[Paper Review] Equality in Brascamp-Lieb-Luttinger Inequalities

Michael Christ|arXiv (Cornell University)|Jun 8, 2017

Advanced Harmonic Analysis Research4 references3 citations

TL;DR

This paper establishes sharp uniqueness and stability results for maximizers of Brascamp-Lieb-Luttinger inequalities in the case of indicator functions on the real line. Under a genericity condition, it proves that equality in the inequality is attained only by intervals centered at the origin, up to translation symmetry, and provides a quantitative stability estimate showing the functional decays quadratically with respect to the symmetric difference from optimal intervals.

ABSTRACT

An inequality of Brascamp-Lieb-Luttinger generalizes the Riesz-Sobolev inequality, stating that certain multilinear functionals, acting on nonnegative functions of one real variable with prescribed distribution functions, are maximized when these functions are symmetrized. It is shown that under certain hypotheses, when the functions are indicator functions of sets of prescribed measures, then up to the natural translation symmetries of the inequality, the maximum is attained only by intervals centered at the origin. Moreover, a quantitative form of this uniqueness is established, sharpening the inequality. The hypotheses include an auxiliary genericity assumption which may not be necessary.

Motivation & Objective

• To characterize the uniqueness of maximizers in Brascamp-Lieb-Luttinger inequalities when the functions are indicator functions of sets with prescribed Lebesgue measures.
• To establish a quantitative stability estimate for the inequality, showing that near-maximizers must be close to origin-centered intervals in symmetric difference.
• To identify conditions under which the maximizer is unique up to translation symmetry, particularly in the interior of the admissible parameter space.
• To analyze the failure of uniqueness on the boundary of the admissible set, where equality can hold for non-interval configurations.

Proposed method

• The analysis uses a decomposition of the symmetric difference between a given set and its optimal interval, quantified by a parameter δ measuring the L1 distance to the nearest interval of equal measure.
• A key step involves expanding the functional ΦL(𝔼) around the maximizer 𝔼⋆ using a first-order perturbation, leading to an expression involving dual functions Kj,𝕀.
• The proof relies on bounding the L∞-norm of the difference between dual functions Kk,𝕀 and Kk,𝔼⋆, showing it is O(δ̃) where δ̃ measures deviation from optimality in the interval approximation.
• A reduction is made via translation symmetry to center the interval I_k at the origin, simplifying the analysis of the dual function and enabling the use of uniform bounds.
• The method exploits the fact that the functional is maximized at symmetric, origin-centered intervals, and uses the non-positivity of the first variation at the maximizer to control the perturbation.
• A quantitative stability estimate is derived by combining the perturbation expansion with the bound on the dual function difference, yielding a quadratic decay in δ.

Experimental results

Research questions

• RQ1 Under what conditions is the maximizer of the Brascamp-Lieb-Luttinger functional unique, up to translation symmetry, when the input functions are indicator functions?
• RQ2 What is the rate of decay of the functional ΦL(𝔼) as the sets 𝔼 deviate from the optimal configuration of origin-centered intervals?
• RQ3 Can a quantitative stability estimate be established that bounds the deficit ΦL(𝔼⋆) − ΦL(𝔼) from below by a quadratic function of the symmetric difference between 𝔼 and 𝔼⋆?
• RQ4 How does the structure of the linear mappings Lj affect the uniqueness and stability of maximizers?
• RQ5 What happens to uniqueness and stability on the boundary of the admissible parameter set, where the Riesz-Sobolev triangle inequality becomes equality?

Key findings

• Equality in the Brascamp-Lieb-Luttinger inequality for indicator functions is attained only when each set is an interval centered at the origin, up to translation symmetry, under a genericity condition.
• The functional ΦL(𝔼) satisfies a sharp stability estimate: ΦL(𝔼⋆) − ΦL(𝔼) ≥ cδ² for some c > 0, where δ measures the symmetric difference between 𝔼 and the optimal configuration 𝔼⋆.
• The stability estimate is quantitative and quadratic, valid in the interior of the admissible set, and fails to hold on the boundary where equality can be achieved by non-interval sets.
• The proof relies on a perturbation expansion of the functional and bounds on the dual functions Kj,𝕀, showing their difference from Kj,𝔼⋆ is O(δ̃) in L∞-norm.
• The analysis shows that the first variation of the functional at the maximizer is non-positive, which is essential for controlling the perturbation term.
• The result establishes that near-maximizers must be close to origin-centered intervals in symmetric difference, with the decay rate controlled by the quadratic deficit.