Stein's square function associated with the Bochner-Riesz means on M\'etivier groups and its applications

Joydwip Singh

arxiv: 2604.27931 · v1 · submitted 2026-04-30 · 🧮 math.AP

Stein's square function associated with the Bochner-Riesz means on M\'etivier groups and its applications

Joydwip Singh This is my paper

Pith reviewed 2026-05-07 06:57 UTC · model grok-4.3

classification 🧮 math.AP

keywords Stein's square functionBochner-Riesz meansMétivier groupssub-Laplacianspectral multipliersmaximal operatorsfractional Schrödinger equationbilinear operators

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The pith

Stein's square function for the sub-Laplacian on Métivier groups is L^p-bounded once smoothness exceeds a threshold set by the group's topological dimension d.

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

The paper proves L^p boundedness of Stein's square function S^α(L) associated to the sub-Laplacian on any Métivier group G whenever the smoothness parameter α exceeds an explicit lower bound written in terms of the topological dimension d of G. This single square-function estimate is then applied to recover sharp multiplier bounds, to control maximal Bochner-Riesz operators, to obtain almost-everywhere convergence of the means, and to derive mixed-norm regularity for solutions of the fractional Schrödinger equation. The same circle of ideas also yields improved L^{p1}×L^{p2}→L^p bounds for bilinear Bochner-Riesz means in the range 2≤p1,p2<∞. A sympathetic reader cares because the threshold depends only on the ordinary topological dimension rather than the homogeneous dimension, giving concrete, checkable ranges for convergence and regularity questions on these non-Euclidean groups.

Core claim

On a Métivier group G of topological dimension d the operator S^α(L) satisfies the L^p bound ||S^α(L)f||_p ≲ ||f||_p for all 1<p<∞ once α is larger than a fixed function of d. The proof uses the special heat-kernel and spectral estimates available on Métivier groups. With this bound in hand the authors obtain an alternate proof of the optimal L^p multiplier theorem, L^p boundedness of the maximal Bochner-Riesz means, pointwise convergence of Bochner-Riesz means at the same α(d), mixed-norm estimates for the fractional Schrödinger equation, and improved bilinear bounds for the corresponding bilinear operators.

What carries the argument

Stein's square function S^α(L), constructed from the time derivatives of the spectral multipliers (1-tL)_+^α for the sub-Laplacian L; it converts square-function control into pointwise and maximal bounds for the associated family of operators.

If this is right

Sharp L^p boundedness for spectral multipliers m(L) is recovered by an alternate argument.
The maximal Bochner-Riesz operator is bounded on L^p, which implies pointwise almost-everywhere convergence of the Bochner-Riesz means with smoothness given in terms of d.
Mixed-norm regularity estimates hold for solutions of the fractional Schrödinger equation i∂_t u + L^β u = 0, with the regularity index again expressed in terms of d.
Bilinear Bochner-Riesz means and their maximal versions map L^{p1}(G)×L^{p2}(G) to L^p(G) for 2≤p1,p2<∞, improving the earlier range.
All of the above statements improve on the earlier results of Mauceri-Meda and Horwich-Martini by replacing homogeneous-dimension dependence with topological-dimension dependence.

Where Pith is reading between the lines

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

Because the threshold depends on topological dimension d rather than homogeneous dimension, the result may be sharp in the Euclidean sense and could serve as a model for square-function estimates on other nilpotent groups that admit comparable kernel decay.
The same square-function technique might extend to variable-coefficient or perturbed sub-Laplacians on Métivier groups, yielding analogous convergence statements for more general dispersive equations.
The bilinear improvement suggests that multilinear versions could be treated by the same method, potentially giving new Strichartz-type estimates for nonlinear Schrödinger equations on these groups.

Load-bearing premise

The underlying group must be Métivier, so that the sub-Laplacian possesses the Gaussian heat-kernel bounds and functional-calculus estimates needed to close the square-function argument; if this structural assumption fails the stated range for α in terms of d no longer guarantees boundedness.

What would settle it

An explicit counter-example, on a concrete Métivier group of known topological dimension d, of a function f in L^p such that the ratio ||S^α(L)f||_p / ||f||_p becomes arbitrarily large for any α strictly below the paper's d-dependent threshold.

Figures

Figures reproduced from arXiv: 2604.27931 by Joydwip Singh.

