Two-weight estimates for sparse square functions and the separated bump conjecture

Spyridon Kakaroumpas

Two-weight estimates for sparse square functions and the separated bump conjecture

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Pith's one-line read Uniform two-weight sparse square function bounds do not imply a two-weight bound for the Hilbert transform.

desk verdict A solid, explicit counterexample that kills the natural sparse-square-function route to the separated bump conjecture, while honestly leaving the conjecture itself open. read the letter →

arxiv 1908.02867 v3 pith:7T6FARW4 submitted 2019-08-07 math.CA

Spyridon Kakaroumpas This is my paper · ORCID

classification math.CA MSC 42B2042B25

keywords two-weightestimatessparsesquarefunctionsHilberttransformseparatedbumpconjectureOrliczbumpstestingconditionsweightednorminequalities

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper constructs explicit weights on the real line for which two-weight $L^p$ bounds for sparse square functions hold uniformly over all sparse families and in both directions, yet the Hilbert transform is unbounded between the same weighted spaces. This disproves the natural Conjecture 1.6 and its stronger variant Conjecture 1.7, which would have made sparse square function control a bridge to singular integral estimates. The same example shows that at $p=2$ these sparse square function bounds do not imply the two separated Orlicz bump conditions, for any Young function satisfying the stated integrability condition. The counterexample is assembled from a known triadic weight construction, rescaled so that a factor of $k$ in the sparse estimates disappears while the Hilbert transform blow-up survives.

What carries the argument

The load-bearing object is the explicit triadic weight $w_k$ on $[0,1)$ recalled and reused from the literature, together with its companion $\sigma_k=w_k^{1-p}$ on the support. Its two quantitative properties are that $|Hw_k|\gtrsim k\,w_k$ on a large set and that the Hardy\u2013Littlewood maximal function $Mw_k$ is comparable to $w_k$ on the support. The paper's central technical work is to verify local testing conditions: for every martingale $\varepsilon$-sparse family $\mathcal{S}$ (a grid subfamily in which the maximal strictly smaller selected intervals occupy at most an $\varepsilon$-fraction of each parent), the sums $\sum_{I\in\mathcal{S},\,I\subseteq L}(\langle w_k\rangle_I)^p\langle\sigma_k\rangle_I\,|I|$ and the symmetric $\sigma_k$-version are bounded by $k\,w_k(L)/(1-\varepsilon)$ and $k\,\sigma_k(L)/(1-\varepsilon)$. Rescaling $w_k$ by $k^{-r}$ with $r\in(\max(1,1/(p-1)),p')$ removes the factor of $k$ from these estimates while preserving the Hilbert transform blow-up because $1-r/p'>0$. A direct-sum-of-singularities construction then assembles the single pair of weights on $\mathbb{R}$.

What would settle it

Take one block $K$ of the triadic lattice and the rescaled weight $\tilde w_k=k^{-r}w_k$ with $\sigma_k$. Compute the two local testing sums of Proposition 4.5 over all selected intervals inside $K$; if either sum exceeds a constant independent of $k$ times $\tilde w_k(K)/(1-\varepsilon)$ (respectively $\sigma_k(K)/(1-\varepsilon)$), the uniform sparse square function bounds collapse. Alternatively, verify on a large set that $|H w_k|\ge (k/3)w_k$; any failure there removes the Hilbert transform blow-up used in the gluing argument.

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Extended reading notes

Core claim

The paper proves that uniform two-weight boundedness of sparse square functions is strictly weaker than two-weight boundedness of the Hilbert transform. For every $1<p<\infty$ there exist weights $w,\sigma$ on $\mathbb{R}$ such that for every $0<\eta<1$ and every $\eta$-sparse family $\mathcal{S}$ of intervals the operators $A_{\mathcal{S},p}(\cdot w)$ and $A_{\mathcal{S},p'}(\cdot\sigma)$ satisfy uniform two-weight bounds, while $H(\cdot w)$ is unbounded from $L^p(w)$ into $L^p(\sigma)$. Thus Conjectures 1.6 and 1.7 are false. For $p=2$, the same example also shows that these sparse square function bounds do not force the separated Orlicz bump conditions: for every Young function $\Phi$ with $\int_c^\infty 1/\Phi(t)\,dt<\infty$, the bump product $\sup_I \|w\|_{L^\Phi(I)}\langle\sigma\rangle_I$ is infinite. The proof verifies local testing conditions for sparse $p$-functions against the constructed weights, upgrades them to full bounds through a testing-condition theorem, and uses a direct-sum-of-singularities gluing to obtain a single pair of weights on the line.

