{"id":"b1eb9a0e-8651-4378-8926-bfd41c05521b","arxiv_id":"2506.02650","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New weighted L2 estimates with exponent 2/9 for broad Fourier extension operators on 1D fractals yield improvements to maximal Schrodinger, maximal extension, circular Lp decay, and an Lp Mizohata-Takeuchi inequality.","lead":"This paper proves new weighted L2 bounds for the Fourier extension operator on one-dimensional fractal sets in the plane, with a sharp-looking exponent 2/9, and applies them to several long-standing Lp problems. The advances matter because they push the maximal Schrodinger range from q=4 to q>18/5, partially settling conjectures of Lee-Rogers-Seeger and extending Wolff's circular decay result.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline L2 bound rests on the two-ends Furstenberg estimate of [WW24], cited rather than proved; the incidence step (2.21) is load-bearing and not independently verified.","rationale":"I read the paper in good faith: the main theorem is a genuine weighted L2 estimate with interesting applications, and the proof strategy is coherent if all cited ingredients are correct. The reader's weakest assumption identified the reliance on Theorem 2.10, a two-ends Furstenberg estimate cited to the author's preprint [WW24]. I agree that this is the most load-bearing concern. The paper explicitly states that Theorem 1.3 uses this estimate, and Proposition 2.2's first method depends on it through Corollary 2.11 at the crucial incidence lower bound (2.21). The line in Section 2.3 setting δ^{-ε1/2}=K and δ^{-ε2/2}=A is terse, and the corollary's hypotheses are not checked in detail; this is a real gap in presentation and verification. I do not treat disagreement with consensus as a problem: the issue is purely that a decisive ingredient is not established in this paper and is cited to an unpublished same-author preprint. I also considered whether the applications have independent internal gaps; they appear to follow once Proposition 3.2 and Corollary 2.12 are granted. The honest non-finding option does not apply because the self-citation is genuinely load-bearing. The fix is conditional: keep the conditional recommendation, but require either a self-contained proof of Theorem 2.10 or a detailed verification of its use in the derivation of (2.21).","tokens_in":17570,"tokens_out":13860,"duration_ms":136379,"concrete_test":"Obtain [WW24] and verify that Theorem 2.1, with its proof, covers exactly the parameters (δ, ε1, ε2, λ) used in Corollary 2.11, and then supply the missing derivation of (2.21) from Corollary 2.11 in the notation of Section 2.3. In particular, trace the identification δ^{-ε1/2}=K and δ^{-ε2/2}=A through the corollary's hypothesis #Tσ(Q) ≲ δ^{ε2}#T(Q) for the pigeonholed tube families T1,Q, and confirm the exponent in (2.21). If the identification is a typo, state the corrected exponent and recompute (2.21)-(2.22).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.3 is proved through Proposition 2.2. In Method 1, Step 2, the lower bound (2.21) on the union of tubes is the only input that converts the per-cap bush size M into a global tube count; it is used to derive the first estimate (2.35). Without (2.21), the interpolation between Methods 1 and 2 does not produce the factor |X|^{4/9}. Equation (2.21) is obtained by applying Corollary 2.11, which is stated as a consequence of Theorem 2.10, 'proved in [WW24, Theorem 2.1]' by the present author and H. Wang. No proof or independent reference is given here. Moreover, the application of Corollary 2.11 is not verbatim: the corollary's two-ends hypothesis is phrased using arcs of length δ^{ε1}, while Step 2 works with K^{-1}-caps and sets δ^{-ε1/2}=K and δ^{-ε2/2}=A. The text does not spell out how the pigeonholed counts (#Tσ(Q) ~ M, #T1(Q) ≲ KM) satisfy the corollary's quantitative hypothesis #Tσ(Q) ≲ δ^{ε2}#T(Q); the identification δ^{ε2}=A^{-2} is tenable only with additional bookkeeping about A≤K and the choice of the cap set Σ_M(Q). Thus, even granting Theorem 2.10, the derivation of (2.21) is not fully verified as written. If either the theorem or this incidence step is incomplete, the main weighted L2 estimate and all applications in Section 3 fail. This is the load-bearing soft spot I identify; I do not see a separate internal contradiction in the applications beyond this dependency.