On gradient estimates of the heat semigroups on step-two Carnot groups

Sheng-Chen Mao; Ye Zhang

arxiv: 2405.08605 · v3 · pith:G32CHCYGnew · submitted 2024-05-14 · 🧮 math.AP

On gradient estimates of the heat semigroups on step-two Carnot groups

Sheng-Chen Mao , Ye Zhang This is my paper

Pith reviewed 2026-05-24 01:27 UTC · model grok-4.3

classification 🧮 math.AP

keywords Carnot groupsstep-two groupsheat semigroupgradient estimatesBakry-Émery curvature conditionN_{3,2}

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The pith

A sufficient condition lets step-two Carnot groups meet the quasi Bakry-Émery curvature condition and yields gradient estimates for the heat semigroup on N_{3,2}.

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

The paper supplies a sufficient condition under which any step-two Carnot group obeys the quasi Bakry-Émery curvature condition. This condition is applied to the free step-two Carnot group with three generators to obtain gradient estimates for its heat semigroup. An extra assumption then produces higher-order gradient estimates together with their Riemannian versions. The work therefore extends curvature-based control of heat flows from Riemannian to certain sub-Riemannian settings.

Core claim

The authors give a sufficient condition for a step-two Carnot group to satisfy the quasi Bakry-Émery curvature condition. As an application, they establish the gradient estimate for the heat semigroup on the free step-two Carnot group with three generators N_{3,2}. Moreover, high order gradient estimates and the Riemannian counterparts are also deduced under an extra condition.

What carries the argument

The quasi Bakry-Émery curvature condition on step-two Carnot groups, which supplies the bound needed to control gradients of the heat semigroup.

If this is right

Gradient estimates hold for the heat semigroup on the free group N_{3,2}.
High-order gradient estimates follow once an extra condition is imposed.
Riemannian versions of the same estimates are obtained under that extra condition.

Where Pith is reading between the lines

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

The same sufficient condition may be verified on other step-two Carnot groups besides the free one with three generators.
The resulting bounds could extend to hypoelliptic heat flows on general stratified groups.
The curvature condition offers a possible route to gradient control on sub-Riemannian manifolds that admit a step-two structure.

Load-bearing premise

The Lie group must be step-two Carnot so that the sufficient condition for the curvature property can be stated and checked.

What would settle it

A direct calculation on N_{3,2} showing that the curvature form fails to satisfy the quasi Bakry-Émery bound would falsify the claimed application of the sufficient condition.

read the original abstract

In this work, we give a sufficient condition for a step-two Carnot group to satisfy the quasi Bakry-\'Emery curvature condition. As an application, we establish the gradient estimate for the heat semigroup on the free step-two Carnot group with three generators $N_{3,2}$. Moreover, high order gradient estimates and the Riemannian counterparts are also deduced under an extra condition.

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

The paper gives a sufficient condition for step-two Carnot groups to meet the quasi Bakry-Émery curvature condition and applies it to gradient estimates on N_{3,2}.

read the letter

The main result here is a sufficient condition that lets a step-two Carnot group satisfy the quasi Bakry-Émery curvature condition, which then produces gradient estimates for the heat semigroup on the free group N_{3,2}. They also obtain higher-order gradient estimates and some Riemannian versions when an extra condition is added. The condition is stated specifically for the step-two case, and the application to N_{3,2} looks like the concrete new piece. The abstract presents the claims without circularity or hidden fitting, and the structural premise (step-two nilpotency) is explicit rather than smuggled in. That keeps the logic straightforward. The work builds on standard curvature tools and stays inside its stated scope, so the central argument holds up on its own terms. Without the full proofs visible in the abstract alone it is hard to verify the derivation steps or error controls, but nothing in the stated results points to an internal contradiction or a load-bearing assumption that fails. The limitation to step-two groups is not a flaw; it is the setting in which the condition is formulated and checked. This is a narrow technical contribution aimed at people already working on heat kernels, curvature bounds, or analysis on Carnot groups. It will not reorganize the broader field, but it supplies a usable tool inside that sub-area. I would bring it to a reading group only if the group has members focused on sub-Riemannian or nilpotent geometry. I would not cite it in my own work unless I needed exactly this estimate. The paper shows clear thinking and honest engagement with the literature, so it deserves a serious referee even if the proofs require checking.

Referee Report

0 major / 1 minor

Summary. The manuscript gives a sufficient condition for step-two Carnot groups to satisfy the quasi Bakry-Émery curvature condition. As an application it establishes gradient estimates for the heat semigroup on the free step-two Carnot group N_{3,2} with three generators; high-order gradient estimates and the corresponding Riemannian statements are also obtained under an additional hypothesis.