**Figure 1.** Figure 1: S α ∗ (L) boundedness: Yellow region [MM90, Corollary 2.8] (any stratified Lie groups), Brown region (Theorem 1.10) ([HM21, Theorem 1.2] + [Mar12]) (M´etivier groups), Red shaded region Corollary 1.11 (Our result on M´etivier groups); Pointwise convergence: Yellow region [MM90, Corollary 2.8] (any stratified Lie groups), Brown + Green region [HM21, Theorem 1.1] (Heisenberg-type groups), Red shaded region … view at source ↗

**Figure 2.** Figure 2: Here α > α∗(p1, p2, n, ps) represents the range of the parameter such that Bα R and Bα ∗ are bounded from L p1 (R n )×L p2 (R n ) to L p (R n ) (see Theorem 1.17 and Theorem 1.18). Now we turn our discussion towards the bilinear Bochner-Riesz means on M´etivier groups. For f, g ∈ S(G) and α ≥ 0, R > 0, the bilinear Bochner-Riesz means of order α associated with the sub-Laplacian L on M´etivier group, denot… view at source ↗

**Figure 3.** Figure 3: Region I = Upper half triangle and Region II = Lower half triangle. Here α > αd(p1, p2) represents that B α R is bounded on L p1 (G) × L p2 (G) → L p (G) for α > αd(p1, p2) (see Theorem 1.19). In a similar vain as in the Euclidean settings, here we consider the maximal bilinear BochnerRiesz means associated with the sub-Laplacian L on M´etivier group, defined as B α ∗ (f, g)(x, u) = sup R>0 |Bα R(f, g)(x,… view at source ↗

**Figure 4.** Figure 4: Left hand side picture represents that B α ∗ and B α R are bounded on L p1 (G)×L p2 (G) → L p (G) for α > α∗(p1, p2, d, pG) (see Theorem 1.20 and Corollary 1.21). Compare this picture with view at source ↗

read the original abstract

In this paper, we study the $L^p$-boundedness of Stein's square function $\mathfrak{S}^{\alpha}(\mathcal{L})$ associated with the sub-Laplacian $\mathcal{L}$ on M\'etivier group $G$. A key aspect of our result is that the smoothness condition is expressed in terms of the topological dimension $d$ of the underlying M\'etivier group $G$. Consequently, we also present several applications of the $L^p$-boundedness of $\mathfrak{S}^{\alpha}(\mathcal{L})$. First, we provide an alternate proof of the sharp $L^p$-boundedness result for spectral multipliers on M\'etivier groups, recently obtained by Niedorf [Niedorf, Studia Math., 2025]. Next we prove $L^p$-boundedness of maximal spectral multipliers and consequently establish sharp $L^p$-boundedness result for the maximal Bochner-Riesz operator on M\'etivier groups, which also yields pointwise almost everywhere convergence of Bochner-Riesz means with smoothness parameter given in terms of the topological dimension of $G$. In case of M\'etivier groups our result improves upon the existing works of Mauceri-Meda [Mauceri, Meda, Rev. Mat. Iberoam., 1990] and Horwich-Martini [Horwich, Martini, J. Lond. Math. Soc., 2021]. Our result further imply the mixed norm regularity estimates for the solution of fractional Schr\"odinger equation on M\'etivier groups, where the regularity index is again expressed in terms of the topological dimension of $G$. Finally, we study the $L^{p_1}(G) \times L^{p_2}(G)$ to $L^p(G)$ boundedness of the bilinear Bochner-Riesz means and its maximal version, associated with the sub-Laplacian on M\'etivier group $G$. Our result improves upon the recent work of the author with Bagchi and Molla [Bagchi, Molla, Singh, J. Funct. Anal., 2026] in the range $2\leq p_1, p_2 <\infty$. In the same range, ......

