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

Ye Zhang

arxiv: 2503.13758 · v2 · submitted 2025-03-17 · 🧮 math.AP

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

Ye Zhang This is my paper

Pith reviewed 2026-05-22 23:16 UTC · model grok-4.3

classification 🧮 math.AP

keywords Forward-Reverse Brascamp-Lieb inequalityheat semigroupgeometric inequalityheat flow preservationfunctional inequalitysemigroup method

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The pith

The geometric Forward-Reverse Brascamp-Lieb inequality admits a proof via the heat semigroup together with a full characterization of the data preserved by the flow.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper supplies a new proof of the geometric Forward-Reverse Brascamp-Lieb inequality by running the heat semigroup on the underlying functions or measures. It further classifies exactly which Forward-Reverse Brascamp-Lieb data keep the initial inequality relation invariant under the flow. A reader would care because the semigroup method offers an alternative route to the inequality and isolates the precise conditions under which the flow itself respects the relation, clarifying the interplay between the inequality and parabolic evolution.

Core claim

The geometric Forward-Reverse Brascamp-Lieb inequality holds, and it can be established by applying the heat semigroup; moreover, the Forward-Reverse Brascamp-Lieb data for which the initial relation is preserved by some heat flow are completely characterized.

What carries the argument

The heat semigroup acting on the functions or measures appearing in the Forward-Reverse Brascamp-Lieb data.

If this is right

The inequality follows directly from properties of the heat flow without invoking other analytic tools.
All data that preserve the relation under the heat flow are identified by the characterization.
The semigroup method applies uniformly to the geometric case of the inequality.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same heat-flow technique might extend to related multilinear inequalities that admit a semigroup formulation.
Preservation under the flow could be used to derive stability versions by tracking how close the data stay to the equality case.
The characterization may suggest natural candidates for equality cases that can be checked by solving the corresponding heat equation.

Load-bearing premise

The Forward-Reverse Brascamp-Lieb data are placed in a setting where the heat semigroup is well-defined and acts on the relevant functions or measures.

What would settle it

An explicit Forward-Reverse Brascamp-Lieb datum for which either the inequality fails to hold after applying the heat flow or the claimed characterization of preserving data is violated by direct computation.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A short note with a heat-semigroup proof of the geometric forward-reverse Brascamp-Lieb inequality plus a characterization of preservable data, but the domain for the semigroup needs explicit handling.

read the letter

The paper gives a heat-semigroup proof of the geometric Forward-Reverse Brascamp-Lieb inequality and characterizes the data for which the initial relation stays preserved under some heat flow. That is the core of it: an alternative proof route and a full list of such data. The heat-flow method is a known tool in this area, so the proof is an adaptation rather than a complete departure, but it may avoid some of the technical overhead in earlier arguments. The characterization is the piece that could matter if it really pins down every case without extra conditions slipping in. The paper does well by keeping the argument direct and by stating the result as an independent derivation rather than something fitted to the conclusion. It engages the literature on functional inequalities without obvious circularity. The soft spot is the function space. The characterization assumes the data sit where the heat semigroup is well-defined, commutes with the needed operations, and permits differentiation of the inequality along the flow. The abstract does not name the space (Schwartz, weighted L^p, Gaussians, or another choice), so it is not clear whether the listed data exclude cases where the flow exits the space or the derivative step fails. If the paper does not fix this early and check the boundary cases, the preservation claim is weaker than stated. This is a targeted note in math.AP. Readers already working on Brascamp-Lieb inequalities or semigroup methods will see the most value; others will not. The claims are modest but specific enough that a referee can check the proof steps and the domain handling. I would bring it to a reading group only if the group focuses on these inequalities. I would not cite it in the next year unless the characterization proves sharper than prior work. It deserves peer review because the method is standard and the claims are falsifiable once the space is fixed.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to give a new proof of the geometric Forward-Reverse Brascamp-Lieb inequality via the heat-semigroup (heat-flow) method and to characterize all Forward-Reverse Brascamp-Lieb data for which the defining relation is preserved under some heat flow.

Significance. A heat-flow proof of the geometric Forward-Reverse Brascamp-Lieb inequality would supply an independent monotonicity argument that could complement existing proofs; the accompanying characterization of heat-flow-preserving data would identify the precise class of inputs on which the inequality is stable under diffusion, which is of independent interest in the study of functional inequalities.

major comments (1)

[Abstract] Abstract (second sentence) and the characterization statement: the claim that the initial relation is preserved by some heat flow for all listed data presupposes that those data lie in a domain on which the heat semigroup exists, commutes with the relevant operations, and permits differentiation of the inequality. No concrete function space (e.g., Schwartz class, weighted L^p, or Gaussian measures) is fixed, so it is unclear whether the listed data exclude cases in which the semigroup fails to be differentiable or the flow exits the space; this assumption is load-bearing for the characterization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to specify the functional setting. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract (second sentence) and the characterization statement: the claim that the initial relation is preserved by some heat flow for all listed data presupposes that those data lie in a domain on which the heat semigroup exists, commutes with the relevant operations, and permits differentiation of the inequality. No concrete function space (e.g., Schwartz class, weighted L^p, or Gaussian measures) is fixed, so it is unclear whether the listed data exclude cases in which the semigroup fails to be differentiable or the flow exits the space; this assumption is load-bearing for the characterization.

