Failure of stability of a maximal operator bound for perturbed Nevo-Thangavelu means

arxiv: 2510.14180 · v2 · submitted 2025-10-16 · 🧮 math.CA · math.FA· math.RT

Failure of stability of a maximal operator bound for perturbed Nevo-Thangavelu means

Jaehyeon Ryu , Andreas Seeger This is my paper

Pith reviewed 2026-05-18 07:01 UTC · model grok-4.3

classification 🧮 math.CA math.FAmath.RT

keywords maximal operatorstwo-step nilpotent groupsNevo-Thangavelu meansL^p boundednesstilted hyperplanesMétivier groupsstability under perturbation

0 comments p. Extension

The pith

Maximal bounds for perturbed spheres on two-step nilpotent groups fail to stay stable under generic tilts except in Métivier cases.

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

The paper studies maximal operators formed by averaging over spheres contained in a d-dimensional linear subspace H of a two-step nilpotent Lie group, with dilations taken from the group's automorphic action. Boundedness results are known when H coincides with the first layer of the Lie algebra. The authors focus on the case of a small tilt of H that breaks invariance under these dilations. They prove that the stability of L^p bounds under such a tilt, which is known to hold for Métivier groups, does not hold for general two-step groups and derive fresh necessary conditions that must be satisfied for boundedness to be possible. The arguments are carried out in the broader setting of tilted submanifolds of the first layer.

Core claim

In two-step nilpotent Lie groups that are not Métivier, the L^p boundedness of maximal functions associated to spheres in a tilted hyperplane H is not stable under small linear perturbations that destroy invariance under the automorphic dilations. This stands in contrast to the Métivier case, and new necessary conditions for any L^p boundedness are obtained, including in the more general setting of tilted submanifolds of g1.

What carries the argument

The maximal averaging operator over dilated spheres lying in a tilted d-dimensional linear subspace H of the Lie algebra that is not preserved by the automorphic dilations.

If this is right

Any proof of L^p boundedness in the tilted setting must incorporate new necessary conditions that reflect the broken invariance under automorphic dilations.
Small linear tilts of the hyperplane can destroy L^p boundedness that holds for the untilted case in non-Métivier two-step groups.
The same failure of stability and the new necessary conditions extend to maximal operators defined on tilted submanifolds of g1.
The distinction between Métivier groups and other two-step groups is essential for the stability question.

Where Pith is reading between the lines

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

Invariance of the subspace under automorphic dilations seems to be the structural feature that permits stability.
The necessary conditions may help classify which two-step groups admit any form of tilted boundedness.
Analogous stability questions arise for maximal operators on nilpotent groups of step greater than two.

Load-bearing premise

The hyperplane is a small tilt of the first layer that is not invariant under the automorphic dilations.

What would settle it

An explicit non-Métivier two-step group together with a concrete small tilt for which the associated maximal operator remains bounded on the same range of L^p as the unperturbed operator would disprove the claimed general failure of stability.

read the original abstract

Let $G$ be a two-step nilpotent Lie group, identified via the exponential map with the Lie-algebra $\mathfrak g=\mathfrak g_1\oplus\mathfrak g_2$, where $[\mathfrak g,\mathfrak g]\subset \mathfrak g_2$. We consider maximal functions associated to spheres in a $d$-dimensional linear subspace $H$, dilated by the automorphic dilations. $L^p$ boundedness results for the case where $H=\mathfrak g_1$ are well understood. Here we consider the case of a tilted hyperplane $H\neq \mathfrak g_1$ which is not invariant under the automorphic dilations. In the case of M\'etivier groups it is known that the $L^p$-boundedness results are stable under a small linear tilt. We show that this is generally not the case for other two-step groups, and provide new necessary conditions for $L^p$ boundedness. We prove these results in a more general setting with tilted versions of submanifolds of $\mathfrak g_1$.