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a weighted L2 estimate for the broad Fourier extension operator on Katz-Tao sets in the plane, and derives several consequences: decay of circular Lp-means of Fourier transforms of fractal measures, maximal Schrödinger and maximal extension operator bounds, and an Lp analogue of the Mizohata–Takeuchi conjecture. The main theorem (Theorem 1.3) is reduced to a scale-induction proposition (Proposition 2.2), whose proof combines wave-packet decompositions, refined decoupling, and a two-ends Furstenberg estimate cited from a preprint by the same author.","tokens_in":17903,"tokens_out":4878,"duration_ms":46895,"significance":"If the central estimate is valid, the applications would genuinely improve known results in several Lp problems (e.g., extending Wolff's L2 circular decay to p>9/5 and pushing the maximal Schrödinger range to q>18/5). The paper is well structured and the application arguments are clearly laid out. However, the central claim is not self-contained: it depends on an unproved, not yet independently refereed two-ends Furstenberg estimate ([WW24]), and the verification of that estimate's hypotheses in the incidence step is incomplete as written. The value of the paper therefore hinges on a load-bearing external input, which the present manuscript neither proves nor fully checks.","major_comments":[{"comment":"The reduction of Theorem 1.3 to Proposition 2.2 is only stated as 'standard arguments' and is not shown in detail. Additionally, the proof of Proposition 2.2 cites Theorem 2.10 from [WW24] (a preprint coauthored by the author) and its consequence Corollary 2.11 without proof. Since equation (2.21) is the only input that turns the per-cap bush size M into a global tube count and is essential for the first estimate (2.35), the main theorem is conditional on an external result that is not established in this manuscript. The paper should either prove Theorem 2.10 for the specific configuration needed here or state clearly which version is used and verify every hypothesis.","section":"Section 2.1, Theorem 1.3 to Proposition 2.2"},{"comment":"The application of Corollary 2.11 is not verbatim. The text sets δ = r^{-1/2+ε0}, δ^{-ε1/2}=K, and δ^{-ε2/2}=A, but Corollary 2.11 requires the two-ends condition #Tσ(Q) ≲ δ^{ε2}#T(Q) for arcs of length δ^{ε1}. The pigeonholing in Step 1 only gives #Tσ(Q) ∼ M for σ ∈ Σ_M(Q) and #T(Q) ≲ KM, and the manuscript does not verify that the chosen index set Σ_M(Q) satisfies the quantitative two-ends hypothesis with the stated constants. In particular, δ^{ε2} = A^{-2}, while the available control is M ≲ A^{-2} K M only if A^2 ≲ K, which is not guaranteed by the hypotheses A ≥ R^{ε'} and K ≤ R^{ε^4} for general ε'. Thus the derivation of (2.21) is not justified as written.","section":"Section 2.3, Step 2, derivation of (2.21)"},{"comment":"The line 'by considering a single bush rooted at Q, where #T_{1,β}(Q) reaches the maximum in (2.46), we have (2.47) r^{α1−α2} ∼ #Q1 ≳ r^{-ε2} M λβ' is not explained. It is not clear why a bush of size M at a single Q forces at least r^{-ε2}Mλβ distinct r^{1/2}-balls Q in Q1. This lower bound on #Q1 is essential for the second estimate (2.50), so a detailed counting argument is needed.","section":"Section 2.3, Method 2, Step 3, inequality (2.47)"},{"comment":"Several intermediate steps are delegated to the preprint [DW25] (also by the present author), including the dyadic pigeonholing after (2.14) and the two-ends reduction in Method 2, Step 1. Since these steps are load-bearing for the induction and for the two-ends/non-two-ends dichotomy, they should be reproduced or at least be replaced by published references. As written, the proof is not verifiable independently of the author's other preprints.","section":"Section 2.3, proof of Proposition 2.2"}],"minor_comments":[{"comment":"The title and abstract contain obvious spacing errors (e.g., 'L2 ESTIMA TES WITH APPLICA TIONS') that should be fixed in the final version.","section":"Title and Abstract"},{"comment":"The citation [Obe23] is presented as the source for the p=4 result of Carbery–Iliopoulou–Shayya on the Lp variant of the Mizohata–Takeuchi conjecture, but the reference is an Oberwolfach workshop report, which seems unlikely to contain that result. Please verify the correct citation.","section":"Section 1.2.3, reference [Obe23]"},{"comment":"In the wave packet decomposition, the notation T = T_{θ,v} and f_T = f_{θ,v} is used, but the distinction between the tube T and the associated