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

Singh gives an alternate proof of the sharp spectral multiplier theorem on Métivier groups and uses the Stein square function to get maximal Bochner-Riesz bounds, bilinear improvements, and mixed-norm Schrödinger estimates all indexed by topological dimension d.

read the letter

The paper's main contribution is an alternate proof of the sharp spectral multiplier theorem on Métivier groups, plus new bounds for the maximal Bochner-Riesz operator and some mixed-norm estimates for the fractional Schrödinger equation, all with smoothness thresholds depending on the topological dimension d of the group. It also improves the bilinear Bochner-Riesz bounds in the range where p1 and p2 are at least 2. These results build on the L^p boundedness of Stein's square function for the sub-Laplacian, with the key feature that the required smoothness α is expressed in terms of d instead of the homogeneous dimension. What stands out is the consistent use of d across the applications, which makes the pointwise convergence and PDE regularity statements more directly applicable to manifold-like settings. The alternate proof for the multiplier result is presented as simpler or more direct than Niedorf's, and the bilinear improvement over the author's prior joint work is explicit. On the downside, everything hinges on the Métivier assumption, which gives the necessary kernel decay and spectral multiplier properties. If the proofs lean heavily on that structure without new ideas for general nilpotent groups, the scope stays narrow. The paper does not appear to contain machine-checked parts or fully explicit constants, so the sharpness claims rest on the standard transference arguments. This work is for specialists in harmonic analysis on stratified groups or sub-Riemannian manifolds. Someone already familiar with Bochner-Riesz means on Lie groups will get the most out of the dimension-dependent thresholds and the listed applications. It is coherent enough on its own terms to merit a full referee report rather than a desk rejection. I would recommend sending it for peer review. The concrete improvements in the bilinear range and the Schrödinger estimates are worth verifying, even if the core square-function bound follows a familiar path.

Referee Report

0 major / 2 minor

Summary. This paper establishes the L^p-boundedness of Stein's square function 𝔖^α(ℒ) associated with the Bochner-Riesz means for the sub-Laplacian ℒ on Métivier groups G, where the required smoothness α is expressed explicitly in terms of the topological dimension d of G. Using this square-function bound, the manuscript derives an alternate proof of sharp L^p spectral multiplier bounds (recovering Niedorf's recent result), L^p bounds for maximal spectral multipliers and maximal Bochner-Riesz operators (yielding pointwise a.e. convergence), mixed-norm regularity estimates for the fractional Schrödinger equation, and L^{p1}×L^{p2}→L^p bounds for bilinear Bochner-Riesz means and their maximal versions, with improvements claimed over Mauceri-Meda, Horwich-Martini, and Bagchi-Molla-Singh in the range 2≤p1,p2<∞.

Significance. If the central boundedness statements hold, the work supplies a unified square-function framework for harmonic analysis on Métivier groups that ties key indices to the topological dimension d rather than the homogeneous dimension Q. This is advantageous for manifold-level conclusions such as pointwise convergence and PDE regularity. The alternate proof of the spectral-multiplier theorem and the stated improvements for maximal and bilinear operators add concrete value; the extension to mixed-norm Schrödinger estimates broadens applicability. The argument is internally consistent once the Métivier assumption is granted, with no evident circularity or parameter-fitting in the high-level program.

minor comments (2)

[Abstract] The abstract is truncated at the end ('In the same range, ......'). The final sentence should be completed so that the precise improvement for the bilinear case is stated.
[Introduction] In the introduction and main theorem statements, the precise lower bound on α (in terms of d) should be displayed explicitly rather than only described qualitatively, to allow immediate comparison with prior thresholds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and careful assessment of our manuscript, including the recommendation for minor revision. No specific major comments were raised in the report. We will address any minor editorial or presentational suggestions in the revised version while preserving the core arguments.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives the L^p boundedness of Stein's square function S^α(L) on Métivier groups from the group's structural properties (spectral and kernel estimates for the sub-Laplacian) and standard transference techniques, with the smoothness threshold stated explicitly in terms of the topological dimension d. This central result is not obtained by fitting parameters to the target applications or by re-deriving prior outputs. Applications (alternate proof of Niedorf's spectral multiplier bounds, maximal Bochner-Riesz, pointwise convergence, mixed-norm Schrödinger estimates, and bilinear bounds) are presented as consequences or improvements, citing independent prior works (Niedorf, Mauceri-Meda, Horwich-Martini) whose results are not presupposed in the square-function argument. The self-citation to the author's earlier joint work appears only in the bilinear section as a range improvement and does not load-bear the main theorem or reduce any claimed prediction to a fitted input by construction. The chain relies on externally verifiable group estimates and does not collapse to self-definition or tautological renaming.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard domain assumptions of harmonic analysis on stratified Lie groups (sub-Laplacian spectral theory, kernel estimates, Fourier analysis on groups). No free parameters or invented entities are mentioned; the smoothness threshold is derived rather than fitted.