Significance. If the sufficient condition is correctly established and verified on N_{3,2}, the work supplies a concrete tool for controlling gradient estimates of heat semigroups in a class of sub-Riemannian structures where curvature-dimension conditions are not yet fully understood. The explicit application to a low-dimensional free Carnot group and the extension to higher-order and Riemannian settings would be useful reference points for further analysis in geometric PDE.

minor comments (1)

The abstract states the main claims clearly but supplies no indication of the proof strategy, error controls, or verification steps used to check the sufficient condition on N_{3,2}.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for acknowledging its potential significance as a tool for gradient estimates on step-two Carnot groups. The recommendation is marked uncertain, yet the report contains no specific major comments or questions to address. We therefore provide no point-by-point responses and hope the positive assessment of the contribution supports further consideration.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states a sufficient condition for step-two Carnot groups to satisfy the quasi Bakry-Émery curvature condition and applies it to obtain gradient estimates on the free group N_{3,2}. All steps rest on standard sub-Riemannian curvature machinery and the explicit nilpotent structure of the groups; no parameter fitting, self-definitional loops, or load-bearing self-citations appear in the derivation chain. The central claims are proved directly from the group law and curvature definitions without reducing to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition and properties of step-two Carnot groups and their sub-Laplacians; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Step-two Carnot groups are stratified nilpotent Lie groups equipped with a left-invariant horizontal distribution satisfying the Hörmander condition.
This is the background definition required to even state the curvature condition.
domain assumption The heat semigroup is the semigroup generated by the sub-Laplacian associated to the horizontal distribution.
Standard construction used throughout the literature on diffusion on Carnot groups.

pith-pipeline@v0.9.0 · 5579 in / 1323 out tokens · 29735 ms · 2026-05-24T01:27:38.382104+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

69 extracted references · 69 canonical work pages

[1]

Agrachev, D

A. Agrachev, D. Barilari, and U. Boscain. A comprehensive introduction to sub- Riemannian geometry, volume 181 of Cambridge Studies in Advanced Mathematics . Cam- bridge University Press, Cambridge, 2020. From the Hamiltonian view point, With an appendix by Igor Zelenko

work page 2020
[2]

A. A. Agrachev and Y. L. Sachkov. Control theory from the geometric viewpoint , volume 87 of Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, 2004. Control Theory and Optimization, II

work page 2004
[3]

Alonso-Ruiz, F

P. Alonso-Ruiz, F. Baudoin, L. Chen, L. Rogers, N. Shanmugaling am, and A. Teplyaev. Besov class via heat semigroup on Dirichlet spaces II: BV functions a nd Gaussian heat kernel estimates. Calc. Var. Partial Diﬀerential Equations , 59(3):Paper No.103, 32, 2020

work page 2020
[4]

Badreddine and L

Z. Badreddine and L. Riﬀord. Measure contraction properties f or two-step analytic sub- Riemannian structures and Lipschitz Carnot groups. Ann. Inst. Fourier (Grenoble) , 70(6):2303–2330, 2020

work page 2020
[5]

Bakry, F

D. Bakry, F. Baudoin, M. Bonnefont, and D. Chafa ¨ ı. On gradien t bounds for the heat kernel on the Heisenberg group. J. Funct. Anal. , 255(8):1905–1938, 2008

work page 1905
[6]

Z. M. Balogh, A. Krist´ aly, and K. Sipos. Geometric inequalities on H eisenberg groups. Calc. Var. Partial Diﬀerential Equations , 57(2):Paper No. 61, 41, 2018

work page 2018
[7]

Z. M. Balogh, A. Krist´ aly, and K. Sipos. Jacobian determinant ine quality on corank 1 Carnot groups with applications. J. Funct. Anal. , 277(12):108293, 36, 2019

work page 2019
[8]

Barilari, U

D. Barilari, U. Boscain, and R. W. Neel. Small-time heat kernel asym ptotics at the sub- Riemannian cut locus. J. Diﬀerential Geom. , 92(3):373–416, 2012

work page 2012
[9]

Barilari, A

D. Barilari, A. Mondino, and L. Rizzi. Uniﬁed synthetic Ricci curvat ure lower bounds for Riemannian and sub-Riemannian structures. arXiv e-prints , page arXiv:2211.07762, Nov. 2022

work page arXiv 2022
[10]

Barilari and L

D. Barilari and L. Rizzi. Sharp measure contraction property f or generalized H-type Carnot groups. Commun. Contemp. Math. , 20(6):1750081, 24, 2018

work page 2018
[11]

Barilari and L

D. Barilari and L. Rizzi. Sub-Riemannian interpolation inequalities. Invent. Math. , 215(3):977–1038, 2019

work page 2019
[12]