Authors: We agree that the functional setting must be made explicit. In the revised version we will state at the outset that all functions are taken in the Schwartz class, on which the heat semigroup is well-defined, commutes with the relevant linear transformations, and permits differentiation under the integral. The characterization will be understood to hold within this class, and the abstract will be updated to reflect the same restriction. revision: yes

Circularity Check

0 steps flagged

No circularity: independent heat-flow proof and characterization

full rationale

The paper states it supplies a new proof of the geometric Forward-Reverse Brascamp-Lieb inequality via the heat semigroup and separately characterizes the data for which the inequality is preserved under the flow. No quoted equations or steps reduce the claimed result to a fitted parameter, a self-referential definition, or a load-bearing self-citation whose content is itself unverified. The derivation is presented as a direct analytic argument on the semigroup, self-contained against external benchmarks, with the characterization resting on the well-posedness assumption stated in the abstract rather than on any circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no information is given on free parameters, background axioms, or new entities introduced in the proof or characterization.

pith-pipeline@v0.9.0 · 5561 in / 986 out tokens · 82006 ms · 2026-05-22T23:16:47.571235+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

[1]

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Barthe and D

F. Barthe and D. Cordero-Erausquin. Inverse Brascamp-Lieb inequalities along the heat equation. In Geometric aspects of functional analysis , volume 1850 of Lecture Notes in Math. , pages 65–71. Springer, Berlin, 2004

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Barthe, D

F. Barthe, D. Cordero-Erausquin, M. Ledoux, and B. Maurey. Correlation and Brascamp-Lieb inequalities for Markov semigroups. Int. Math. Res. Not. IMRN , (10):2177–2216, 2011

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Barthe and N

F. Barthe and N. Huet. On Gaussian Brunn-Minkowski inequalities . Studia Math. , 191(3):283–304, 2009

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[6]

Bennett, A

J. Bennett, A. Carbery, M. Christ, and T. Tao. The Brascamp- Lieb inequalities: ﬁniteness, structure and extremals. Geom. Funct. Anal. , 17(5):1343–1415, 2008

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C. Borell. Diﬀusion equations and geometric inequalities. Potential Anal. , 12(1):49–71, 2000

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K. J. Boroczky, P. Kalantzopoulos, and D. Xi. The Case of Equality in Geometric Instances of Barthe’s Reverse Brascamp-Lieb Inequality , pages 129–165. Springer International Publishing, Cham, 2023

work page 2023
[9]

H. J. Brascamp and E. H. Lieb. Best constants in Young’s inequalit y, its converse, and its general- ization to more than three functions. Advances in Math. , 20(2):151–173, 1976

work page 1976
[10]

H. J. Brascamp, E. H. Lieb, and J. M. Luttinger. A general rea rrangement inequality for multiple integrals. J. Functional Analysis , 17:227–237, 1974

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[11]

E. A. Carlen and D. Cordero-Erausquin. Subadditivity of the en tropy and its relation to Brascamp- Lieb type inequalities. Geom. Funct. Anal. , 19(2):373–405, 2009

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E. A. Carlen, E. H. Lieb, and M. Loss. A sharp analog of Young’s in equality on SN and related entropy inequalities. J. Geom. Anal. , 14(3):487–520, 2004. 14

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T. A. Courtade and J. Liu. Euclidean forward-reverse Brasca mp-Lieb inequalities: ﬁniteness, struc- ture, and extremals. J. Geom. Anal. , 31(4):3300–3350, 2021

work page 2021
[14]

Grigor’yan

A. Grigor’yan. Heat kernel and analysis on manifolds , volume 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009

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[15]

Ishige, Q

K. Ishige, Q. Liu, and P. Salani. A parabolic PDE-based approach to Borell–Brascamp–Lieb in- equality. arXiv e-prints , page arXiv:2405.16721, May 2024

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[16]

J. Lehec. Short probabilistic proof of the Brascamp-Lieb and B arthe theorems. Canad. Math. Bull. , 57(3):585–597, 2014

work page 2014
[17]

E. H. Lieb. Gaussian kernels have only Gaussian maximizers. Invent. Math. , 102(1):179–208, 1990

work page 1990
[18]

J. Liu, T. A. Courtade, P. W. Cuﬀ, and S. Verd´ u. A forward-r everse Brascamp-Lieb inequality: entropic duality and Gaussian optimality. Entropy, 20(6):Paper No. 418, 32, 2018

work page 2018
[19]