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

Stability of these maximal bounds fails under small tilts for non-Métivier two-step groups, along with some new necessary conditions.

read the letter

The main takeaway is that L^p boundedness for these maximal operators tied to spheres in a tilted hyperplane does not survive small linear perturbations when the two-step nilpotent group is not Métivier. The authors also derive fresh necessary conditions for boundedness that apply in this perturbed setting. This stands in contrast to the stability known for Métivier groups and extends the analysis to cases where the hyperplane loses invariance under the automorphic dilations. They frame the work more generally for tilted submanifolds of the first layer, which keeps the results from being too narrowly scoped. That extension and the direct comparison to prior stability results are the clearest contributions here. The arguments appear to rest on analyzing the operators and checking the necessary conditions against the tilted geometry rather than relying on fitted quantities or circular reductions. On the softer side, the critical step is verifying that the tilt actually violates those necessary conditions enough to force unboundedness. If the test functions or decay estimates do not fully reflect the broken scaling, the failure-of-stability claim could rest on thinner evidence than it first appears. The abstract suggests the proofs handle the general case directly, but the details of how the counterexamples capture the non-homogeneous behavior would need close checking. This is specialized work for people already following maximal operators and harmonic analysis on Lie groups. A reader interested in stability questions or necessary conditions for nilpotent settings would get concrete value from the necessary conditions and the counterexamples to stability. It engages the existing literature on Métivier groups without overclaiming, so the thinking looks honest on its own terms. I would send it to peer review. The claims are specific enough and the gap they address is real, so referees can sort out the technical steps without it being a waste of time.

Referee Report

2 major / 2 minor

Summary. The paper studies maximal operators associated to dilated spheres in a d-dimensional linear subspace H of a two-step nilpotent Lie group G, where H is a tilted hyperplane not invariant under the automorphic dilations. While L^p boundedness is known to be stable under small tilts when G is Métivier, the authors show this stability generally fails for other two-step groups. They derive new necessary conditions for L^p boundedness and extend the analysis to tilted submanifolds of g1.

Significance. If the central claims hold, the work clarifies the role of the Métivier condition in preserving boundedness under perturbations and supplies new necessary conditions that can be checked in concrete examples. The direct analysis of the operators (rather than reduction to fitted quantities) is a strength, as is the generalization beyond hyperplanes.

major comments (2)

[§3] §3 (new necessary conditions): the derivation of the necessary condition via testing on suitable functions or Fourier decay of the surface measure must be shown to apply verbatim once H is tilted and the homogeneity is broken; if any step tacitly re-uses dilation invariance that survives only for the un-tilted case, the necessity claim does not transfer.
[§4.2] §4.2 (verification for small tilts): the explicit check that the new necessary condition is violated for arbitrarily small tilts in a non-Métivier group is load-bearing; the manuscript should supply a concrete computation (e.g., the decay rate of the Fourier transform of the dilated surface measure on the tilted H) rather than an appeal to the general statement.

minor comments (2)

[Introduction] Notation for the decomposition g = g1 ⊕ g2 and the tilted hyperplane H should be introduced once and used uniformly; several places switch between “H” and “perturbed H” without re-definition.
[Theorem 1.3] The statement of the main theorem in the general submanifold setting would benefit from an explicit list of the assumptions that replace the hyperplane condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised help strengthen the presentation of the new necessary conditions and the verification of instability under small tilts. We respond to each major comment below and will incorporate revisions as indicated.

read point-by-point responses

Referee: [§3] §3 (new necessary conditions): the derivation of the necessary condition via testing on suitable functions or Fourier decay of the surface measure must be shown to apply verbatim once H is tilted and the homogeneity is broken; if any step tacitly re-uses dilation invariance that survives only for the un-tilted case, the necessity claim does not transfer.

Authors: The necessary conditions in §3 are derived directly in the tilted setting. We test the maximal operator against functions adapted to the tilted hyperplane H (or more generally to tilted submanifolds of g1) and obtain the required Fourier decay estimates from the two-step group law and the explicit tilt parameters, without invoking the full automorphic dilation invariance that holds only when H = g1. We will revise §3 to include a short clarifying paragraph at the outset that isolates the steps relying solely on the tilted geometry and confirms that no broken homogeneity is tacitly reused. revision: yes
Referee: [§4.2] §4.2 (verification for small tilts): the explicit check that the new necessary condition is violated for arbitrarily small tilts in a non-Métivier group is load-bearing; the manuscript should supply a concrete computation (e.g., the decay rate of the Fourier transform of the dilated surface measure on the tilted H) rather than an appeal to the general statement.

