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arxiv: 2607.02189 · v1 · pith:TG5DPMLUnew · submitted 2026-07-02 · 🧮 math.CA

Bochner-Riesz means on the Heisenberg group

Detlef M\"uller , Lars Niedorf , Andreas Seeger , Betsy Stovall This is my paper

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classification 🧮 math.CA
keywords Bochner-Riesz meansHeisenberg groupsub-Laplacianspectral multiplierssquare functionswave operatorL^p boundednessspectral decomposition
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The pith

Bochner-Riesz means on the Heisenberg group satisfy L^p bounds for p up to a threshold p_n that approaches 2 with rising dimension.

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

The paper establishes L^p boundedness for Bochner-Riesz means tied to the spectral decomposition of the sub-Laplacian on the Heisenberg group H_n, holding for all p in the interval from 1 to a value p_n that tends to 2 as the dimension n increases. Earlier approaches based on Euclidean-style restriction theorems were blocked because no Stein-Tomas type result holds on H_n, leaving only the endpoint cases p=1 and p=∞. The new range follows from a general p-sensitive spectral multiplier theorem, which is the paper's main result and rests on L^p estimates for square functions built from the Heisenberg wave operator.

Core claim

We prove a p-sensitive spectral multiplier theorem for the sub-Laplacian on H_n that yields L^p boundedness of the associated Bochner-Riesz means for 1 ≤ p ≤ p_n with p_n → 2 as n → ∞. These multiplier bounds are derived from L^p estimates on square functions associated with the Heisenberg wave operator.

What carries the argument

The p-sensitive spectral multiplier theorem, obtained directly from L^p estimates for square functions of the Heisenberg wave operator.

If this is right

  • Bochner-Riesz means of the sub-Laplacian are bounded on L^p(H_n) for every p in [1, p_n].
  • The allowable range of p for spectral multipliers expands beyond the endpoints previously known.
  • The same square-function estimates control a wider class of spectral multipliers than those treated by earlier restriction-based methods.
  • The size of the p-interval shrinks toward the single point p=2 as the dimension n grows.

Where Pith is reading between the lines

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

  • If future work improves the range of the square-function estimates, the multiplier theorem would automatically extend the Bochner-Riesz interval as well.
  • The same square-function technique may be adaptable to other step-two nilpotent groups where Euclidean restriction fails.
  • The results highlight that wave-operator square functions can serve as a substitute for missing restriction theorems when studying spectral multipliers on stratified groups.

Load-bearing premise

The L^p estimates for square functions associated with the Heisenberg wave operator hold throughout the stated range of p.

What would settle it

An explicit counterexample showing that the square-function estimates fail for some p strictly between 1 and p_n would disprove both the multiplier theorem and the claimed Bochner-Riesz bounds.

Figures

Figures reproduced from arXiv: 2607.02189 by Andreas Seeger, Betsy Stovall, Detlef M\"uller, Lars Niedorf.

Figure 1
Figure 1. Figure 1: A plot of the singular support Σ1 of the convolution kernel of e i √ L. The curve admits infinitely many zigzags near the origin. Our analysis uses a more precise parametrix derived by the first and third authors in [41]; this involves a subordination argument using the Schr¨odinger type operators e itL, and a decomposition in the joint spectrum of L and i∂u. To begin with, we note that the proof of Theore… view at source ↗
Figure 2
Figure 2. Figure 2: The localization regions Dk,ℓ associated with the kth zigzag of the (|x|, |u|)-profile of the singular support Σ1. Case I corresponds to ℓ ≥ 2, and Case II to ℓ = 1. The choice and the further analysis of such frozen operators will be different in the cases ℓ ≥ 2 and ℓ = 1 (corresponding to Case I and Case II in [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A plot indicating horizontal points of the singular sup￾port Σ1 of the convolution kernel of e i √ L. Note that these hori￾zontal points are distinct from the cusp points visible in the plot. possible by means of error estimates for the T k,ℓ λ,t for k ̸= 0. The error estimates can be found in Appendix A (see also [41] for a different treatment). For k ̸= 0 these localizations force us to decompose the tim… view at source ↗
read the original abstract

We prove new $L^p$ boundedness results for Bochner-Riesz means associated with the spectral decomposition of the sub-Laplacian on the Heisenberg group $\mathbb H_n$. Our results hold for a range $1\le p\le p_n$ where $p_n\to 2$ as $n\to\infty$. As shown by the first named author in 1990 a Stein-Tomas type Fourier restriction theorem fails to hold on $\mathbb H_n$ and thus previous results based on the approach by Fefferman and Stein from the Euclidean setting only allowed to cover the cases $p=1$ and $p=\infty$. Our results on Bochner-Riesz means follow from a more general $p$-sensitive spectral multiplier theorem which is the main result of this article. This is obtained as a consequence of $L^p$ estimates for square functions associated with the Heisenberg wave operator.

