Regularized Brascamp-Lieb inequalities via Optimal Transport and Study of Equality Cases

1. Problem Statement and Context

The paper addresses the Brascamp-Lieb (BL) inequalities, a powerful generalization of classical inequalities such as Hölder's, Young's convolution, and the Loomis-Whitney inequality. While the standard BL inequalities are well-understood regarding their finiteness conditions and the existence of Gaussian extremizers (functions that achieve equality), this work focuses on regularized versions.

The Regularization:
The author imposes structural constraints on the input functions $f_i$. Specifically, the functions are required to be more log-convex (or more log-concave) than a specific Gaussian kernel $g_Q(x) = \exp(-\pi \langle Qx, x \rangle)$.

• A function $f$ is "more log-convex than $g_Q$" if $f = g_Q \cdot h$ where $h$ is log-convex.
• This restriction allows for the derivation of improved inequalities with sharper constants and provides a framework to study equality cases more precisely.

The Goal:

1. Prove regularized forward and reverse BL inequalities.
2. Establish necessary and sufficient conditions for the finiteness of the regularized BL constant.
3. Characterize the existence of Gaussian extremizers.
4. Fully describe the structure of all optimizers (equality cases) for these inequalities.

2. Methodology

The paper employs a synthesis of Optimal Transport (OT) theory and Heat Flow methods.

• Optimal Transport & Caffarelli's Contraction:
  The core geometric tool is an anisotropic extension of Caffarelli's contraction theorem. The author proves that if a source measure has a Hessian of the potential bounded above by a matrix $A$ and the target measure has a Hessian bounded below by a matrix $B$, the optimal transport map (Brenier map) $\nabla \Phi$ satisfies a specific Hessian bound:
  $$\nabla^2 \Phi \leq A^{1/2}(A^{1/2}BA^{1/2})^{-1/2}A^{1/2}$$
  This result is used to derive the regularized BL inequalities by mapping general log-convex/concave functions to Gaussian functions via transport maps, preserving the structural constraints.

• Heat Flow (Parabolic Monotonicity):
  To study the existence of extremizers and the equality cases, the author utilizes the heat flow approach (inspired by Carlen, Lieb, and Loss). By evolving functions via the heat equation, the paper demonstrates that the BL functional is monotonic. This allows the reduction of general extremizers to Gaussian extremizers via the Central Limit Theorem (CLT) applied to convolutions.

• Algebraic Decomposition:
  For the equality cases, the paper introduces a decomposition of the underlying Euclidean space into generalized independent subspaces and critical subspaces. This algebraic structure, combined with the heat flow analysis, allows for a precise characterization of the Hessian of the logarithm of the optimizer.

3. Key Contributions and Results

A. Regularized Inequalities (Theorem 2.4)

The paper establishes the regularized forward and reverse BL inequalities.

• Forward Inequality: For functions $f_i$ that are more log-convex than $g_{Q_i}$, the integral of the product is bounded by a supremum over Gaussian kernels $g_{B_i}$ where $0 < B_i \leq Q_i$.
• Reverse Inequality: A dual inequality holds for functions that are more log-concave than $g_{Q_i^{-1}}$, involving an infimum over $B_i \geq Q_i^{-1}$.
• Significance: These results generalize previous work on "Type G" functions (convolutions with Gaussians) to a broader class of log-convex/concave functions.

B. Finiteness and Gaussian Extremizability (Theorems 3.2 & 3.12)

• Finiteness Condition: The regularized BL constant is finite if and only if a dimension inequality holds for all subspaces $V \subset H$:
  $$\dim(V) \leq \sum_{j=1}^m p_j \dim(C_j V)$$
  (with modifications if $Q$ is not positive definite).
• Gaussian Extremizability: A major result is that if the BL constant is attained by any function in the regularized class, it is also attained by Gaussian functions. This is proven using the heat flow and the fact that the class of log-convex functions is closed under convolution and scaling.

C. Structure of Optimizers (Theorems 4.6 & 4.8)

The paper provides a complete characterization of the functions $f_j$ that achieve equality.

• Reduction to Geometric Data: Any extremizable BL datum is equivalent (via linear transformations) to a "generalized geometric" datum where the constant is 1.
• Form of Optimizers: The optimizers are shown to be of a specific form involving:
  1. Gaussian components: Determined by the critical subspaces of the data.
  2. Arbitrary functions: On "generalized independent" subspaces, the functions can be arbitrary (subject to log-convexity), similar to the equality case in Hölder's inequality.
  3. Quadratic constraints: On the dependent parts of the space, the functions must be Gaussian.
• This generalizes Valdimarsson's results on non-regularized BL inequalities to the regularized setting.

D. Applications (Section 5)

The theory is applied to improve existing results in:

• Hypercontractivity: The paper derives improved forward regularized hypercontractivity inequalities, refining bounds found in previous literature (e.g., Chen-Dafnis-Paouris).
• Gaussian Measures: The results are reformulated for Gaussian measures, providing new inequalities for expectations of products of functions under joint Gaussian distributions.

4. Significance and Impact

• Unification: The work unifies the study of BL inequalities with optimal transport and log-convexity/concavity, providing a robust framework for "regularized" functional inequalities.
• Sharpness: By characterizing the equality cases, the paper identifies exactly when the standard Gaussian extremizers are sufficient and when more complex structures (involving arbitrary functions on specific subspaces) are necessary.
• Generalization of Caffarelli's Theorem: The appendix provides a new, sharp anisotropic version of Caffarelli's contraction theorem for optimal transport maps between measures with general Hessian bounds, which is a standalone contribution to the field of Optimal Transport.
• Methodological Advancement: The successful application of heat flow methods to regularized inequalities demonstrates a powerful technique for proving existence and structure of extremizers in geometric analysis.

In summary, Bader Ammari's paper significantly advances the theory of Brascamp-Lieb inequalities by introducing a regularization scheme based on log-convexity, proving the sufficiency of Gaussian extremizers, and providing a complete algebraic and analytic description of all functions that achieve equality.