function f_T is not consistently maintained; for example, in (2.19)–(2.20) the factor K appears without a clear definition of how T_{1,Q} is related to the cap partition after pigeonholing.","section":"Equation (2.7) and surrounding text"},{"comment":"The set eX_λ (the R-dilate of X_λ) is used without definition, and the factor λR^2 in (3.16) is not derived. Please clarify the normalization of the measure µ_λ and the dilation step.","section":"Proof of Theorem 3.3, after (3.15)"},{"comment":"The identity w'_{R/K^2}(L_σ(X)) = K^{-3} sup_T ... ||T ∩ X| is asserted without proof; a short explanation of how the tube dimensions and the weight scale under L_σ would help the reader.","section":"Equation (3.36)"},{"comment":"The notation rO(ε0) and KO(1) is used informally; in several places it is unclear whether O(·) is independent of ε. For instance, the line 'r^{O(ε0)} ≤ K^{O(1)}' after (2.22) should be justified with the explicit choice of ε0 and the relation r ≥ R^{ε^2}.","section":"Throughout Section 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's main theorem is conditional on Theorem 2.10 from [WW24], a preprint coauthored by the author, and on several technical reductions delegated to [DW25], another preprint by the author. While self-citation is not problematic per se, the lack of independent verification of these inputs is a significant risk. The editor may wish to solicit a separate opinion on the two-ends Furstenberg estimate. Also, the reference [Obe23] appears to be incorrect; the claimed Carbery–Iliopoulou–Shayya result is not in that workshop report. The paper is otherwise well written and the applications are plausible if the central estimate holds."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real news is Theorem 1.3: a weighted L2 bound \\|Br_A E_S f\\|_{L^2(X)} \\lesssim R^\\epsilon |X|^{2/9}\\|f\\|_2 for 1D fractal sets X, improving the bilinear exponent 1/4 to 2/9 in the broad setting. The applications are genuinely new ranges: maximal Schrödinger q>18/5, circular Lp decay for p in [9/5,2], and an Lp Mizohata–Takeuchi bound. If the main estimate survives scrutiny, this is a solid step forward.\n\nThe paper is honestly written. The proof of Proposition 2.2 is detailed and uses standard modern machinery (refined decoupling, wave packets, pigeonholing). The author is transparent that the proof depends on the two-ends Furstenberg estimate from [WW24], a preprint coauthored by the same author. That is not by itself a flaw, but it is a load-bearing external input that is not proved here.\n\nThere is one spot that worried me more than the self-citation. In Method 1, Step 2, Corollary 2.11 is invoked with δ = r^{-1/2+ε0}, δ^{-ε1/2}=K, δ^{-ε2/2}=A. The corollary's hypothesis is #Tσ(Q) ≲ δ^{ε2}#T(Q), i.e. #Tσ(Q) ≲ A^{-2}#T(Q). In the text we only know #Tσ(Q) ~ M and #T(Q) ≲ KM. So the hypothesis reduces to M ≲ A^{-2}KM, which holds only if A ≳ K^{1/2}. The assumptions allow A to be much smaller than K^{1/2} (e.g. K = R^{ε^4}, A = R^{ε'} with ε' just below ε^4). The text simply asserts the configuration obeys the assumption. That is a genuine gap as written. I suspect it can be fixed by a more careful choice of the cap set Σ_M(Q) or a different parametrization, but it is not a cosmetic detail — equation (2.21) is what converts the bush size M into the global tube count and ultimately produces the |X|^{4/9} factor.\n\nEverything after Proposition 2.2 is fairly standard broad-narrow reduction; the applications are conditional on the main estimate. So the load-bearing question is whether Theorem 2.10 and the incidence step can be made rigorous.\n\nWho gets value from this paper: harmonic analysts working on restriction, maximal Schrödinger estimates, and Mizohata–Takeuchi type problems. It deserves a serious referee, not a desk rejection. I would tell the editor to send it out, but with a referee who understands the two-ends Furstenberg literature and will press the author on the application of Corollary 2.11. Conditional acceptance is the right default unless the gap proves fatal.","headline":"New weighted L2 broad estimate with exponent 2/9 and several nice applications, but the load-bearing two-ends Furstenberg input is cited from a same-author preprint and one incidence step has a real hypothesis gap.","tokens_in":18499,"tokens_out":2431,"would_cite":false,"duration_ms":26333,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B10","42B25","42B37"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a