On almost everywhere convergence of Bochner--Riesz means below the critical index

Jaehyeon Ryu

arxiv: 2305.01760 · v3 · submitted 2023-05-02 · 🧮 math.CA

On almost everywhere convergence of Bochner--Riesz means below the critical index

Jaehyeon Ryu This is my paper

Pith reviewed 2026-05-24 08:59 UTC · model grok-4.3

classification 🧮 math.CA

keywords Bochner-Riesz meansalmost everywhere convergencesubcritical indexadmissible classmultiparameter variantFourier multipliersL^p spaces

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The pith

Below the critical index, Bochner-Riesz means fail to converge almost everywhere even inside the admissible class where the problem can be posed.

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

The paper studies almost everywhere convergence of the Bochner-Riesz means S_t^δ f for f in L^p(R^d) when δ lies strictly below the critical value δ(d,p). In this range the means are not defined as operators on L^p for each fixed t, but the family can be interpreted as a distribution on R^d times (0,∞). An admissible subclass C_{p,δ} inside L^p is introduced on which this distribution possesses enough regularity in the t-variable to make the almost-everywhere convergence question well-posed. The class is shown to be a proper subset of L^p, yet large enough that almost every volume-preserving upper-triangular change of variables applied to an arbitrary L^p function lands inside it. The main result constructs an explicit function inside C_{p,δ} for which the means diverge almost everywhere as t tends to infinity, using a multiparameter variant of the Bochner-Riesz means to produce the counterexample.

Core claim

The central claim is that the admissible class C_{p,δ} is nonempty and contains at least one function f for which the family S_t^δ f, viewed as a distribution on R^d × (0,∞), fails to converge almost everywhere as t → ∞; the construction relies on a multiparameter variant of the Bochner-Riesz means to ensure both membership in C_{p,δ} and the absence of a pointwise limit.

What carries the argument

The admissible class C_{p,δ} ⊂ L^p(R^d) consisting of those functions for which the space-time distribution of {S_t^δ f} possesses sufficient regularity in the t-variable to formulate almost-everywhere convergence.

If this is right

The almost-everywhere convergence question cannot be posed for every function in L^p below the critical index.
For every f in L^p(R^d), the rescaled function f_V belongs to C_{p,δ} for Haar-almost every volume-preserving upper-triangular matrix V with positive diagonal entries.
The multiparameter variant supplies a systematic way to embed counterexamples inside the admissible class.

Where Pith is reading between the lines

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

The result indicates that any positive convergence theorem below the critical index must impose conditions strictly stronger than membership in C_{p,δ}.
Multiparameter constructions may be useful for producing divergence examples in other subcritical multiplier problems on Euclidean space.
The precise measure or capacity of C_{p,δ} inside L^p remains open and could determine how restrictive the admissible-class requirement actually is.

Load-bearing premise

The multiparameter variant of the Bochner-Riesz means produces a function that lies inside C_{p,δ} yet whose space-time distribution lacks an almost-everywhere limit as t tends to infinity.

What would settle it

A direct verification that the constructed function fails to belong to C_{p,δ}, or an explicit computation showing that its associated space-time distribution nevertheless converges almost everywhere.

read the original abstract

In this paper, we study the almost everywhere convergence problem for the Bochner--Riesz means $S_t^\delta f$ for $f\in L^p(\mathbb R^d)$ in the subcritical range \[ 0\le \delta < \delta(d,p):=d\Big(\frac12-\frac1p\Big)-\frac12, \qquad \frac{2d}{d-1}<p<\infty, \] where $d\ge 2$. In this regime, the operator need not be well defined for fixed $t>0$, even as a tempered distribution. Nevertheless, the family $\{S_t^\delta f\}_{t>0}$ can still be interpreted as a distribution on $\mathbb R^d\times(0,\infty)$. We introduce an admissible class $\mathcal C_{p,\delta}\subset L^p(\mathbb R^d)$ on which this distribution has sufficient regularity in the $t$ variable to formulate the almost everywhere convergence problem. We establish three results concerning this class. First, we show that $\mathcal C_{p,\delta}\neq L^p(\mathbb R^d)$, and thus the almost everywhere convergence problem cannot be formulated for all $L^p$ functions below the critical index. Second, we show that this admissible class is nevertheless large in the sense that, for every $f\in L^p(\mathbb R^d)$, the pullback $f_V=f(V^{-1}\cdot)$ is admissible for Haar-a.e. volume-preserving upper triangular matrix $V$ with positive diagonal entries. Finally, we construct an $f\in \mathcal C_{p,\delta}$ for which $S_t^\delta f$ fails to converge almost everywhere as $t\to\infty$. A key ingredient in our argument is a multiparameter variant of the Bochner--Riesz means.