Load-bearing premise

The counterexample inherits two quantitative properties of the triadic weight construction: the Hilbert transform of $w_k$ must exceed a constant times $k\,w_k$ on a large set, and the Hardy\u2013Littlewood maximal function must stay comparable to $w_k$ on its support; if either estimate fails, the sparse bounds or the Hilbert transform blow-up cannot both go through.

Editorial extensions

If this is right

Conjecture 1.6 and the stronger Conjecture 1.7 are false for the Hilbert transform on the real line.
For every $1<p<\infty$, uniform two-weight $L^p$ bounds for both $A_{\mathcal{S},p}(\cdot w)$ and $A_{\mathcal{S},p'}(\cdot\sigma)$ over all $\eta$-sparse families do not imply boundedness of the Hilbert transform between the same weighted spaces.
At $p=2$, the same sparse square function bounds do not imply the two separated Orlicz bump conditions for any Young function with $\int_c^\infty 1/\Phi(t)\,dt<\infty$.
The constructed weights do satisfy separated bump conditions on triadic intervals for the Young function $\Phi(t)=t\log(e+t)(\log(\log(ee+t)))^{1+\delta}$, so the obstruction to separated bumps is carried by non-triadic intervals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The mechanism is probably not specific to sparse square functions: any sparse positive operator whose local testing conditions obey the same scaling would inherit the obstruction, including sparse maximal functions.
The triadic-lattice improvement suggests a testable refinement: if sparse families are required to lie in a fixed grid, separated bump conditions can hold simultaneously with Hilbert transform blow-up; the paper establishes this for triadic intervals.
Because the counterexample is one-dimensional and directional, whether the same failure occurs for higher-dimensional or vector-valued singular integrals is an open question that the methods here do not settle.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: this paper closes a specific route, not the whole problem. Kakaroumpas builds explicit weights for which sparse square functions satisfy uniform two-weight bounds in both directions but the Hilbert transform is unbounded. That directly refutes Conjectures 1.6 and 1.7, which the paper states clearly. The separated bump conjecture itself stays open, and the paper says so openly in Proposition 1.9. That honesty is rare and welcome.

What is genuinely new: the counterexample. The base construction is from Reguera–Thiele and Reguera–Scurry, but the twist is new. The rescaling w̃_k = k^{-r} w_k with r in a carefully chosen interval makes the sparse bounds uniform in k without destroying the Hilbert transform blow-up. The proof then verifies Sawyer-type testing conditions, invokes Culiuc's theorem to push them to sparse square function bounds, and uses a direct sum argument to pass from [0,1) to R. I read the testing condition verification in Section 4 closely enough to see that the work is real: the local lemmas (4.8–4.12) handle the triadic and general interval cases, and the appendix reduces general sparse families to martingale sparse families. The Hilbert transform blow-up in (2.7) survives the rescaling because r < p'. The logic holds together.

Soft spots: the paper inherits the Reguera–Thiele weight construction, and the sparse bounds are verified through a chain of long estimates that I did not machine-check. But the construction is recalled in enough detail that I could follow the structure, and the published Reguera–Scurry estimates are used faithfully. There is also a mild mismatch between the abstract's promised scope and the modest conclusion: this is an important counterexample within the subfield, not a resolution of the separated bump conjecture. The paper never overclaims, so this is fair enough.

Who this is for: harmonic analysts working on two-weight theory, weighted inequalities, or sparse domination. If that is your area, you should read it. If not, it is still a clean example of how sparse bounds can fail to capture singular integral behavior.

My recommendation: engage with it. Send it to a serious referee. The central claim is explicit, the proof is substantial, and the counterexample is likely to be cited. The referee will need to check the long estimates carefully, but the paper deserves that time.

Referee Report

0 major / 6 minor

Summary. The paper constructs explicit weights w, σ on R such that for any 1 < p < ∞ and any η-sparse family S of intervals, the two-weight norms of the sparse p-function A_{S,p}(·w) from L^p(w) to L^p(σ) and of A_{S,p′}(·σ) from L^{p′}(σ) to L^{p′}(w) are bounded by a constant depending only on η and p, while the Hilbert transform H(·w) is unbounded from L^p(w) to L^p(σ). This disproves Conjectures 1.6 and 1.7. The author also shows that for p = 2 this example fails the separated Orlicz bump conditions of Conjecture 1.2 for every Young function Φ with ∫^∞ 1/Φ < ∞, while for triadic intervals it satisfies a logarithmic bump condition with exponent below the threshold of Theorem 1.3. The proof uses the Reguera–Thiele weights, a rescaling by k^{-r}, local testing conditions verified in Section 4, Culiuc's theorem to pass to global sparse bounds, and a direct-sum argument.