weighted L^2 bound for the broad Fourier extension operator on one-dimensional fractal sets in R^2, with exponent |X|^{2/9}, and uses it to prove new circular-decay, maximal Schrödinger, maximal extension, and…","keywords":["Fourier extension operator","weighted L2 estimates","broad extension operator","Katz-Tao sets","decoupling","maximal Schrödinger operator","circular means","Mizohata-Takeuchi conjecture"],"falsifier":"Construct a Katz–Tao $(\\delta,1)$-set of $\\delta$-balls and a set of $\\delta$-separated lines with $\\lambda$-dense, two-ends shadings for which the union of shaded balls is smaller than the lower bound asserted by the two-ends Furstenberg estimate; this would invalidate the incidence step behind equation (2.21). Alternatively, find a fractal unit-ball set $X$ with $|X|\\sim R$ and a large $A$ for which $\\|\\mathrm{Br}_A E_S f\\|_{L^2(X)}$ exceeds $C R^\\varepsilon |X|^{2/9}\\|f\\|_2$.","tokens_in":17322,"feed_emoji":"📐","tokens_out":20365,"duration_ms":167014,"temperature":0.7,"pith_summary":"This paper establishes weighted $L^2$ estimates for the Fourier extension operator on a compact nonzero-curvature curve $S$ in $\\mathbb{R}^2$, with a weight that is a one-dimensional fractal set: the broad extension operator satisfies $\\|\\mathrm{Br}_A E_S f\\|_{L^2(X)} \\leq C_{\\varepsilon,\\varepsilon'} R^{\\varepsilon} |X|^{2/9} \\|f\\|_2$ for $A \\geq R^{\\varepsilon'}$ and $K \\leq R^{\\varepsilon^4}$. This improves the classical bilinear-restriction exponent $1/4$ for fractal weights, which is the feature the applications need. From that estimate, the paper derives four $L^p$ results: decay of circular $L^p$-means of Fourier transforms of Frostman measures with exponent $1/(2p)$, estimates for the maximal Schrödinger and maximal extension operators for $q>18/5$, and an $L^p$ analogue of the Mizohata–Takeuchi conjecture for $p \\geq 18/5$. The paper derives the four applications from this bound by standard broad-narrow analysis.","feed_headline":"Four Lp estimates follow from one weighted L2 bound","feed_subtitle":"For fractal sets in R2, one bound yields new circular means, maximal Schrödinger, and Mizohata–Takeuchi results.","key_machinery":"The load-bearing object is the broad extension operator $\\mathrm{Br}_A E_S f(x) = \\max_{\\Sigma':\\#\\Sigma'=A} \\min_{\\sigma\\in\\Sigma'} |E_S f_\\sigma(x)|$, which isolates inputs whose Fourier support spreads across many $K^{-1}$-arcs, together with the Katz–Tao $(\\delta,1)$-condition $\\#(E\\cap B(x,r)) \\leq C(r/\\delta)$ that encodes the one-dimensional fractal geometry of the weight. The proof machinery is a wave packet decomposition into $R^{1/2}\\times R$ tubes; a refined decoupling theorem converts $L^2$ control into $L^6$ control in the first method, and a two-ends Furstenberg incidence estimate supplies the lower bound on the number of tubes that meet the fractal set. Induction on scales gives the second method. Interpolating the first method's $r^{2/5}$ estimate with the second method's $r^{\\alpha_1-1/2}$ estimate yields the $|X_1|^{4/9}$ exponent in the squared $L^2$ inequality, equivalently $|X|^{2/9}$ in the $L^2$ norm.","core_discovery":"The paper's central claim is Theorem 1.3: for a compact curve $S$ in $\\mathbb{R}^2$ with nonzero curvature, and for a union $X \\subset B_R$ of unit balls whose $R^{-1}$-dilate is a Katz–Tao $(R^{-1},1)$-set (a one-dimensional fractal condition at unit scale), the broad extension operator satisfies $\\|\\mathrm{Br}_A E_S f\\|_{L^2(X)} \\leq C_{\\varepsilon,\\varepsilon'} R^{\\varepsilon} |X|^{2/9} \\|f\\|_2$ whenever $A \\geq R^{\\varepsilon'}$ and $K \\leq R^{\\varepsilon^4}$. Here $\\mathrm{Br}_A E_S$ is the maximum, over collections of $A$ arcs of length $K^{-1}$, of the minimum of the extensions of the pieces of $f$ localized to those arcs. The proof reduces to a local induction statement, decomposes the input into wave packets, runs two independent estimates (one via refined decoupling plus a two-ends Furstenberg incidence lower bound, one via induction on scales), and interpolates them. The four applications are then derived from this single weighted $L^2$ bound by the standard broad-narrow reduction.","pith_inferences":["Editorial inference: because the proof imports the two-ends Furstenberg estimate from the companion paper [WW24] without proving it, that incidence theorem is the single external point on which Theorem 1.3 currently rests; an