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arxiv: 2605.25919 · v3 · pith:JQ3I3FRJnew · submitted 2026-05-25 · 🧮 math.CA · math.FA

Sparse domination of Calder\'on-Zygmund operators by mean oscillations

Andrei K. Lerner This is my paper

Pith reviewed 2026-06-29 19:22 UTC · model grok-4.3

classification 🧮 math.CA math.FA
keywords Calderón-Zygmund operatorssparse dominationmean oscillationsDini continuityT(1) conditionSobolev inequalitiespointwise estimates
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The pith

Dini-continuous Calderón-Zygmund operators with T(1)=0 admit sparse domination by local mean oscillations instead of averages.

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

The paper establishes a sharpened form of sparse domination for Calderón-Zygmund operators. It proves that if the operator is Dini-continuous and annihilates constants, then the domination inequality can use mean oscillations of the function instead of its averages. This extends earlier work that required smoother kernels. The result yields a characterization of which such operators satisfy a pointwise Sobolev-type inequality, namely precisely when the image of the constant function one is bounded.

Core claim

If T is a Dini-continuous Calderón–Zygmund operator satisfying T(1)=0, then T admits sparse domination in which the controlling terms are local mean oscillations rather than local averages. This extends the Benea–Bernicot result from smoother kernels to the Dini-continuous setting. As a consequence, the Calderón–Zygmund operators for which a pointwise Sobolev-type inequality holds are exactly those with T(1) belonging to L^∞.

What carries the argument

Sparse domination inequality for T controlled by the local mean oscillation of the input function, under the Dini continuity assumption on the kernel.

If this is right

  • The pointwise Sobolev-type inequality holds for T if and only if T(1) belongs to L^∞.
  • The sharpened domination applies to the full class of Dini-continuous kernels rather than only smoother ones.
  • Any estimate previously derived from sparse bounds for such T can now be controlled by mean oscillations.
  • The result answers the question of Hoang, Moen and Pérez on the precise condition for the Sobolev inequality.

Where Pith is reading between the lines

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

  • The oscillation-based control may yield improved dependence on constants when deriving weighted inequalities from the sparse bound.
  • The same replacement of averages by oscillations could be tested on other classes of singular integrals that satisfy a T(1)=0 condition.
  • If the Dini condition can be weakened further, the characterization of Sobolev bounds would extend to an even larger family of operators.

Load-bearing premise

The kernel satisfies the Dini continuity condition, which is the regularity needed to carry the sparse domination argument through.

What would settle it

A concrete Dini-continuous Calderón–Zygmund operator T with T(1) not in L^∞ for which the pointwise Sobolev inequality still holds would falsify the claimed characterization.

read the original abstract

We show that if $T$ is a Dini-continuous Calder\'on--Zygmund operator satisfying $T(1)=0$, then the usual sparse domination for $T$ can be sharpened by replacing local averages with local mean oscillations. This extends a result of Benea and Bernicot for smoother kernels to the more general Dini-continuous setting. As an application, we characterize the Calder\'on--Zygmund operators for which a pointwise Sobolev-type inequality holds: this is the case if and only if $T(1)\in L^\infty$. This answers a recent question of Hoang, Moen and P\'erez.

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

0 major / 2 minor

Summary. The manuscript proves that if T is a Dini-continuous Calderón-Zygmund operator with T(1)=0, then the standard sparse domination for T can be sharpened by replacing local averages with local mean oscillations; this extends Benea-Bernicot from smoother kernels. As an application, the authors characterize the CZ operators admitting a pointwise Sobolev-type inequality as precisely those with T(1) ∈ L^∞, answering a question of Hoang-Moen-Pérez.

