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On the heat semigroup approach to the geometric Forward-Reverse Brascamp-Lieb inequality

Ye Zhang

TL;DR

This work introduces a heat-flow (heat semigroup) proof of the geometric Forward-Reverse Brascamp–Lieb inequality and characterizes precisely which Forward-Reverse Brascamp–Lieb data maintain the initial inequality under the flow. The main result shows that, for geometric datum, the inequality is preserved by $(e^{t\Delta_{E_i}})_{t>0}$ and that the optimal constant satisfies $C(\bold{c},\bold{d},\mathbf Q)=1$; moreover, data equivalent to geometric ones are exactly those for which the preservation holds, with Gaussian extremizers providing a sharp attainment. The proof hinges on a parabolic maximum principle tailored to semicontinuous initial data, a set of preparatory matrix-analytic lemmas, and a Vitali–Carathéodory-based approximation to extend from smooth to general integrable data. Overall, the paper furnishes a new, analytically transparent route to FRBL inequalities in the geometric regime and offers a complete characterization of data compatible with heat-flow preservation, underscoring the central role of Gaussian extremizers in this framework.

Abstract

In this short paper we provide a new proof of the geometric Forward-Reverse Brascamp-Lieb inequality, using the approach of the heat semigroup, or the heat flow. Furthermore, we characterize all the Forward-Reverse Brascamp-Lieb data such that the initial relation can be preserved by some heat flow.

On the heat semigroup approach to the geometric Forward-Reverse Brascamp-Lieb inequality

TL;DR

This work introduces a heat-flow (heat semigroup) proof of the geometric Forward-Reverse Brascamp–Lieb inequality and characterizes precisely which Forward-Reverse Brascamp–Lieb data maintain the initial inequality under the flow. The main result shows that, for geometric datum, the inequality is preserved by and that the optimal constant satisfies ; moreover, data equivalent to geometric ones are exactly those for which the preservation holds, with Gaussian extremizers providing a sharp attainment. The proof hinges on a parabolic maximum principle tailored to semicontinuous initial data, a set of preparatory matrix-analytic lemmas, and a Vitali–Carathéodory-based approximation to extend from smooth to general integrable data. Overall, the paper furnishes a new, analytically transparent route to FRBL inequalities in the geometric regime and offers a complete characterization of data compatible with heat-flow preservation, underscoring the central role of Gaussian extremizers in this framework.

Abstract

In this short paper we provide a new proof of the geometric Forward-Reverse Brascamp-Lieb inequality, using the approach of the heat semigroup, or the heat flow. Furthermore, we characterize all the Forward-Reverse Brascamp-Lieb data such that the initial relation can be preserved by some heat flow.

Paper Structure

This paper contains 11 sections, 10 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Forward-Reverse Brascamp--Lieb inequality
  4. Geometric Forward-Reverse Brascamp--Lieb datum
  5. Main result
  6. Proof of Theorem \ref{['mthm']}
  7. Maximum principle for parabolic equations
  8. Some preparation lemmas
  9. The proof under suitable regularity
  10. Approximation process
  11. Proof of Theorem \ref{['mthm2']}

Key Result

Theorem 1

Given a geometric Forward-Reverse Brascamp--Lieb datum $(\bold{c}, \bold{d}, \mathbf Q)$, if all the functions $(f_i)_{1 \le i \le k}, (g_j)_{1 \le j \le m}$ are non-negative and integrable, and ass3 holds, namely then we have

Theorems & Definitions (24)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • Remark 4
  • Corollary 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 14 more