Authors: We agree that an explicit computation strengthens the argument. In the revised version we will insert a self-contained calculation for a concrete non-Métivier two-step group, using the Baker-Campbell-Hausdorff formula to compute the leading asymptotic decay of the Fourier transform of the dilated surface measure on the tilted H as the tilt parameter tends to zero. This will show directly that the necessary condition is violated for arbitrarily small tilts. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation is self-contained via direct operator analysis.

full rationale

The paper establishes new necessary conditions for L^p boundedness of maximal operators associated to dilated spheres in tilted hyperplanes H of two-step nilpotent groups by explicit testing on suitable functions and analysis of Fourier decay of surface measures. These conditions are then shown to fail under small tilts away from dilation-invariant cases when the group is not Métivier. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the arguments rely on independent Lie-algebraic and harmonic-analytic estimates that do not presuppose the target stability failure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of two-step nilpotent Lie groups and their automorphic dilations; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Two-step nilpotent Lie groups identified with Lie algebra g = g1 ⊕ g2 where [g, g] ⊂ g2.
Basic structural assumption for the groups under study.
standard math Existence of automorphic dilations used to scale spheres in subspace H.
Standard tool for defining the dilated maximal operators.

pith-pipeline@v0.9.0 · 5724 in / 1291 out tokens · 51383 ms · 2026-05-18T07:01:39.495395+00:00 · methodology

discussion (0)

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We show that this is generally not the case for other two-step groups, and provide new necessary conditions for L^p boundedness... Hypothesis H(r). There exist ϑ∈g∗2 and ω∘∈Σ... dim(V_Λ,ϑ)=r.

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Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages

[1]

Anderson, Laura Cladek, Malabika Pramanik, and Andreas Seeger

Theresa C. Anderson, Laura Cladek, Malabika Pramanik, and Andreas Seeger. Spherical means on the Heisenberg group: stability of a maximal function estimate. J. Anal. Math. 145 (2021), no. 1, 1–28. MR 4361901

work page 2021
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David Beltran, Shaoming Guo, Jonathan Hickman, and Andreas Seeger.The circular maximal operator on Heisenberg radial functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), no. 2, 501–568. MR 4453958

work page 2022
[3]

Differentiation of integrals inR n

Miguel de Guzm´ an. Differentiation of integrals inR n. With appendices by Antonio C´ ordoba, and Robert Fefferman, and two by Roberto Moriy´ on. Lecture Notes in Math., Vol. 481, Springer-Verlag, Berlin-Heidelberg-New York, 1975. xii+266 pp

work page 1975
[4]

Joonil Kim.Annulus maximal averages on variable hyperplanes, arXiv:1906.03797, 2019

work page arXiv 1906
[5]

Juyoung Lee and Sanghyuk Lee.L p →L q estimates for the circular maximal op- erator on Heisenberg radial functions.Math. Ann. 385 (2023), no. 3-4, 1521–1544. MR 4566682

work page 2023
[6]

Naijia Liu and Lixin Yan.Singular spherical maximal operators on a class of degen- erate two-step nilpotent Lie groups, Math. Z. 304 (2023), no. 1, 16. MR 4581163

work page 2023
[7]

Detlef M¨ uller and Andreas Seeger,Singular spherical maximal operators on a class of two step nilpotent Lie groups, Israel J. Math. 141 (2004), 315–340. MR 2063040

work page 2004
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E. K. Narayanan and Sundaram Thangavelu.An optimal theorem for the spheri- cal maximal operator on the Heisenberg group, Israel J. Math. 144 (2004), 211–219. MR 2121541

work page 2004
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Amos Nevo and Sundaram Thangavelu.Pointwise ergodic theorems for radial aver- ages on the Heisenberg group, Adv. Math. 127 (1997), no. 2, 307–334. MR 1448717

work page 1997
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Otton Martin Nikod´ ym.Sur la mesure des ensembles planes dont tous les points sont rectilin´ eairement accessible.Fund. Math. 10 (1927), 116-168

work page 1927
[11]

Malabika Pramanik and Andreas Seeger.L p-Sobolev estimates for a class of integral operators with folding canonical relations, J. Geom. Anal., 31, no. 7 (2021), 6725– 6765. ON PERTURBED NEVO–THANGAVELU MEANS 11

work page 2021
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Joris Roos, Andreas Seeger, and Rajula Srivastava.Lebesgue space estimates for spherical maximal functions on Heisenberg groups, Int. Math. Res. Not., IMRN 2022, no. 24, 19222–19257. MR 4523247

work page 2022
[13]

