[Paper Review] Maximizers of Rogers-Brascamp-Lieb-Luttinger functionals in higher dimensions

Michael Christ, Kevin M. O'neill|arXiv (Cornell University)|Nov 30, 2017

Nonlinear Partial Differential Equations16 references3 citations

TL;DR

This paper characterizes all maximizers of Rogers-Brascamp-Lieb-Luttinger functionals in dimensions greater than one, proving that under natural structural and admissibility conditions, the only maximizers are tuples of sets that are images of origin-centered balls under volume-preserving linear transformations and translations. The result establishes a sharp inequality with explicit uniqueness up to symmetry.

ABSTRACT

A symmetrization inequality of Rogers and of Brascamp-Lieb-Luttinger states that for a certain class of multilinear integral expressions, among tuples of sets of prescribed Lebesgue measures, tuples of balls centered at the origin are among the maximizers. Under natural hypotheses, we characterize all maximizing tuples for these inequalities for dimensions strictly greater than 1. We establish a sharpened form of the inequality.

Motivation & Objective

• To determine all maximizers of multilinear integral functionals of the form $\Phi_{\mathcal{L}}(\mathbf{E}) = \int_{\mathbb{R}^D} \prod_{j\in J} \mathbf{1}_{E_j} \circ L_j $ under fixed Lebesgue measures for $d > 1$.
• To extend previous results on the Riesz-Sobolev and Brascamp-Lieb-Luttinger inequalities from dimension one to higher dimensions.
• To establish a sharp inequality by characterizing all configurations that achieve equality in the symmetrization inequality.
• To identify the full set of symmetries—translations and special linear transformations—under which maximizers are invariant.

Proposed method

• The authors use a structural hypothesis (1.1) that ensures compatibility between the linear mappings $L_j^d$ and the action of $\mathrm{GL}(d)$, particularly $\mathrm{SL}(d)$.
• They apply a stability argument based on perturbations of the $d=1$ case, using a parameter $s$ to deform configurations near the symmetric case.
• A key technique involves analyzing the distance from a configuration to the orbit of its symmetric rearrangement, using the centers of intervals in the $d=1$ case.
• The method relies on a stability theorem from prior work [12], applied to the $d=1$ case, to derive a quadratic lower bound on the deficit in the inequality.
• They use polynomial approximation of center functions $c_j(y'_j,s)$ to relate the deficit to the squared distance from symmetry, yielding a quantitative bound.
• Integration over a set $\Omega(s)$ of positive measure ensures that the deficit is bounded below by a positive multiple of $s^2$, proving strict suboptimality of non-symmetric configurations.

Experimental results

Research questions

• RQ1 Which tuples of measurable sets with prescribed Lebesgue measures maximize the Rogers-Brascamp-Lieb-Luttinger functional in dimensions $d > 1$?
• RQ2 Under what conditions is the symmetric configuration—tuples of origin-centered balls—the unique maximizer up to symmetry?
• RQ3 Can a quantitative stability estimate be established for the symmetrization inequality in higher dimensions?
• RQ4 How do the symmetries of translation and $\mathrm{SL}(d)$-action affect the set of maximizers?
• RQ5 What role does the genericity condition on the linear data $({\mathcal{L}}^1, \mathbf{e})$ play in ensuring uniqueness of maximizers?

Key findings

• The only maximizers of $\Phi_{\mathcal{L}}(\mathbf{E})$ under fixed measures are those tuples $\mathbf{E}$ that are obtained from origin-centered balls via a volume-preserving linear transformation $\psi \in \mathrm{SL}(d)$ and a translation $\mathbf{v} \in \mathbb{R}^{md}$.
• The maximizers are characterized by the condition $E_j = \psi(E_j^\star) + L_j(\mathbf{v})$ for all $j \in J$, proving uniqueness up to the action of $\mathrm{SL}(d)$ and translations.
• A sharp stability inequality holds: $\Phi_{\mathcal{L}}(\mathbf{E}) \leq \Phi_{\mathcal{L}}(\mathbf{E}^\star) - c \cdot \mathrm{dist}(\mathbf{E}, \mathrm{orbit}(\mathbf{E}^\star))^2$ for some $c > 0$, with the distance measured in terms of center displacements.
• The deficit in the symmetrization inequality is bounded below by a positive multiple of $s^2$ for perturbations of size $s$, proving strict suboptimality of non-symmetric configurations.
• The set of perturbation parameters $\mathbf{y}'$ for which the deficit is bounded below has measure uniformly bounded away from zero, ensuring the stability estimate holds on a positive-measure set.
• The result extends the classical symmetrization inequality to higher dimensions with full characterization of equality cases, resolving the inverse problem for this class of functionals.