Significance. If the proofs are correct, the sharpened sparse bound under the minimal Dini hypothesis strengthens a key tool in harmonic analysis and yields a clean characterization of the Sobolev inequality. The result is parameter-free in its statement and directly resolves an open question; these features increase its potential impact on weighted estimates and related inequalities.

minor comments (2)
  1. The statement of the main sparse-domination theorem (presumably Theorem 1.1 or 2.1) should explicitly record the dependence of the implicit constant on the Dini modulus of continuity of the kernel.
  2. In the application section, the precise form of the pointwise Sobolev inequality (including the range of exponents) should be restated for the reader's convenience before the characterization is proved.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation extends external prior result

full rationale

The paper's central steps consist of extending the Benea-Bernicot sparse domination theorem from smoother kernels to the Dini-continuous case (a standard minimal regularity hypothesis in CZ theory) and then deriving the T(1) in L^infty characterization of the pointwise Sobolev inequality as a direct application. No quoted equation or step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the argument is positioned as building on independent external work and answers an open question posed by other authors. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of Calderón–Zygmund operators, the Dini continuity condition on the kernel, and the existing sparse-domination framework of Benea–Bernicot; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption T is a Calderón–Zygmund operator whose kernel satisfies the Dini continuity condition
    Invoked to extend the Benea–Bernicot result to the stated setting.
  • domain assumption Standard properties of sparse domination and mean oscillations hold in the Dini-continuous case
    Required for the sharpened inequality to be valid.

pith-pipeline@v0.9.1-grok · 5632 in / 1321 out tokens · 37437 ms · 2026-06-29T19:22:22.007533+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

10 extracted references · 1 canonical work pages · 1 internal anchor

  1. [1]

    Cancellative sparse domination

    J.M. Conde-Alonso, E. Lorist and G. Rey,Cancellative sparse domination, preprint. Available at https://arxiv.org/pdf/2603.08446

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Cruz-Uribe and K

    D. Cruz-Uribe and K. Moen,One and two weight norm inequalities for Riesz potentials, Illinois J. Math.57(2013), no. 1, 295-–323

    2013

  3. [3]

    David and J-L

    G. David and J-L. Journ´ e,A boundedness criterion for generalized Calder´ on–Zygmund oper- ators, Ann. of Math. (2)120(1984), no. 2, 371–397

    1984

  4. [4]

    Grafakos, Modern Fourier analysis

    L. Grafakos, Modern Fourier analysis. Third edition. Graduate Texts in Mathematics, 250. Springer, New York, 2014

    2014

  5. [5]

    Hoang, K

    C. Hoang, K. Moen and C. P´ erez,Pointwise estimates for rough operators with applications to Sobolev inequalities, J. Anal. Math.155(2025), no. 1, 43–74

    2025

  6. [6]

    Hoang, K

    C. Hoang, K. Moen and C. P´ erez,An homage to Guido Weiss and his leadership of the Saint Louis team: commutators of singular integrals and Sobolev inequalities, The mathematical heritage of Guido Weiss, 293–311, Appl. Numer. Harmon. Anal., Birkh¨ auser/Springer, Cham, 2025

    2025

  7. [7]

    Hoang, K

    C. Hoang, K. Moen and C. P´ erez,New pointwise bounds by Riesz potential type operators, J. Funct. Anal.289(2025), no. 9, Paper No. 111060, 35 pp

    2025

  8. [8]

    Lerner, E

    A.K. Lerner, E. Lorist and S. Ombrosi,Operator-free sparse domination, Forum Math. Sigma 10(2022), Paper No. e15, 28 pp

    2022

  9. [9]

    Lerner and F

    A.K. Lerner and F. Nazarov,Intuitive dyadic calculus: the basics, Expo. Math.37(2019), no. 3, 225–265

    2019

  10. [10]

    Lerner and S

    A.K. Lerner and S. Ombrosi,Some remarks on the pointwise sparse domination, J. Geom. Anal.30(2020), no. 1, 1011–1027. (A.K. Lerner)Department of Mathematics, Bar-Ilan University, 5290002 Ramat Gan, Israel Email address:lernera@math.biu.ac.il

    2020