Significance. This is a well-written and technically careful paper that clarifies the structural relations between two-weight estimates for sparse square functions, separated bump conditions, and singular integrals. The main counterexample is explicit and the proof is essentially self-contained, including a detailed recollection of the Reguera–Thiele construction and a complete verification of the required Sawyer-type testing conditions. The companion results on Orlicz and Lorentz bumps in Sections 5.3–5.6 are informative and establish a sharp contrast with the triadic case. The paper is an original contribution of interest to the weighted theory community. I found no internal inconsistency or gap in the main line of argument.

minor comments (6)

[Section 4.3.2, Lemma 4.10(a)] The displayed inequality "N ≤ k log 3 − log 4 / log(1/ε) + 1" contains a sign error: from the preceding chain one obtains (N−1) log(1/ε) ≤ k log 3 + log 4, so the log 4 term should have a positive sign; the subsequent bound N ≲ k/(1−ε) is unaffected.
[Section 2.3 and Section 5.3.1] There are several typos: "dractic simpliﬁcation" should be "drastic simpliﬁcation", "tha Φ" should be "that Φ", and "is and only if" should be "is if and only if".
[Section 4.2, Eq. (4.5)] The justification "Since pr > r+1 and r > 1" is equivalent to r > max(1, 1/(p−1)), the condition actually used; rephrasing this line would make the parameter choice clearer to the reader.
[Section 4.4, proof of Proposition 4.13] For intervals in S^2_m the bound ⟨\tilde{w}⟩_I, ⟨σ⟩_I ≲_p m2^{-m} is true, but the reason is that the total masses ∑ k^{-r} and ∑ 3^{-k} are finite, so the averages are O(2^{-m}); mentioning this would make the argument more transparent.
[Proposition 1.10 and Section 5.5] Proposition 1.10 uses a Young function with exponent 1+δ, while Section 5.5 uses a parameter r ∈ (1,2); the relation δ = r−1 is not stated explicitly and should be added.
[Theorem 4.6 and Section 4.2] When applying Culiuc's theorem, the coefficients a_Q are implicitly taken to be a_Q = 1_Q; making this identification explicit would help the reader follow the application of the theorem.

Circularity Check

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No significant circularity: the counterexample is built from the external Reguera–Thiele construction and the sparse bounds are verified independently.

full rationale

The paper is a counterexample construction against external conjectures, not a derivation that assumes its own conclusion. Its load-bearing input is the published Reguera–Thiele/Reguera–Scurry weight construction, recalled in Section 3 and cited as [17],[18]; the paper does not refit those weights to force the sparse estimates. The sparse square-function bounds are derived from explicitly verified Sawyer-type testing conditions (Proposition 4.5, Lemma 4.12, Proposition 4.13) via Culiuc's theorem (Theorem 4.6), an independent parameter-free result, and the reduction from general sparse families to martingale sparse families is proved in the appendix rather than imported by citation. The Hilbert-transform blow-up is inherited from estimate (2.3) and survives the rescaling through (2.7), with no assumption of the desired conclusion. The only auxiliary parameter, the rescaling exponent r, is chosen freely in the open interval (max(1,1/(p-1)), p'), and the estimates (4.5) hold for any such choice, so no fitted parameter is renamed as a prediction. The external results used (Reguera–Thiele construction, Culiuc's testing-condition theorem, Treil–Volberg bump domination) are either recalled with proof details or have stated hypotheses that do not include the target result; the author's own work is not used as load-bearing support. Thus the derivation chain is self-contained relative to its external assumptions, and no circular step is present.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on external published constructions and theorems, not on new postulated objects. The only hand-chosen quantity is the rescaling exponent r, which is arbitrary in an open interval and not fitted to data.

free parameters (1)

rescaling exponent r = any in (max(1, 1/(p-1)), p')
Introduced in Section 2.2 and Section 4.2 to renormalize w_k into w_tilde_k = k^{-r} w_k, making sparse p-function testing estimates uniform in k. The result holds for any such r; it is a proof parameter, not fitted to data.