W. Rudin. Real and complex analysis . McGraw-Hill Book Co., New York, third edition, 1987

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[20]

S. I. Valdimarsson. Optimisers for the Brascamp-Lieb inequality . Israel J. Math. , 168:253–274, 2008. Ye Zhang Scuola Internazionale Superiore di Studi Avanzati (SISSA) Via Bonomea 265, 34136 Trieste, Italy E-mail: yezhang@sissa.it or zhangye0217@gmail.com 15

work page 2008

[1] [1]

F. Barthe. On a reverse form of the Brascamp-Lieb inequality. Invent. Math. , 134(2):335–361, 1998

work page 1998

[2] [2]

F. Barthe. Optimal Young’s inequality and its converse: a simple pr oof. Geom. Funct. Anal. , 8(2):234–242, 1998

work page 1998

[3] [3]

Barthe and D

F. Barthe and D. Cordero-Erausquin. Inverse Brascamp-Lieb inequalities along the heat equation. In Geometric aspects of functional analysis , volume 1850 of Lecture Notes in Math. , pages 65–71. Springer, Berlin, 2004

work page 2004

[4] [4]

Barthe, D

F. Barthe, D. Cordero-Erausquin, M. Ledoux, and B. Maurey. Correlation and Brascamp-Lieb inequalities for Markov semigroups. Int. Math. Res. Not. IMRN , (10):2177–2216, 2011

work page 2011

[5] [5]

Barthe and N

F. Barthe and N. Huet. On Gaussian Brunn-Minkowski inequalities . Studia Math. , 191(3):283–304, 2009

work page 2009

[6] [6]

Bennett, A

J. Bennett, A. Carbery, M. Christ, and T. Tao. The Brascamp- Lieb inequalities: ﬁniteness, structure and extremals. Geom. Funct. Anal. , 17(5):1343–1415, 2008

work page 2008

[7] [7]

C. Borell. Diﬀusion equations and geometric inequalities. Potential Anal. , 12(1):49–71, 2000

work page 2000

[8] [8]

K. J. Boroczky, P. Kalantzopoulos, and D. Xi. The Case of Equality in Geometric Instances of Barthe’s Reverse Brascamp-Lieb Inequality , pages 129–165. Springer International Publishing, Cham, 2023

work page 2023

[9] [9]

H. J. Brascamp and E. H. Lieb. Best constants in Young’s inequalit y, its converse, and its general- ization to more than three functions. Advances in Math. , 20(2):151–173, 1976

work page 1976

[10] [10]

H. J. Brascamp, E. H. Lieb, and J. M. Luttinger. A general rea rrangement inequality for multiple integrals. J. Functional Analysis , 17:227–237, 1974

work page 1974

[11] [11]

E. A. Carlen and D. Cordero-Erausquin. Subadditivity of the en tropy and its relation to Brascamp- Lieb type inequalities. Geom. Funct. Anal. , 19(2):373–405, 2009

work page 2009

[12] [12]

E. A. Carlen, E. H. Lieb, and M. Loss. A sharp analog of Young’s in equality on SN and related entropy inequalities. J. Geom. Anal. , 14(3):487–520, 2004. 14

work page 2004

[13] [13]

T. A. Courtade and J. Liu. Euclidean forward-reverse Brasca mp-Lieb inequalities: ﬁniteness, struc- ture, and extremals. J. Geom. Anal. , 31(4):3300–3350, 2021

work page 2021

[14] [14]

Grigor’yan

A. Grigor’yan. Heat kernel and analysis on manifolds , volume 47 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009

work page 2009

[15] [15]

Ishige, Q

K. Ishige, Q. Liu, and P. Salani. A parabolic PDE-based approach to Borell–Brascamp–Lieb in- equality. arXiv e-prints , page arXiv:2405.16721, May 2024

work page arXiv 2024

[16] [16]

J. Lehec. Short probabilistic proof of the Brascamp-Lieb and B arthe theorems. Canad. Math. Bull. , 57(3):585–597, 2014

work page 2014

[17] [17]

E. H. Lieb. Gaussian kernels have only Gaussian maximizers. Invent. Math. , 102(1):179–208, 1990

work page 1990

[18] [18]

J. Liu, T. A. Courtade, P. W. Cuﬀ, and S. Verd´ u. A forward-r everse Brascamp-Lieb inequality: entropic duality and Gaussian optimality. Entropy, 20(6):Paper No. 418, 32, 2018

work page 2018

[19] [19]

W. Rudin. Real and complex analysis . McGraw-Hill Book Co., New York, third edition, 1987

work page 1987

[20] [20]

S. I. Valdimarsson. Optimisers for the Brascamp-Lieb inequality . Israel J. Math. , 168:253–274, 2008. Ye Zhang Scuola Internazionale Superiore di Studi Avanzati (SISSA) Via Bonomea 265, 34136 Trieste, Italy E-mail: yezhang@sissa.it or zhangye0217@gmail.com 15

work page 2008