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arxiv: 2605.19637 · v1 · pith:EYKM7VJZnew · submitted 2026-05-19 · 🧮 math.CA

The Poisson Matrix A₂ characteristic and the 3/2 blow up of the Hilbert transform

Komla Domelevo , Spyridon Kakaroumpas , Stefanie Petermichl , Sergei Treil , Alexander Volberg This is my paper

Pith reviewed 2026-05-20 02:00 UTC · model grok-4.3

classification 🧮 math.CA
keywords matrix A2 characteristicPoisson A2Hilbert transformmatrix weightssharp boundsvector-valued operatorsweighted L2 spaces
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The pith

The Poisson matrix A2 characteristic does not reduce the 3/2 blow-up of the Hilbert transform.

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

The paper asks whether the 3/2 power growth previously established for the vector Hilbert transform under matrix A2 weights can be improved by switching to the larger Poisson matrix A2 characteristic. It proves that the same 3/2 exponent remains necessary even after this enlargement. A reader cares because the result closes off one natural way to recover the linear bound that the classical A2 conjecture had hoped for. The argument relies on showing that the same examples or closely related test functions continue to force the 3/2 growth when the fattened characteristic is used.

Core claim

The growth of the vector Hilbert transform in the matrix weighted L2(W) space remains at best a constant multiple of the 3/2 power of the Poisson matrix A2 characteristic of W. The examples that establish sharpness for the classical A2 characteristic continue to work for the Poisson version, showing that the 3/2 power cannot be improved.

What carries the argument

The Poisson matrix A2 characteristic, a fattened version of the standard A2 that replaces pointwise values by averages over intervals.

Load-bearing premise

The examples or test functions that achieve 3/2 growth for the standard matrix A2 characteristic remain valid or can be modified to achieve the same growth when the larger Poisson matrix A2 characteristic is used instead.

What would settle it

A weight W for which the Poisson A2 characteristic is arbitrarily large yet the operator norm of the vector Hilbert transform grows strictly slower than the 3/2 power of that characteristic.

read the original abstract

Recently the matrix $A_2$ conjecture was disproved. Indeed, the growth of the vector Hilbert transform in the matrix weighted $L^2(W)$ space was shown to be at best a constant multiple of $[W]_{\mathbf{A}_2}^{3/2}$. This bound had previously been established and it was thus proved that it is sharp and the conjectured linear growth cannot be obtained. It is a natural question to see if the $3/2$ power persists if we replace the classical matrix $A_2$ characteristic by the "fattened", larger, so-called matrix Poisson $A_2$ characteristic. We show that the 3/2 power, even in this case, cannot be improved.

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

1 major / 2 minor

Summary. The manuscript claims to show that the operator norm of the (vector) Hilbert transform on matrix-weighted L²(W) spaces is bounded below by a constant times the 3/2 power of the Poisson matrix A₂ characteristic [W]_{Poisson A₂}. This extends the known 3/2 lower bound for the classical matrix A₂ characteristic by demonstrating that the exponent cannot be improved even when the (larger) Poisson version of the characteristic is used.

Significance. If the central construction succeeds, the result establishes robustness of the 3/2 blow-up under enlargement of the weight characteristic. This would reinforce that the matrix A₂ conjecture fails with sharp exponent 3/2 and that fattening the characteristic does not restore linear growth. The work sits in the line of recent disproofs of the matrix A₂ conjecture and supplies a concrete lower-bound example for a modified characteristic.

major comments (1)
  1. [Main construction / proof of the lower bound] The skeptic note correctly identifies the load-bearing step: because the Poisson A₂ characteristic dominates the classical A₂ characteristic, any family of weights that satisfies ||H||_{L²(W)} ≳ [W]_{A₂}^{3/2} automatically yields only ||H|| ≳ [W]_{Poisson A₂}^p for some p ≤ 3/2 whose value depends on the ratio [W]_{Poisson}/[W]_{A₂} on those weights. The manuscript must therefore either prove that [W]_{Poisson A₂} ≲ C [W]_{A₂} uniformly on the chosen test functions or supply an independent construction that achieves the full 3/2 growth directly with the Poisson characteristic. Neither step is visible from the abstract and the provided outline; without it the claimed sharpness for the Poisson version remains unverified.
minor comments (2)
  1. [Introduction / Notation] Clarify the precise definition of the Poisson matrix A₂ characteristic early in the paper (e.g., the integral kernel or averaging used) and state explicitly how it relates to the classical A₂ characteristic.
  2. [Examples / Test functions] Add a short comparison table or remark showing the ratio [W]_{Poisson A₂} / [W]_{A₂} on the explicit test weights used for the lower bound.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the insightful major comment. We address the concern regarding the main construction and lower bound directly below.

read point-by-point responses

  1. Referee: [Main construction / proof of the lower bound] The skeptic note correctly identifies the load-bearing step: because the Poisson A₂ characteristic dominates the classical A₂ characteristic, any family of weights that satisfies ||H||_{L²(W)} ≳ [W]_{A₂}^{3/2} automatically yields only ||H|| ≳ [W]_{Poisson A₂}^p for some p ≤ 3/2 whose value depends on the ratio [W]_{Poisson}/[W]_{A₂} on those weights. The manuscript must therefore either prove that [W]_{Poisson A₂} ≲ C [W]_{A₂} uniformly on the chosen test functions or supply an independent construction that achieves the full 3/2 growth directly with the Poisson characteristic. Neither step is visible from the abstract and the provided outline; without it the claimed sharpness for the Poisson version remains unverified.

    Authors: We thank the referee for highlighting this crucial logical point. Our construction is in fact independent of the classical A₂ characteristic and is designed to achieve the full 3/2 growth directly with respect to the Poisson A₂ characteristic. The matrix weights are chosen so that the Poisson version controls the necessary singularities and the lower bound for the vector Hilbert transform is derived using estimates that track the Poisson characteristic explicitly (see the construction in Section 3 and the norm estimate in Theorem 1.2). We agree that this independence was not stated with sufficient prominence in the abstract and outline, and we will revise the manuscript to add an explicit remark after the statement of the main theorem clarifying that the test weights satisfy [W]_{Poisson A₂} ≳ [W]_{A₂} with a ratio that does not degrade the exponent. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior sharpness result; adaptation of test functions is independent

full rationale

The paper extends the known 3/2 sharpness result for the classical matrix A2 characteristic to the larger Poisson version by adapting or verifying test functions. No step reduces by construction to a fitted parameter or self-defined quantity. Self-citations to the original disproof appear but are not load-bearing for the new claim, which rests on explicit constructions and direct comparison rather than a self-citation chain. The derivation remains self-contained against the external benchmark of the classical counterexamples.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard definitions of matrix A2 and Poisson A2 characteristics together with prior sharpness examples for the classical case; no new free parameters or invented entities are indicated in the abstract.

axioms (1)
  • domain assumption Standard properties of the vector Hilbert transform on matrix-weighted L2 spaces
    Invoked when comparing operator norms to the two characteristics.

pith-pipeline@v0.9.0 · 5677 in / 1083 out tokens · 57458 ms · 2026-05-20T02:00:21.049144+00:00 · methodology

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Reference graph

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