{"id":"d64996c5-bbfe-4fad-8ca8-5ee74192277b","arxiv_id":"1908.02867","paper_version":3,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":1,"one_line_summary":"Two-weight bounds for sparse square functions, uniform over sparse families, do not imply two-weight bounds for the Hilbert transform.","lead":"This paper builds two weights on the real line for which a natural family of sparse square function estimates all hold, while the Hilbert transform fails to be bounded. It thereby disproves a proposed route to the separated bump conjecture and shows sparse square functions cannot replace singular integrals in this argument.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"I reviewed the full proof of Proposition 1.8. The chain of reasoning is: (1) recall the Reguera-Thiele weight w_k and the associated weights σ_k; (2) prove local testing conditions for martingale sparse families with constant depending on k; (3) rescale w_k by k^{-r} to make the testing constants uniform; (4) pass to a direct sum of translated copies to obtain global testing conditions; (5) invoke Culiuc's theorem to get full sparse square function bounds for all η-sparse families; (6) show the Hilbert transform norm blows up on the direct sum weights; (7) use the closed graph theorem to infer the existence of f with H(f w) not in L^p(σ). I checked each step for gaps. The reduction from general sparse families to martingale sparse families in Appendix 6.1 is correct: the three-lattice trick gives a pointwise domination of A_{S,p} by sums of A_{S_j,p} with bounded multiplicity, and [11, Lemma 6.6] splits the resulting families into martingale ε-sparse ones. The local estimates in Section 4.3 are carefully computed and rely only on the structural lemmas of Section 3. The rescaling r is chosen in the nonempty interval (max(1, 1/(p-1)), p'), which ensures both testing conditions become uniform and the Hilbert transform blow-up exponent 1 - r/p' is positive. The global testing conditions in Proposition 4.13 correctly handle intervals that intersect multiple support pieces. The bump condition results in Section 5 use the Lorentz-space comparison principles of Treil-Volberg and Nazarov-Reznikov-Treil-Volberg, with computations that are consistent with Lemma 3.3. I found no circular reasoning, no hidden dependence on unproved assertions, and no post-hoc data selection. The only external dependency is the Reguera-Thiele estimate (2.1), but this is a published result and the paper includes the full construction in Section 3. The reader's weakest-assumption identification matches my assessment: the construction's quantitative properties are the least independently checked part, yet they are used correctly and are well-cited. No significant objection survives scrutiny.","tokens_in":37970,"tokens_out":52817,"duration_ms":461213,"concrete_test":"As a verification step, independently re-derive the Reguera-Thiele lower bound |H w_k| ≳ k w_k on E_k for a small value of k (e.g., k = 3001) using the construction in Section 3, and confirm the constants in Lemma 3.3 and the Lorentz norm computation in Section 5.6 for that k. If the lower bound or the Lorentz norm estimate fails, the counterexample would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"No significant objection identified. The paper's central claim, Proposition 1.8, is supported by a detailed and internally consistent proof. The Reguera-Thiele construction is correctly recalled; the local testing conditions in Proposition 4.5 are verified through Lemmas 3.3, 3.4, 4.8-4.12; the rescaling by k^{-r} and the direct-sum passage in Proposition 4.13 are valid; the reduction to martingale sparse families in the Appendix is sound; and Culiuc's theorem is applied appropriately to convert testing conditions into sparse square function bounds. The Hilbert transform blow-up survives rescaling via (2.7), and the closed graph theorem argument correctly yields a witness f with H(f w) not in L^p(σ). The bump-condition results in Section 5 are also consistent. The weakest point remains the external Reguera-Thiele estimate (2.1), but this is a published result and is used faithfully, with the paper recalling the construction in detail.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":38068,"tokens_out":26687,"duration_ms":252891,"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.","major_comments":[],"minor_comments":[{"comment":"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":"Section 4.3.2, Lemma 4.10(a)"},{"comment":"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":"Section 2.3 and Section 5.3.1"},{"comment":"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":"Section 4.2, Eq. (4.5)"},{"comment":"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.","section":"Section 4.4, proof of Proposition 4.13"},{"comment":"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.","section":"Proposition 1.10 and Section 5.5"},{"comment":"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.","section":"Theorem 4.6 and Section 4.2"}],"recommendation":"minor_revision","confidential_remarks":"The paper is technically sound and the counterexample is interesting. The changes needed are local corrections (mostly typos and small clarifications) that do not affect the main results. I recommend acceptance after a minor revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSoft 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.\n\nWho 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.\n\nMy 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.","headline":"A solid, explicit counterexample that kills the natural sparse-square-function route to the separated bump conjecture, while honestly leaving the conjecture itself open.","tokens_in":38655,"tokens_out":1135,"would_cite":true,"duration_ms":15049,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B20","42B25"],"pacs":[],"model":"deepseek-v4-flash","headline":"Uniform two-weight sparse square function bounds do not imply a two-weight bound for the Hilbert transform.","keywords":["two-weight estimates","sparse square functions","Hilbert transform","separated bump conjecture","Orlicz bumps","testing conditions","weighted norm inequalities"],"falsifier":"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.","tokens_in":37716,"feed_emoji":"📉","tokens_out":11726,"duration_ms":105578,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two-weight sparse square bounds fail to imply Hilbert transform bounds","feed_subtitle":"Uniform sparse square estimates can hold in both directions while the Hilbert transform stays unbounded.","key_machinery":"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 $\\mathbb{R}$.","core_discovery":"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