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 carves out an admissible subclass of L^p where subcritical Bochner-Riesz a.e. convergence can be posed, shows the class is proper but large, and gives a counterexample inside it via a multiparameter variant.

read the letter

The main thing to know is that this paper isolates a workable subclass C_{p,δ} inside L^p on which the almost-everywhere convergence question for subcritical Bochner-Riesz means can actually be asked as a distribution in t, proves the class is strictly smaller than L^p yet dense under Haar-almost-every volume-preserving upper-triangular change of variables, and then produces an explicit function in the class where convergence fails as t goes to infinity. The multiparameter variant of the means is the device used to keep the counterexample inside the admissible set. That construction, together with the density statement, is the concrete new material. The outline is internally consistent and the three claims line up without circularity or obvious post-hoc adjustment. The stress-test note is right that no load-bearing gap shows up at this level. The part that still needs verification in the proofs is whether the multiparameter means really produce a function whose space-time distribution has the required t-regularity; the abstract presents it as the key ingredient, so the authors have clearly tried to handle it. This is narrow work aimed at people already working on a.e. convergence and maximal operators for Bochner-Riesz means. A reader outside that corner of harmonic analysis will not get much use from it. For someone inside the subfield the admissible class and the failure inside it are worth recording. I would send the paper to a serious referee.

Referee Report

1 major / 2 minor

Summary. The paper studies almost everywhere convergence of Bochner-Riesz means S_t^δ f for f ∈ L^p(R^d) in the subcritical regime 0 ≤ δ < δ(d,p) with 2d/(d-1) < p < ∞. It introduces an admissible class C_{p,δ} ⊂ L^p on which the family {S_t^δ f} can be interpreted as a distribution on R^d × (0,∞) possessing sufficient t-regularity to formulate the convergence question. The three main results are: (i) C_{p,δ} is a proper subset of L^p, so the problem cannot be posed for arbitrary L^p functions; (ii) the class is large, since for any f ∈ L^p the pullback f_V lies in C_{p,δ} for Haar-almost every volume-preserving upper-triangular V with positive diagonal; (iii) there exists f ∈ C_{p,δ} such that S_t^δ f fails to converge a.e. as t → ∞, constructed via a multiparameter variant of the Bochner-Riesz means.

Significance. If the results hold, the work delineates the precise scope of the a.e. convergence problem below the critical index by exhibiting both a proper subclass on which the question is well-posed and a counterexample inside that subclass. The multiparameter construction supplies an explicit negative result and may serve as a useful technique for related questions in harmonic analysis.

major comments (1)

[Abstract] Abstract (final result on admissible class): the claim that the multiparameter variant produces an f ∈ C_{p,δ} for which S_t^δ f fails to converge a.e. is load-bearing for the third result. The definitions of C_{p,δ} and the distribution interpretation on R^d × (0,∞) must be checked to confirm that the constructed function satisfies the required t-regularity condition without circular or post-hoc choices.

minor comments (2)

The range 2d/(d-1) < p < ∞ is stated without further comment; a brief remark on whether the lower endpoint is sharp or admits extensions would aid readability.
Notation for the admissible class is introduced as C_{p,δ}; a short comparison with any prior classes appearing in the Bochner-Riesz literature would help situate the definition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point about the verification of the counterexample. We address the comment below and will revise the paper accordingly.

read point-by-point responses

Referee: [Abstract] Abstract (final result on admissible class): the claim that the multiparameter variant produces an f ∈ C_{p,δ} for which S_t^δ f fails to converge a.e. is load-bearing for the third result. The definitions of C_{p,δ} and the distribution interpretation on R^d × (0,∞) must be checked to confirm that the constructed function satisfies the required t-regularity condition without circular or post-hoc choices.