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.

Referee Report

2 major / 0 minor

Summary. The manuscript establishes new L^p boundedness results for Bochner-Riesz means associated with the spectral decomposition of the sub-Laplacian on the Heisenberg group H_n. The results hold for 1 ≤ p ≤ p_n where p_n → 2 as n → ∞. These bounds are obtained from a p-sensitive spectral multiplier theorem, which is derived as a consequence of L^p estimates for square functions associated with the Heisenberg wave operator. This approach is motivated by the failure of Stein-Tomas type restriction theorems on H_n, which previously limited results to the endpoints p=1 and p=∞.

Significance. If the square-function estimates for the wave operator hold in the stated range and transfer to the multiplier theorem without introducing further losses, the result would constitute a meaningful advance in harmonic analysis on stratified Lie groups by furnishing the first non-trivial interval of p for which Bochner-Riesz means are bounded, where the admissible range necessarily shrinks with dimension.

major comments (2)
  1. [Abstract] Abstract: the central claim that the p-sensitive spectral multiplier theorem follows from the L^p square-function estimates for the Heisenberg wave operator is asserted without any derivation steps, error estimates, or range verification; the abstract supplies no information on the proof of those square functions or the precise transfer argument, leaving the load-bearing implication uncheckable.
  2. [Abstract] Abstract, final sentence: the range 1 ≤ p ≤ p_n is stated to be controlled by the square-function bounds, yet no explicit dependence of p_n on n or on the constants appearing in the square-function estimates is indicated; without this, it cannot be verified whether the transfer preserves the claimed interval or introduces shrinkage.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and comments regarding the abstract. We address each major comment below, directing to the relevant sections of the full manuscript for the detailed arguments.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the p-sensitive spectral multiplier theorem follows from the L^p square-function estimates for the Heisenberg wave operator is asserted without any derivation steps, error estimates, or range verification; the abstract supplies no information on the proof of those square functions or the precise transfer argument, leaving the load-bearing implication uncheckable.

    Authors: The abstract is a concise high-level summary of the main results, their motivation, and the overall strategy. The full derivation of the p-sensitive spectral multiplier theorem from the square-function estimates—including all steps, error estimates, range verification, and the precise transfer argument—is contained in Sections 3 and 4, with the square-function estimates for the Heisenberg wave operator proved in Section 2 (Theorem 2.1) and transferred in the proof of Theorem 1.3. revision: no

  2. Referee: [Abstract] Abstract, final sentence: the range 1 ≤ p ≤ p_n is stated to be controlled by the square-function bounds, yet no explicit dependence of p_n on n or on the constants appearing in the square-function estimates is indicated; without this, it cannot be verified whether the transfer preserves the claimed interval or introduces shrinkage.

    Authors: The explicit dependence of p_n on n and on the constants appearing in the square-function estimates is stated in Theorem 1.1 together with the remarks immediately following it in the introduction; this dependence is chosen precisely so that the transfer from the square-function bounds preserves the interval 1 ≤ p ≤ p_n without additional shrinkage. The abstract only records the asymptotic p_n → 2 as n → ∞. revision: no

Circularity Check

1 steps flagged

Minor self-citation to 1990 restriction failure; derivation otherwise independent

specific steps
  1. self citation load bearing [abstract]
    "As shown by the first named author in 1990 a Stein-Tomas type Fourier restriction theorem fails to hold on H_n and thus previous results based on the approach by Fefferman and Stein from the Euclidean setting only allowed to cover the cases p=1 and p=∞."

    This is a self-citation to prior work by one of the authors, used to motivate the new approach. However, it only documents a negative result and does not support or derive the new Bochner-Riesz or multiplier bounds, which are instead claimed to follow from square-function estimates. The citation is minor and not load-bearing for the central positive claims.

full rationale

The abstract presents the main results as following from a p-sensitive spectral multiplier theorem obtained from L^p square function estimates for the Heisenberg wave operator. The sole self-citation notes the failure of a Stein-Tomas restriction theorem (by the first author in 1990) to explain why prior Euclidean methods only reach p=1,∞. This citation is not load-bearing for the positive claims. No self-definitional equations, fitted inputs renamed as predictions, or other reductions by construction appear. The derivation chain is presented as one-way and self-contained against the square-function estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated.

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discussion (0)

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