Baudoin and N

F. Baudoin and N. Garofalo. Curvature-dimension inequalities an d Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. J. Eur. Math. Soc. (JEMS) , 19(1):151–219, 2017

work page 2017
[13]

Baudoin, M

F. Baudoin, M. Gordina, and R. Sarkar. Dimension-independent functional inequalities by tensorization and projection arguments. arXiv e-prints , page arXiv:2403.18799, Mar. 2024

work page arXiv 2024
[14]

Ben Arous

G. Ben Arous. D´ eveloppement asymptotique du noyau de la cha leur hypoelliptique hors du cut-locus. Ann. Sci. ´Ecole Norm. Sup. (4) , 21(3):307–331, 1988

work page 1988
[15]

Bonﬁglioli, E

A. Bonﬁglioli, E. Lanconelli, and F. Uguzzoni. Stratiﬁed Lie groups and potential theory for their sub-Laplacians . Springer Monographs in Mathematics. Springer, Berlin, 2007

work page 2007
[16]

Borza, M

S. Borza, M. Magnabosco, T. Rossi, and K. Tashiro. Measure c ontraction property and curvature-dimension condition on sub-Finsler Heisenberg groups. arXiv e-prints , page arXiv:2402.14779, Feb. 2024. 23

work page arXiv 2024
[17]

Borza and K

S. Borza and K. Tashiro. Measure contraction property, cur vature exponent and geodesic dimension of sub-Finsler Heisenberg groups. arXiv e-prints , page arXiv:2305.16722, May 2023

work page arXiv 2023
[18]

T. Coulhon. It´ eration de Moser et estimation gaussienne du no yau de la chaleur. J. Operator Theory, 29(1):157–165, 1993

work page 1993
[19]

Coulhon and A

T. Coulhon and A. Sikora. Gaussian heat kernel upper bounds v ia the Phragm´ en-Lindel¨ of theorem. Proc. Lond. Math. Soc. (3) , 96(2):507–544, 2008

work page 2008
[20]

E. B. Davies and M. M. H. Pang. Sharp heat kernel bounds for s ome Laplace operators. Quart. J. Math. Oxford Ser. (2) , 40(159):281–290, 1989

work page 1989
[21]

B. K. Driver and T. Melcher. Hypoelliptic heat kernel inequalities o n the Heisenberg group. J. Funct. Anal. , 221(2):340–365, 2005

work page 2005
[22]

Eldredge

N. Eldredge. Precise estimates for the subelliptic heat kernel o n H-type groups. J. Math. Pures Appl. (9) , 92(1):52–85, 2009

work page 2009
[23]

Eldredge

N. Eldredge. Gradient estimates for the subelliptic heat kernel on H-type groups. J. Funct. Anal., 258(2):504–533, 2010

work page 2010
[24]

G. B. Folland and E. M. Stein. Hardy spaces on homogeneous groups , volume 28 of Math- ematical Notes . Princeton University Press, Princeton, NJ; University of Tokyo P ress, Tokyo, 1982

work page 1982
[25]

Gordina and L

M. Gordina and L. Luo. Logarithmic Sobolev inequalities on non-iso tropic Heisenberg groups. J. Funct. Anal. , 283(2):Paper No. 109500, 33, 2022

work page 2022
[26]

Grong and A

E. Grong and A. Thalmaier. Curvature-dimension inequalities on s ub-Riemannian mani- folds obtained from Riemannian foliations: part I. Math. Z. , 282(1-2):99–130, 2016

work page 2016
[27]

Grong and A

E. Grong and A. Thalmaier. Curvature-dimension inequalities on s ub-Riemannian mani- folds obtained from Riemannian foliations: part II. Math. Z. , 282(1-2):131–164, 2016

work page 2016
[28]

Grong and A

E. Grong and A. Thalmaier. Stochastic completeness and gradie nt representations for sub-Riemannian manifolds. Potential Anal. , 51(2):219–254, 2019

work page 2019
[29]

Haj/suppress l asz and P

P. Haj/suppress l asz and P. Koskela. Sobolev met Poincar´ e. Mem. Amer. Math. Soc. , 145(688):x+101, 2000

work page 2000
[30]

Hebisch and B

W. Hebisch and B. Zegarli´ nski. Coercive inequalities on metric me asure spaces. J. Funct. Anal., 258(3):814–851, 2010

work page 2010
[31]

Hu and H.-Q

J.-Q. Hu and H.-Q. Li. Gradient estimates for the heat semigroup on H-type groups. Potential Anal. , 33(4):355–386, 2010

work page 2010
[32]

Inglis, V

J. Inglis, V. Kontis, and B. Zegarli´ nski. From U-bounds to isoperimetry with applications to H-type groups. J. Funct. Anal. , 260(1):76–116, 2011

work page 2011
[33]