independent proof or counterexample there would settle the status of the whole chain.","Editorial inference: the endpoint $q=18/5$ in the applications is forced by interpolating the two methods; if either the refined decoupling input or the two-ends incidence bound were improved, the thresholds in all four applications would likely move together.","Editorial inference: the author's lower-bound example shows the weighted exponent cannot beat $1/6$, leaving a gap between $1/6$ and $2/9$; closing that gap would require a mechanism other than the two-method interpolation used here."],"forward_implications":["For every one-dimensional Frostman measure $\\mu$ on $[0,1]^2$, the circular $L^p$-means obey $(\\int_{S^1}|\\widehat{\\mu}(R\\xi)|^p\\,d\\sigma(\\xi))^{1/p} \\lesssim R^{-1/(2p)}$ for $p\\in[9/5,2]$, and a fixed-$R$ example shows this decay rate is sharp.","The maximal Schrödinger operator is bounded in the predicted Besov spaces for $q>18/5$ in the plane, verifying Conjectures 1.6 and 1.7 in that range.","The maximal extension operator in two dimensions satisfies the conjectured $L^q_x L^\\infty_{x_2}$ estimate for $q>18/5$.","For $p\\geq 18/5$, any set $X\\subset B_R$ has $\\|E f\\|_{L^p(X)}^p \\leq C_\\varepsilon R^\\varepsilon w_R(X)\\|f\\|_2^p$, the $L^p$ analogue of the Mizohata–Takeuchi conjecture, with $w_R(X)$ the maximal $1\\times R$-tube overlap.","As an intermediate corollary, for $q\\geq 18/5$ the broad extension operator satisfies $\\|\\mathrm{Br}_A E f\\|_{L^q(X)} \\leq C_\\varepsilon R^\\varepsilon \\|f\\|_2$ on the same class of fractal sets $X$."],"supporting_citations":[{"why":"Supplies the two-ends Furstenberg incidence estimate (Theorem 2.10) that yields the lower bound in equation (2.21), essential for the first method.","marker":"[WW24]"},{"why":"Supplies the refined decoupling theorem (Theorem 2.5) used in the first method to pass from L2 to L6 control.","marker":"[GIOW20]"},{"why":"Proves the ell2 decoupling inequality that [GIOW20] refines and that underlies the wave packet framework.","marker":"[BD15]"},{"why":"Introduces the broad operator Br_A E_S and the subadditivity lemma used in the pigeonholing steps.","marker":"[Gut16]"},{"why":"Formulates the maximal Schrödinger conjectures and provides the local reduction (Proposition 5.1) used in the proof of Theorem 1.8.","marker":"[LRS13]"},{"why":"Poses the circular Lp-means problem and proves the L2 case that Theorem 1.5 generalizes.","marker":"[Wol99]"},{"why":"Supplies the model proof of the dyadic pigeonholing structure (Proposition 4.13) that the main argument follows.","marker":"[DW25]"},{"why":"Raises the maximal extension operator problem and gives the epsilon-removal argument used to derive Theorem 1.10 from Proposition 3.2.","marker":"[Wu21]"}],"fun_headline_variants":["One weighted L2 bound yields four Lp estimates","From a single L2 estimate to maximal Schrödinger and more","Broad-narrow reduction turns L2 into four Lp results","Fractal sets in R2: one L2 bound, four Lp theorems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The main bound rests on a combinatorial incidence estimate taken from a companion preprint that the paper does not prove; if that estimate is false or incomplete, Theorem 1.3 does not follow.","fun_headline_variants_meta":{"raw":{"variants":["One weighted L2 bound yields four Lp estimates","From a single L2 estimate to maximal Schrödinger and more","Broad-narrow reduction turns L2 into four Lp results","Fractal sets in R2: one L2 bound, four Lp theorems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001438,"raw_usage":{"total_tokens":5762,"prompt_tokens":873,"completion_tokens":4889,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":4815}},"tokens_in":489,"tokens_out":4889,"duration_ms":35757,"temperature":1.0,"reasoning_tokens":4815,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:19:03.674432+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a Katz–Tao $(\\delta,1)$-set of $\\delta$-balls and a set of $\\delta$-separated lines with $\\lambda$-dense, two-ends shadings for which the union of shaded balls is smaller than the lower bound asserted by the two-ends Furstenberg estimate; this would invalidate the incidence step behind equation (2.21). Alternatively, find a fractal unit-ball set $X$ with $|X|\\sim R$ and a large $A$ for which $\\|\\mathrm{Br}_A E_S f\\|_{L^2(X)}$ exceeds $C R^\\varepsilon |X|^{2/9}\\|f\\|_2$.","supporting_citations":[],"review_version":1}