assumptions (5)

domain assumption Reguera-Thiele and Reguera-Scurry weights w_k, sigma_k satisfy (2.1), (2.2), and the average estimates in Lemmas 3.3-3.7.
The counterexample is built on these published estimates from [17] and [18]. They are recalled in Section 3 but not reproved from scratch.
domain assumption Culiuc's theorem: localized testing conditions imply global boundedness for generalized sparse operators.
Used as Theorem 4.6 to convert the testing conditions in (4.6) into full two-weight L^p bounds for sparse p-functions.
domain assumption Estimates for martingale sparse families transfer to all eta-sparse families.
Appendix 6.1 gives the reduction, relying on the three-lattices trick and [11, Lemma 6.6]. This is needed to state the sparse bounds for arbitrary eta-sparse families.
standard math Carleson Embedding Theorem for grids of cubes.
Used in Lemma 4.3 and in the proof of testing conditions to control weighted sums over sparse families.
domain assumption Treil-Volberg and Nazarov-Reznikov-Treil-Volberg comparisons between Orlicz norms and Lorentz norms.
Used in Section 5 to replace Orlicz bump estimates with Lorentz norm computations for the weights w_k and sigma_k.

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Pith. "Pith review of Two-weight estimates for sparse square functions and the separated bump conjecture." pith.science (2026). https://pith.science/paper/7T6FARW4

@misc{pith2026190802867,
  author       = {Pith},
  title        = {Pith review of: Two-weight estimates for sparse square functions and the separated bump conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7T6FARW4}},
  note         = {Machine review of arXiv:1908.02867}
}

abstract

We show that two-weight $L^2$ bounds for sparse square functions, uniformly with respect to the sparseness constant of the underlying sparse family, and in both directions, do not imply a two-weight $L^2$ bound for the Hilbert transform. We present an explicit example, making use of the construction due to Reguera--Thiele from [18]. At the same time, we show that such two-weight bounds for sparse square functions do not imply both separated Orlicz bump conditions of the involved weights for $p=2$ (and for Young functions satisfying an appropriate integrability condition). We rely on the domination of $L\log L$ bumps by Orlicz bumps (for Young functions satisfying an appropriate integrability condition) observed by Treil--Volberg in [20].

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Anderson, David V

Theresa C. Anderson, David V. Cruz-Uribe, and Kabe Moen, Logarithmic Bump Conditions for Calder´ on–Zygmund Operators on Spaces of Homogeneous type, Publ. Mat. , Volume 59, Number 1 (2015), 17-43

work page 2015

[2] [2]

129, Academic Press Inc., Boston, MA, 1988

Colin Bennett, Robert Sharpley, Interpolation of operators , Pure and Applied Mathematics, vol. 129, Academic Press Inc., Boston, MA, 1988

work page 1988

[3] [3]

Cruz-Uribe, Jos´ e Maria Martell, and Carlos P´ erez,Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol

David V. Cruz-Uribe, Jos´ e Maria Martell, and Carlos P´ erez,Weights, extrapolation and the theory of Rubio de Francia, Operator Theory: Advances and Applications, vol. 215, Birkh¨ auser/Springer Basel AG, Basel, 2011

work page 2011

[4] [4]

Cruz-Uribe, Carlos P´ erez, Two-weight, weak-type norm inequalities for fractional in te- grals, Calder´ on–Zygmund operators and commutators, Indiana Univ

David V. Cruz-Uribe, Carlos P´ erez, Two-weight, weak-type norm inequalities for fractional in te- grals, Calder´ on–Zygmund operators and commutators, Indiana Univ. Math. J. , 49 (2000), no. 2, 697–721

work page 2000

[5] [5]

Cruz-Uribe, Alexander Reznikov, and Alexander Volberg , Logarithmic bump conditions and the two-weight boundedness of Calder´ on–Zygmund operato rs, Adv

David V. Cruz-Uribe, Alexander Reznikov, and Alexander Volberg , Logarithmic bump conditions and the two-weight boundedness of Calder´ on–Zygmund operato rs, Adv. Math. 255 (2014), 706– 729, DOI 10.1016/j.aim.2014.01.016. MR3167497

work page doi:10.1016/j.aim.2014.01.016 2014

[6] [6]

A note on two weight bounds for the generalized Hardy-Littlewood Maximal operator

Amalia V. Culiuc, A note on two weight bounds for the generalized Ha rdy–Littlewood Maximal operator, arXiv:1506.07125v1

work page Pith review arXiv

[7] [7]

Lacey, On the separated bumps conjecture for Calder´ on–Zygmund operators, Hokkaido Math

Michael T. Lacey, On the separated bumps conjecture for Calder´ on–Zygmund operators, Hokkaido Math. J. , Volume 45, Number 2 (2016), 223-242

work page 2016

[8] [8]

Lacey, An elementary proof of the A2 bound, Israel J

Michael T. Lacey, An elementary proof of the A2 bound, Israel J. of Math. , March 2017, 217(1), 181–195

work page 2017