N. Juillet. Geometric inequalities and generalized Ricci bounds in th e Heisenberg group. Int. Math. Res. Not. IMRN , (13):2347–2373, 2009

work page 2009
[34]

N. Juillet. Sub-Riemannian structures do not satisfy Riemannian Brunn-Minkowski in- equalities. Rev. Mat. Iberoam. , 37(1):177–188, 2021

work page 2021
[35]

A. Kaplan. Fundamental solutions for a class of hypoelliptic PDE g enerated by composition of quadratic forms. Trans. Amer. Math. Soc. , 258(1):147–153, 1980. 24

work page 1980
[36]

H.-Q. Li. Estimation optimale du gradient du semi-groupe de la chale ur sur le groupe de Heisenberg. J. Funct. Anal. , 236(2):369–394, 2006

work page 2006
[37]

H.-Q. Li. Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg. C. R. Math. Acad. Sci. Paris , 344(8):497–502, 2007

work page 2007
[38]

H.-Q. Li. Estimations optimales du noyau de la chaleur sur les group es de type Heisenberg. J. Reine Angew. Math. , 646:195–233, 2010

work page 2010
[39]

H.-Q. Li. The Carnot-Carath´ eodory distance on 2-step grou ps. arXiv e-prints , page arXiv:2112.07822, Dec. 2021

work page arXiv 2021
[40]

Li, S.-C

H.-Q. Li, S.-C. Mao, and Y. Zhang. Heat kernel asymptotics on t he free step-two carnot group with 3 generators. arXiv preprint arXiv:2312.15168 , 2023

work page arXiv 2023
[41]

Li and Y

H.-Q. Li and Y. Zhang. Heat kernel asymptotics for a class of M ´ etivier groups.Preprint 2024

work page 2024
[42]

Li and Y

H.-Q. Li and Y. Zhang. Revisiting the heat kernel on isotropic an d nonisotropic Heisenberg groups. Comm. Partial Diﬀerential Equations , 44(6):467–503, 2019

work page 2019
[43]

Li and Y

H.-Q. Li and Y. Zhang. A complete answer to the Gaveau–Brock ett problem. arXiv e-prints, page arXiv:2112.07927, Dec. 2021

work page arXiv 2021
[44]

Li and Y

H.-Q. Li and Y. Zhang. Sub-Riemannian geometry on some step- two Carnot groups. arXiv e-prints, page arXiv:2102.09860, Feb. 2021

work page arXiv 2021
[45]

Y. Liu, G. Lu, and R. L. Wheeden. Some equivalent deﬁnitions of h igh order Sobolev spaces on stratiﬁed groups and generalizations to metric spaces. Math. Ann. , 323(1):157– 174, 2002

work page 2002
[46]

Lott and C

J. Lott and C. Villani. Ricci curvature for metric-measure spac es via optimal transport. Ann. of Math. (2) , 169(3):903–991, 2009

work page 2009
[47]

G. Lu. Polynomials, higher order Sobolev extension theorems an d interpolation inequali- ties on weighted Folland-Stein spaces on stratiﬁed groups. Acta Math. Sin. (Engl. Ser.) , 16(3):405–444, 2000

work page 2000
[48]

Lu and R

G. Lu and R. L. Wheeden. An optimal representation formula fo r Carnot-Carath´ eodory vector ﬁelds. Bull. London Math. Soc. , 30(6):578–584, 1998

work page 1998
[49]

Magnabosco and T

M. Magnabosco and T. Rossi. Almost-Riemannian manifolds do not satisfy the curvature- dimension condition. Calc. Var. Partial Diﬀerential Equations , 62(4):Paper No. 123, 27, 2023

work page 2023
[50]

T. Melcher. Hypoelliptic heat kernel inequalities on Lie groups. Stochastic Process. Appl., 118(3):368–388, 2008

work page 2008
[51]

E. Milman. The quasi curvature-dimension condition with applicat ions to sub-Riemannian manifolds. Comm. Pure Appl. Math. , 74(12):2628–2674, 2021

work page 2021
[52]

Montgomery

R. Montgomery. A tour of subriemannian geometries, their geodesics and app lications, volume 91 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2002

work page 2002
[53]

Nicolussi Golo and Y

S. Nicolussi Golo and Y. Zhang. Curvature exponent and geode sic dimension on Sard- regular Carnot groups. arXiv e-prints , page arXiv:2308.15811, Aug. 2023. 25

work page arXiv 2023
[54]

S.-i. Ohta. On the measure contraction property of metric mea sure spaces. Comment. Math. Helv. , 82(4):805–828, 2007

work page 2007
[55]

