}{\int f^2\,d\mu}\right)\,d\mu \leq c \int_X H_q\left(\frac{|\nabla f|}{f}\right) f^2\,d\mu.$$

The key step is to verify that a weighted U-bound holds for $d(x)^{q(p-1)}$, which combines the gradient structure and geometry of $X$. This, together with Sobolev/Poincaré inequalities on the reference measure, triggers the MLSI via a two-step process:

• First, derive a defective MLSI with an additional $L^2$ defect,
• Then remove the defect using spectral-gap arguments and concentration–reverse tricks (Barthe–Kolesnikov theorem).

5. Outline of the Proof Strategy

The mechanism for establishing MLSI is multistep:

1. Weighted U-bound:

$$\int d(x)^{q(p-1)} f^2\,d\mu \leq C \int |\nabla f|^q\,|f|^{2-q} d\mu + D \int f^2\,d\mu$$

(Proposition 2.4).

1. Upgrading the inequality: Extends to $f^2|\nabla \Phi|^2 + f^2\Phi$ weights (Proposition 2.6), matching the hypothesis required for a defective MLSI.
2. Classical Sobolev and Jensen: Application yields a defective MLSI with lower-order $L^2$ term.
3. Spectral-gap term: Use of Poincaré inequality and already-established U-bound assures that the defective MLSI is eligible for defect removal.
4. Defect-removal and conclusion: By invoking Barthe–Kolesnikov’s defect-removal theorem, the $L^2$ term can be absorbed to yield the full MLSI$(H_q)$:

$$\int_X f^2 \log\left(\frac{f^2}{\int f^2\,d\mu}\right) d\mu \leq c \int_X H_q\left(\frac{|\nabla f|}{f}\right) f^2\,d\mu$$

6. Illustrative Examples: Non-Log-Concave Measures

As a demonstration, consider $X = \mathbb{R}^n$ and the potential

$$\Phi(x) = d(x)^p + a(x) \cos(d(x)), \;\; a(x) = k d(x)^{p-1}, \;\; 0 < k < 1, \;\; p \ge q'.$$

Then the corresponding measure

$d\mu(x) = Z^{-1} e^{-\Phi(x)}\,dx$

is non-log-concave and supports the same MLSI$(H_q)$: $$\int f^2 \log\left(\frac{f^2}{\int f^2\,d\mu}\right)\,d\mu \leq c \int H_q\left(\frac{|\nabla f|}{f}\right) f^2\,d\mu.$$ This extends MLSI to oscillatory measures, exemplifying the flexibility of Papageorgiou’s criterion and U-bound methodology.

7. Comparison, Implications, and Broader Connections

The U-bound-based MLSI paradigm provides a robust method for proving functional inequalities in contexts where classical convexity fails, such as when dealing with sub-quadratic, oscillatory, or strongly perturbed potentials. It connects directly to techniques for defective log-Sobolev inequalities (Barthe–Kolesnikov defect-removal), Sobolev/Poincaré controls, and analytic approaches for concentration of measure and spectral-gap-type decay. The applicability to non-log-concave probability measures significantly enlarges the landscape for entropy–energy inequalities, with implications ranging from analysis of metric measure spaces to statistical physics, Markov semigroup theory, and geometric analysis.

A plausible implication is that the U-bound technique, adapted to higher-order $q$ and more exotic potentials, could be employed to paper MLSI and associated concentration phenomena in systems exhibiting complex nonlinearities, non-convex Hamiltonians, and non-Euclidean geometries.