axioms (1)

domain assumption Sub-Laplacian on a Métivier group admits the usual spectral and heat-kernel estimates used in harmonic analysis on nilpotent groups
Invoked implicitly when defining the Bochner-Riesz means and square function; standard in the field but not proved inside the paper.

pith-pipeline@v0.9.0 · 5717 in / 1498 out tokens · 61965 ms · 2026-05-07T06:57:56.108861+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

On Spectral multiplier theorem for sub-Laplacians with drift on M\'etivier groups
math.AP 2026-05 unverdicted novelty 6.0

Spectral multiplier theorem for sub-Laplacians with drift on Métivier groups, with multiplier smoothness reduced from homogeneous dimension to topological dimension.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 1 Pith paper

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[BMS26] ,Bilinear Bochner-Riesz means on M´ etivier groups, J

MR 4439910 [BMS25] Sayan Bagchi, Md Nurul Molla, and Joydwip Singh,Bilinear bochner-riesz means for grushin operators, arXiv preprint arXiv:2505.18509 (2025). [BMS26] ,Bilinear Bochner-Riesz means on M´ etivier groups, J. Funct. Anal.290(2026), no. 10, Paper No. 111397. MR 5032907 [Bou91] J. Bourgain,Besicovitch type maximal operators and applications to ...

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MR 4905420 [GOW+25b] Shaoming Guo, Changkeun Oh, Hong Wang, Shukun Wu, and Ruixiang Zhang,The Bochner-Riesz problem: an old approach revisited, Peking Math. J.8(2025), no. 2, 201–270 (English). [GRY20] Shaoming Guo, Joris Roos, and Po-Lam Yung,Sharp variation-norm estimates for oscillatory integrals related to Carleson’s theorem, Anal. PDE13(2020), no. 5,...

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MR 4881571 [Heb93] Waldemar Hebisch,Multiplier theorem on generalized Heisenberg groups, Colloq. Math.65(1993), no. 2, 231–239. MR 1240169 [HM21] Adam D. Horwich and Alessio Martini,Almost everywhere convergence of Bochner-Riesz means on Heisenberg-type groups, J. Lond. Math. Soc. (2)103(2021), no. 3, 1066–1119. MR 4245831 [H¨ or60] Lars H¨ ormander,Estim...

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SINGH [JLV18] Eunhee Jeong, Sanghyuk Lee, and Ana Vargas,Improved bound for the bilinear Bochner-Riesz operator, Math

MR 4103874 68 J. SINGH [JLV18] Eunhee Jeong, Sanghyuk Lee, and Ana Vargas,Improved bound for the bilinear Bochner-Riesz operator, Math. Ann.372(2018), no. 1-2, 581–609. MR 3856823 [JS22] K. Jotsaroop and Saurabh Shrivastava,Maximal estimates for bilinear Bochner-Riesz means, Adv. Math.395(2022), Paper No. 108100,

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J.122(2004), no

MR 4363590 [Lee04] Sanghyuk Lee,Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators, Duke Math. J.122(2004), no. 1, 205–232. MR 2046812 [Lee18] ,Square function estimates for the Bochner-Riesz means, Anal. PDE11(2018), no. 6, 1535–1586. MR 3803718 [LRS12] Sanghyuk Lee, Keith M. Rogers, and Andreas Seeger,Improved bounds for Stein’s squar...

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[MMNG23] Alessio Martini, Detlef M¨ uller, and Sebastiano Nicolussi Golo,Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type, J. Eur. Math. Soc. (JEMS)25(2023), no. 3, 785–843. MR 4577953 [MS94] D. M¨ uller and E. M. Stein,On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. (9)73...

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Fourier Anal

[Nie24] Lars Niedorf,AnL p-spectral multiplier theorem with sharpp-specific regularity bound on Heisenberg type groups, J. Fourier Anal. Appl.30(2024), no. 2, Paper No. 22,

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[Nie25b] Lars Niedorf,Spectral multipliers on M´ etivier groups, Studia Math.282(2025), no

MR 4728249 [Nie25a] Lars Niedorf,Restriction type estimates on general two-step stratified lie groups, https://arxiv.org/abs/2304.12960 (2025). [Nie25b] Lars Niedorf,Spectral multipliers on M´ etivier groups, Studia Math.282(2025), no. 2, 149–197. MR 4908550 [Ouh05] El Maati Ouhabaz,Analysis of heat equations on domains, London Mathematical Society Monogr...

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