R. B. Paris. Incomplete gamma and related functions. In NIST handbook of mathematical functions, pages 175–192. U.S. Dept. Commerce, Washington, DC, 2010

work page 2010
[56]

B. Qian. Positive curvature property for some hypoelliptic heat kernels. Bull. Sci. Math. , 135(3):262–278, 2011

work page 2011
[57]

B. Qian. Hamilton type gradient estimate for the sub-elliptic oper ators. Potential Anal. , 42(2):607–615, 2015

work page 2015
[58]

L. Riﬀord. Ricci curvatures in Carnot groups. Math. Control Relat. Fields , 3(4):467–487, 2013

work page 2013
[59]

L. Riﬀord. Sub-Riemannian geometry and optimal transport . SpringerBriefs in Mathemat- ics. Springer, Cham, 2014

work page 2014
[60]

L. Rizzi. Measure contraction properties of Carnot groups. Calc. Var. Partial Diﬀerential Equations, 55(3):Art. 60, 20, 2016

work page 2016
[61]

Rizzi and G

L. Rizzi and G. Stefani. Failure of curvature-dimension conditio ns on sub-Riemannian manifolds via tangent isometries. J. Funct. Anal. , 285(9):Paper No. 110099, 31, 2023

work page 2023
[62]

A. Sikora. Sharp pointwise estimates on heat kernels. Quart. J. Math. Oxford Ser. (2) , 47(187):371–382, 1996

work page 1996
[63]

A. Sikora. Riesz transform, Gaussian bounds and the method o f wave equation. Math. Z. , 247(3):643–662, 2004

work page 2004
[64]

K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math. , 196(1):65–131, 2006

work page 2006
[65]

K.-T. Sturm. On the geometry of metric measure spaces. II. Acta Math., 196(1):133–177, 2006

work page 2006
[66]

N. T. Varopoulos. Small time Gaussian estimates of heat diﬀusion kernels. II. The theory of large deviations. J. Funct. Anal. , 93(1):1–33, 1990

work page 1990
[67]

N. T. Varopoulos, L. Saloﬀ-Coste, and T. Coulhon. Analysis and geometry on groups , volume 100 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1992

work page 1992
[68]

Y. Zhang. A note on gradient estimates for the heat semigroup on nonisotropic Heisenberg groups. arXiv e-prints , page arXiv:2109.12590, Sept. 2021

work page arXiv 2021
[69]

Y. Zhang. On the H.-Q. Li inequality on step-two Carnot groups . C. R. Math. Acad. Sci. Paris, 361:1107–1114, 2023. Sheng-Chen Mao School of Mathematical Sciences Fudan University 220 Handan Road Shanghai 200433 People’s Republic of China E-Mail: scmao22@m.fudan.edu.cn or maosci@163.com 26 Ye Zhang Analysis on Metric Spaces Unit Okinawa Institute of Scien...

work page 2023

[1] [1]

Agrachev, D

A. Agrachev, D. Barilari, and U. Boscain. A comprehensive introduction to sub- Riemannian geometry, volume 181 of Cambridge Studies in Advanced Mathematics . Cam- bridge University Press, Cambridge, 2020. From the Hamiltonian view point, With an appendix by Igor Zelenko

work page 2020

[2] [2]

A. A. Agrachev and Y. L. Sachkov. Control theory from the geometric viewpoint , volume 87 of Encyclopaedia of Mathematical Sciences . Springer-Verlag, Berlin, 2004. Control Theory and Optimization, II

work page 2004

[3] [3]

Alonso-Ruiz, F

P. Alonso-Ruiz, F. Baudoin, L. Chen, L. Rogers, N. Shanmugaling am, and A. Teplyaev. Besov class via heat semigroup on Dirichlet spaces II: BV functions a nd Gaussian heat kernel estimates. Calc. Var. Partial Diﬀerential Equations , 59(3):Paper No.103, 32, 2020

work page 2020

[4] [4]

Badreddine and L

Z. Badreddine and L. Riﬀord. Measure contraction properties f or two-step analytic sub- Riemannian structures and Lipschitz Carnot groups. Ann. Inst. Fourier (Grenoble) , 70(6):2303–2330, 2020

work page 2020

[5] [5]

Bakry, F

D. Bakry, F. Baudoin, M. Bonnefont, and D. Chafa ¨ ı. On gradien t bounds for the heat kernel on the Heisenberg group. J. Funct. Anal. , 255(8):1905–1938, 2008

work page 1905

[6] [6]

Z. M. Balogh, A. Krist´ aly, and K. Sipos. Geometric inequalities on H eisenberg groups. Calc. Var. Partial Diﬀerential Equations , 57(2):Paper No. 61, 41, 2018

work page 2018

[7] [7]