Jaehyeon Ryu and Andreas Seeger.Spherical maximal functions on two step nilpotent groups, Advances in Mathematics, 453 (2024), 109846

work page 2024
[14]

Stein.Maximal functions

Elias M. Stein.Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116 Jaehyeon Ryu: Department of Mathematics, Ewha Womans University, Seoul 03760, Korea Email address:jhryu67@ewha.ac.kr Andreas Seeger: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706, USA. Email ...

work page 1976

[1] [1]

Anderson, Laura Cladek, Malabika Pramanik, and Andreas Seeger

Theresa C. Anderson, Laura Cladek, Malabika Pramanik, and Andreas Seeger. Spherical means on the Heisenberg group: stability of a maximal function estimate. J. Anal. Math. 145 (2021), no. 1, 1–28. MR 4361901

work page 2021

[2] [2]

David Beltran, Shaoming Guo, Jonathan Hickman, and Andreas Seeger.The circular maximal operator on Heisenberg radial functions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), no. 2, 501–568. MR 4453958

work page 2022

[3] [3]

Differentiation of integrals inR n

Miguel de Guzm´ an. Differentiation of integrals inR n. With appendices by Antonio C´ ordoba, and Robert Fefferman, and two by Roberto Moriy´ on. Lecture Notes in Math., Vol. 481, Springer-Verlag, Berlin-Heidelberg-New York, 1975. xii+266 pp

work page 1975

[4] [4]

Joonil Kim.Annulus maximal averages on variable hyperplanes, arXiv:1906.03797, 2019

work page arXiv 1906

[5] [5]

Juyoung Lee and Sanghyuk Lee.L p →L q estimates for the circular maximal op- erator on Heisenberg radial functions.Math. Ann. 385 (2023), no. 3-4, 1521–1544. MR 4566682

work page 2023

[6] [6]

Naijia Liu and Lixin Yan.Singular spherical maximal operators on a class of degen- erate two-step nilpotent Lie groups, Math. Z. 304 (2023), no. 1, 16. MR 4581163

work page 2023

[7] [7]

Detlef M¨ uller and Andreas Seeger,Singular spherical maximal operators on a class of two step nilpotent Lie groups, Israel J. Math. 141 (2004), 315–340. MR 2063040

work page 2004

[8] [8]

E. K. Narayanan and Sundaram Thangavelu.An optimal theorem for the spheri- cal maximal operator on the Heisenberg group, Israel J. Math. 144 (2004), 211–219. MR 2121541

work page 2004

[9] [9]

Amos Nevo and Sundaram Thangavelu.Pointwise ergodic theorems for radial aver- ages on the Heisenberg group, Adv. Math. 127 (1997), no. 2, 307–334. MR 1448717

work page 1997

[10] [10]

Otton Martin Nikod´ ym.Sur la mesure des ensembles planes dont tous les points sont rectilin´ eairement accessible.Fund. Math. 10 (1927), 116-168

work page 1927

[11] [11]

Malabika Pramanik and Andreas Seeger.L p-Sobolev estimates for a class of integral operators with folding canonical relations, J. Geom. Anal., 31, no. 7 (2021), 6725– 6765. ON PERTURBED NEVO–THANGAVELU MEANS 11

work page 2021

[12] [12]

Joris Roos, Andreas Seeger, and Rajula Srivastava.Lebesgue space estimates for spherical maximal functions on Heisenberg groups, Int. Math. Res. Not., IMRN 2022, no. 24, 19222–19257. MR 4523247

work page 2022

[13] [13]

Jaehyeon Ryu and Andreas Seeger.Spherical maximal functions on two step nilpotent groups, Advances in Mathematics, 453 (2024), 109846

work page 2024

[14] [14]

Stein.Maximal functions

Elias M. Stein.Maximal functions. I. Spherical means, Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. MR 420116 Jaehyeon Ryu: Department of Mathematics, Ewha Womans University, Seoul 03760, Korea Email address:jhryu67@ewha.ac.kr Andreas Seeger: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI, 53706, USA. Email ...

work page 1976