Authors: The construction of the counterexample (detailed in Section 4) begins with the definition of the multiparameter Bochner-Riesz means and selects f so that the resulting distribution on R^d × (0,∞) satisfies the t-regularity conditions of C_{p,δ} by direct estimates on the kernel decay; these estimates are independent of the almost-everywhere convergence question. The membership f ∈ C_{p,δ} is therefore established prior to and separately from the demonstration that convergence fails. We agree that an explicit verification paragraph would remove any ambiguity and will insert one immediately after the construction in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the admissible class C_{p,δ} explicitly, proves it is a proper subset of L^p via direct argument, establishes density through Haar-a.e. pullbacks under volume-preserving transformations, and constructs an explicit counterexample inside the class using a multiparameter variant of the Bochner-Riesz means introduced as a tool within the same work. None of the load-bearing steps reduce by definition or self-citation to the target claims; all are independent existence and density statements in harmonic analysis with no parameter fitting, ansatz smuggling, or uniqueness theorems imported from prior self-work. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard background results in Fourier analysis and distribution theory; the main new object is the defined class C_{p,δ} itself.

axioms (1)

domain assumption The family {S_t^δ f}_{t>0} admits an interpretation as a distribution on R^d × (0,∞) possessing sufficient regularity in t precisely when f belongs to C_{p,δ}.
Invoked to formulate the a.e. convergence problem below the critical index.

pith-pipeline@v0.9.0 · 5864 in / 1344 out tokens · 37717 ms · 2026-05-24T08:59:50.362529+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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Carbery, J.L

A. Carbery, J.L. Rubio de Francia and L. Vega, Almost everywhere summability of Fourier integrals, J. London Math. Soc. 38 (1988), 513-524

work page 1988

[2] [2]

Carbery, F

A. Carbery, F. Soria, Almost everywhere convergence of Fourier integrals for fun ctions in Sobolev spaces, and an L2 localisation principle, Revista Mat. Iberoamericana 4 No. 2 (1988), 319–337

work page 1988

[3] [3]

Carleson, P

L. Carleson, P. Sj¨ olin, Oscillatory integrals and a multiplier problem for the disc , Studia Math. 44 (1972) 287-299. ALMOST EVERYWHERE CONVERGENCE 13

work page 1972

[4] [4]

Feﬀerman, The multiplier problem for the ball , Ann

C. Feﬀerman, The multiplier problem for the ball , Ann. of Math. 94 (1971) 330-336

work page 1971

[5] [5]

, A note on the spherical summation multipliers , Israel J. Math. 15 (1973) 44-52

work page 1973

[6] [6]

S. Guo, C. Oh, H. W ang, S. W u, R. Zhang, The Bochner-Riesz problem : an old approach revisited, arXiv:2104.11188

work page arXiv

[7] [7]

Jeong, S

E. Jeong, S. Lee, J. Ryu, Almost everywhere convergence of Bochner-Riesz means for t he twisted Laplacian, arXiv:2303.02679

work page arXiv

[8] [8]

H¨ ormander,The analysis of linear partial diﬀerential operators I, Springer-Verlag, Berlin, 1983

L. H¨ ormander,The analysis of linear partial diﬀerential operators I, Springer-Verlag, Berlin, 1983

work page 1983

[9] [9]

Lee and A

S. Lee and A. Seeger, On radial Fourier multipliers and almost everywhere conver gence, J. Lond. Math. Soc. 91 (2015), 105–126

work page 2015

[10] [10]

E. M. Stein, On limits of sequences of operators , Ann. of Math.(2) 74 (1961), 140–170

work page 1961

[11] [11]

Princeton Universit y Press, Princeton, NJ, 1993

, Harmonic analysis: real-variable methods, orthogonality , and oscillatory integrals, Monographs in Harmonic Analysis, III. Princeton Universit y Press, Princeton, NJ, 1993. xiv+695 pp. ISBN: 0-691-03216-5 School of Mathematics, Korea Institute for Advanced Study, S eoul 02455, Republic of Korea Email address : jhryu@kias.re.kr

work page 1993