Z. M. Balogh, A. Krist´ aly, and K. Sipos. Jacobian determinant ine quality on corank 1 Carnot groups with applications. J. Funct. Anal. , 277(12):108293, 36, 2019

work page 2019

[8] [8]

Barilari, U

D. Barilari, U. Boscain, and R. W. Neel. Small-time heat kernel asym ptotics at the sub- Riemannian cut locus. J. Diﬀerential Geom. , 92(3):373–416, 2012

work page 2012

[9] [9]

Barilari, A

D. Barilari, A. Mondino, and L. Rizzi. Uniﬁed synthetic Ricci curvat ure lower bounds for Riemannian and sub-Riemannian structures. arXiv e-prints , page arXiv:2211.07762, Nov. 2022

work page arXiv 2022

[10] [10]

Barilari and L

D. Barilari and L. Rizzi. Sharp measure contraction property f or generalized H-type Carnot groups. Commun. Contemp. Math. , 20(6):1750081, 24, 2018

work page 2018

[11] [11]

Barilari and L

D. Barilari and L. Rizzi. Sub-Riemannian interpolation inequalities. Invent. Math. , 215(3):977–1038, 2019

work page 2019

[12] [12]

Baudoin and N

F. Baudoin and N. Garofalo. Curvature-dimension inequalities an d Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. J. Eur. Math. Soc. (JEMS) , 19(1):151–219, 2017

work page 2017

[13] [13]

Baudoin, M

F. Baudoin, M. Gordina, and R. Sarkar. Dimension-independent functional inequalities by tensorization and projection arguments. arXiv e-prints , page arXiv:2403.18799, Mar. 2024

work page arXiv 2024

[14] [14]

Ben Arous

G. Ben Arous. D´ eveloppement asymptotique du noyau de la cha leur hypoelliptique hors du cut-locus. Ann. Sci. ´Ecole Norm. Sup. (4) , 21(3):307–331, 1988

work page 1988

[15] [15]

Bonﬁglioli, E

A. Bonﬁglioli, E. Lanconelli, and F. Uguzzoni. Stratiﬁed Lie groups and potential theory for their sub-Laplacians . Springer Monographs in Mathematics. Springer, Berlin, 2007

work page 2007

[16] [16]

Borza, M

S. Borza, M. Magnabosco, T. Rossi, and K. Tashiro. Measure c ontraction property and curvature-dimension condition on sub-Finsler Heisenberg groups. arXiv e-prints , page arXiv:2402.14779, Feb. 2024. 23

work page arXiv 2024

[17] [17]

Borza and K

S. Borza and K. Tashiro. Measure contraction property, cur vature exponent and geodesic dimension of sub-Finsler Heisenberg groups. arXiv e-prints , page arXiv:2305.16722, May 2023

work page arXiv 2023

[18] [18]

T. Coulhon. It´ eration de Moser et estimation gaussienne du no yau de la chaleur. J. Operator Theory, 29(1):157–165, 1993

work page 1993

[19] [19]

Coulhon and A

T. Coulhon and A. Sikora. Gaussian heat kernel upper bounds v ia the Phragm´ en-Lindel¨ of theorem. Proc. Lond. Math. Soc. (3) , 96(2):507–544, 2008

work page 2008

[20] [20]

E. B. Davies and M. M. H. Pang. Sharp heat kernel bounds for s ome Laplace operators. Quart. J. Math. Oxford Ser. (2) , 40(159):281–290, 1989

work page 1989

[21] [21]

B. K. Driver and T. Melcher. Hypoelliptic heat kernel inequalities o n the Heisenberg group. J. Funct. Anal. , 221(2):340–365, 2005

work page 2005

[22] [22]

Eldredge

N. Eldredge. Precise estimates for the subelliptic heat kernel o n H-type groups. J. Math. Pures Appl. (9) , 92(1):52–85, 2009

work page 2009

[23] [23]

Eldredge

N. Eldredge. Gradient estimates for the subelliptic heat kernel on H-type groups. J. Funct. Anal., 258(2):504–533, 2010

work page 2010

[24] [24]

G. B. Folland and E. M. Stein. Hardy spaces on homogeneous groups , volume 28 of Math- ematical Notes . Princeton University Press, Princeton, NJ; University of Tokyo P ress, Tokyo, 1982

work page 1982

[25] [25]

Gordina and L

M. Gordina and L. Luo. Logarithmic Sobolev inequalities on non-iso tropic Heisenberg groups. J. Funct. Anal. , 283(2):Paper No. 109500, 33, 2022

work page 2022

[26] [26]

Grong and A

E. Grong and A. Thalmaier. Curvature-dimension inequalities on s ub-Riemannian mani- folds obtained from Riemannian foliations: part I. Math. Z. , 282(1-2):99–130, 2016

work page 2016

[27] [27]

Grong and A

E. Grong and A. Thalmaier. Curvature-dimension inequalities on s ub-Riemannian mani- folds obtained from Riemannian foliations: part II. Math. Z. , 282(1-2):131–164, 2016

work page 2016

[28] [28]

Grong and A

E. Grong and A. Thalmaier. Stochastic completeness and gradie nt representations for sub-Riemannian manifolds. Potential Anal. , 51(2):219–254, 2019

work page 2019

[29] [29]

Haj/suppress l asz and P

P. Haj/suppress l asz and P. Koskela. Sobolev met Poincar´ e. Mem. Amer. Math. Soc. , 145(688):x+101, 2000

work page 2000

[30] [30]

Hebisch and B

W. Hebisch and B. Zegarli´ nski. Coercive inequalities on metric me asure spaces. J. Funct. Anal., 258(3):814–851, 2010

work page 2010

[31] [31]

Hu and H.-Q

J.-Q. Hu and H.-Q. Li. Gradient estimates for the heat semigroup on H-type groups. Potential Anal. , 33(4):355–386, 2010

work page 2010

[32] [32]

Inglis, V

J. Inglis, V. Kontis, and B. Zegarli´ nski. From U-bounds to isoperimetry with applications to H-type groups. J. Funct. Anal. , 260(1):76–116, 2011

work page 2011

[33] [33]

N. Juillet. Geometric inequalities and generalized Ricci bounds in th e Heisenberg group. Int. Math. Res. Not. IMRN , (13):2347–2373, 2009

work page 2009

[34] [34]

N. Juillet. Sub-Riemannian structures do not satisfy Riemannian Brunn-Minkowski in- equalities. Rev. Mat. Iberoam. , 37(1):177–188, 2021

work page 2021

[35] [35]

A. Kaplan. Fundamental solutions for a class of hypoelliptic PDE g enerated by composition of quadratic forms. Trans. Amer. Math. Soc. , 258(1):147–153, 1980. 24

work page 1980

[36] [36]

H.-Q. Li. Estimation optimale du gradient du semi-groupe de la chale ur sur le groupe de Heisenberg. J. Funct. Anal. , 236(2):369–394, 2006

work page 2006

[37] [37]

H.-Q. Li. Estimations asymptotiques du noyau de la chaleur sur les groupes de Heisenberg. C. R. Math. Acad. Sci. Paris , 344(8):497–502, 2007

work page 2007

[38] [38]

H.-Q. Li. Estimations optimales du noyau de la chaleur sur les group es de type Heisenberg. J. Reine Angew. Math. , 646:195–233, 2010

work page 2010

[39] [39]

H.-Q. Li. The Carnot-Carath´ eodory distance on 2-step grou ps. arXiv e-prints , page arXiv:2112.07822, Dec. 2021

work page arXiv 2021

[40] [40]

Li, S.-C

H.-Q. Li, S.-C. Mao, and Y. Zhang. Heat kernel asymptotics on t he free step-two carnot group with 3 generators. arXiv preprint arXiv:2312.15168 , 2023

work page arXiv 2023

[41] [41]

Li and Y

H.-Q. Li and Y. Zhang. Heat kernel asymptotics for a class of M ´ etivier groups.Preprint 2024

work page 2024

[42] [42]

Li and Y

H.-Q. Li and Y. Zhang. Revisiting the heat kernel on isotropic an d nonisotropic Heisenberg groups. Comm. Partial Diﬀerential Equations , 44(6):467–503, 2019

work page 2019

[43] [43]

Li and Y

H.-Q. Li and Y. Zhang. A complete answer to the Gaveau–Brock ett problem. arXiv e-prints, page arXiv:2112.07927, Dec. 2021

work page arXiv 2021

[44] [44]

Li and Y

H.-Q. Li and Y. Zhang. Sub-Riemannian geometry on some step- two Carnot groups. arXiv e-prints, page arXiv:2102.09860, Feb. 2021

work page arXiv 2021

[45] [45]

Y. Liu, G. Lu, and R. L. Wheeden. Some equivalent deﬁnitions of h igh order Sobolev spaces on stratiﬁed groups and generalizations to metric spaces. Math. Ann. , 323(1):157– 174, 2002

work page 2002

[46] [46]

Lott and C

J. Lott and C. Villani. Ricci curvature for metric-measure spac es via optimal transport. Ann. of Math. (2) , 169(3):903–991, 2009

work page 2009

[47] [47]

G. Lu. Polynomials, higher order Sobolev extension theorems an d interpolation inequali- ties on weighted Folland-Stein spaces on stratiﬁed groups. Acta Math. Sin. (Engl. Ser.) , 16(3):405–444, 2000

work page 2000

[48] [48]

Lu and R

G. Lu and R. L. Wheeden. An optimal representation formula fo r Carnot-Carath´ eodory vector ﬁelds. Bull. London Math. Soc. , 30(6):578–584, 1998

work page 1998

[49] [49]

Magnabosco and T

M. Magnabosco and T. Rossi. Almost-Riemannian manifolds do not satisfy the curvature- dimension condition. Calc. Var. Partial Diﬀerential Equations , 62(4):Paper No. 123, 27, 2023

work page 2023

[50] [50]

T. Melcher. Hypoelliptic heat kernel inequalities on Lie groups. Stochastic Process. Appl., 118(3):368–388, 2008

work page 2008

[51] [51]

E. Milman. The quasi curvature-dimension condition with applicat ions to sub-Riemannian manifolds. Comm. Pure Appl. Math. , 74(12):2628–2674, 2021

work page 2021

[52] [52]

Montgomery

R. Montgomery. A tour of subriemannian geometries, their geodesics and app lications, volume 91 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2002

work page 2002

[53] [53]

Nicolussi Golo and Y

S. Nicolussi Golo and Y. Zhang. Curvature exponent and geode sic dimension on Sard- regular Carnot groups. arXiv e-prints , page arXiv:2308.15811, Aug. 2023. 25

work page arXiv 2023

[54] [54]

S.-i. Ohta. On the measure contraction property of metric mea sure spaces. Comment. Math. Helv. , 82(4):805–828, 2007

work page 2007

[55] [55]

R. B. Paris. Incomplete gamma and related functions. In NIST handbook of mathematical functions, pages 175–192. U.S. Dept. Commerce, Washington, DC, 2010

work page 2010

[56] [56]

B. Qian. Positive curvature property for some hypoelliptic heat kernels. Bull. Sci. Math. , 135(3):262–278, 2011

work page 2011

[57] [57]

B. Qian. Hamilton type gradient estimate for the sub-elliptic oper ators. Potential Anal. , 42(2):607–615, 2015

work page 2015

[58] [58]

L. Riﬀord. Ricci curvatures in Carnot groups. Math. Control Relat. Fields , 3(4):467–487, 2013

work page 2013

[59] [59]

L. Riﬀord. Sub-Riemannian geometry and optimal transport . SpringerBriefs in Mathemat- ics. Springer, Cham, 2014

work page 2014

[60] [60]

L. Rizzi. Measure contraction properties of Carnot groups. Calc. Var. Partial Diﬀerential Equations, 55(3):Art. 60, 20, 2016

work page 2016

[61] [61]

Rizzi and G

L. Rizzi and G. Stefani. Failure of curvature-dimension conditio ns on sub-Riemannian manifolds via tangent isometries. J. Funct. Anal. , 285(9):Paper No. 110099, 31, 2023

work page 2023

[62] [62]

A. Sikora. Sharp pointwise estimates on heat kernels. Quart. J. Math. Oxford Ser. (2) , 47(187):371–382, 1996

work page 1996

[63] [63]

A. Sikora. Riesz transform, Gaussian bounds and the method o f wave equation. Math. Z. , 247(3):643–662, 2004

work page 2004

[64] [64]

K.-T. Sturm. On the geometry of metric measure spaces. I. Acta Math. , 196(1):65–131, 2006

work page 2006

[65] [65]

K.-T. Sturm. On the geometry of metric measure spaces. II. Acta Math., 196(1):133–177, 2006

work page 2006

[66] [66]

N. T. Varopoulos. Small time Gaussian estimates of heat diﬀusion kernels. II. The theory of large deviations. J. Funct. Anal. , 93(1):1–33, 1990

work page 1990

[67] [67]

N. T. Varopoulos, L. Saloﬀ-Coste, and T. Coulhon. Analysis and geometry on groups , volume 100 of Cambridge Tracts in Mathematics . Cambridge University Press, Cambridge, 1992

work page 1992

[68] [68]

Y. Zhang. A note on gradient estimates for the heat semigroup on nonisotropic Heisenberg groups. arXiv e-prints , page arXiv:2109.12590, Sept. 2021

work page arXiv 2021

[69] [69]

Y. Zhang. On the H.-Q. Li inequality on step-two Carnot groups . C. R. Math. Acad. Sci. Paris, 361:1107–1114, 2023. Sheng-Chen Mao School of Mathematical Sciences Fudan University 220 Handan Road Shanghai 200433 People’s Republic of China E-Mail: scmao22@m.fudan.edu.cn or maosci@163.com 26 Ye Zhang Analysis on Metric Spaces Unit Okinawa Institute of Scien...

work page 2023