{"id":"b69371d4-6fec-4861-aea6-6d6d56b30000","arxiv_id":"2607.11280","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For radial Schrödinger operators with inverse-square asymptotics, the kernel of H^{-s/2} admits a clean two-sided estimate precisely for 0<s<min{d,d-2σ₀,d-2σ∞}, with a complete N&S classification of broken-power Lorentz estimates.","lead":"The paper pins down the exact range of fractional powers where a radial Schrödinger operator’s integral kernel has a clean two-sided size estimate, and fully classifies the weighted Lorentz-space bounds that follow. Specialists in Schrödinger operators and weighted inequalities get sharp necessary-and-sufficient conditions covering endpoints and q < p.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"The claimed maximal s-range and complete N&S Lorentz classification rest on an external heat-kernel hypothesis whose conversion into sharp kernel and endpoint statements cannot be audited from the abstract alone.","rationale":"The Reader correctly isolated the Ishige–Kabeya–Ouhabaz heat-kernel bound as the weakest assumption and correctly returned UNVERDICTED/LOW because only the abstract is available. My concern coincides with that assessment: the paper’s distinctive content—the maximality of the s-interval and the completeness of the N&S list, including the delicate scale and corner cases—cannot be verified without the proofs that convert the external heat-kernel estimate into the clean kernel bound and then into the Lorentz classification. No internal contradiction is visible from the abstract, and the claimed techniques are standard within the subfield, so there is no ground for a stronger verdict. The verdict therefore remains UNVERDICTED.","tokens_in":2357,"tokens_out":631,"duration_ms":14168,"concrete_test":"Obtain the full text and check whether the necessity proofs for the scale condition u ≤ v (including q < p) and for the corner pair (u,v)=(1,∞) are realized by explicit counter-examples or rank-one/duality arguments that remain valid for every s inside the claimed open interval and for both signs of the ground-state exponents; if any necessity argument silently requires a smaller range of s or an extra restriction on V, the completeness claim fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim asserts both that the clean two-sided kernel bound K_s^H(x,y) ≃ |x−y|^{s−d} U(|x|)U(|y|)/[U(|x|+|x−y|)U(|y|+|x−y|)] holds precisely for 0 < s < min{d, d−2σ₀, d−2σ∞} and that a complete list of necessary-and-sufficient conditions (including all one-sided equalities, both scale equalities u ≤ v even when q < p, and simultaneous corners only for (u,v)=(1,∞)) governs the broken-power Lorentz estimate. Both assertions are derived under the standing external hypothesis of the Ishige–Kabeya–Ouhabaz two-sided ground-state heat-kernel estimate. The abstract lists the tools (clean-kernel analysis, local Lorentz-HLS, rank-one endpoints, annular sequence spaces, triangular matrix theorem, nine-block decomposition) but supplies none of the arguments. Consequently it is impossible to confirm that the open interval for s is maximal, that necessity of every listed equality case is established inside that interval, or that the nine-block decomposition closes all gaps when signs of σ₀, σ∞ and the relation q ≶ p vary. The load-bearing step is therefore the unexamined passage from the external heat-kernel bound to these sharpness statements.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The manuscript studies fractional powers H^{-s/2} of nonnegative radial Schrödinger operators H = −Δ + V(|x|) on R^d (d ≥ 2) whose positive harmonic function U has inverse-square asymptotics U(r) ≃ r^{−σ_0} near zero and U(r) ≃ r^{−σ_∞} at infinity, with −d/2 < σ_0, σ_∞ < d/2. Under the standing two-sided ground-state heat-kernel estimate of Ishige–Kabeya–Ouhabaz, it claims that the integral kernel of H^{-s/2} admits the clean two-sided bound K_s^H(x,y) ≃ |x−y|^{s−d} U(|x|)U(|y|)/[U(|x|+|x−y|)U(|y|+|x−y|)] precisely for 0 < s < min{d, d−2σ_0, d−2σ_∞}. In that range it asserts a complete necessary-and-sufficient classification of the broken-power Lorentz estimate ‖w_{−β_0,−β_∞} H^{−s/2} f‖_{L^{q,v}} ≲ ‖w_{α_0,α_∞} f‖_{L^{p,u}} for 1 < p,q < ∞ and 1 ≤ u,v ≤ ∞, covering signed ground-state exponents, the full range q < p, all one-sided weight equalities, both scale equalities (governed by u ≤ v even when q < p), and simultaneous endpoint corners (only (u,v) = (1,∞) admissible).","tokens_in":2654,"tokens_out":1274,"duration_ms":14688,"significance":"If the claimed maximal s-range and the complete N&S Lorentz classification are correct, the paper would supply a definitive sharp theory for fractional powers of radial Schrödinger operators with inverse-square asymptotics in Lorentz spaces with broken power weights. The inclusion of signed exponents, the full range q < p, scale equality under u ≤ v, and the precise corner restrictions would substantially extend classical HLS-type results and clarify endpoint phenomena for this class of operators. The abstract lists a coherent toolkit (clean-kernel analysis, local Lorentz–HLS, rank-one endpoints, annular sequence spaces, triangular matrix theorem, nine-block decomposition) that is standard and appropriate for such sharpness statements; machine-checked or fully reproducible arguments for the necessity of every listed equality case would be a genuine contribution.","major_comments":[{"comment":"The central claims—the maximality of the open interval 0 < s < min{d, d−2σ_0, d−2σ_∞} for the clean two-sided kernel bound, and the necessity of every listed equality case (one-sided weight equalities, both scale equalities including u ≤ v when q < p, and simultaneous corners only for (u,v)=(1,∞))—are load-bearing and rest on the conversion of the external Ishige–Kabeya–Ouhabaz heat-kernel hypothesis into sharp kernel and endpoint statements. Only the abstract is available for review; the detailed arguments (clean-kernel analysis, nine-block decomposition, triangular matrix theorem, rank-one endpoints) cannot be audited. Without those arguments it is impossible to confirm that the s-range is maximal or that the N&S list is complete and free of gaps when the signs of σ_0, σ_∞ and the relation q ≶ p vary.","section":"Abstract (claimed s-range and N&S classification)"},{"comment":"All subsequent kernel bounds and Lorentz estimates are derived under the standing external hypothesis of the two-sided ground-state heat-kernel estimate of Ishige–Kabeya–Ouhabaz. The abstract does not delineate the precise class of radial potentials V for which this hypothesis is known to hold, nor does it indicate whether any part of the N&S classification can be obtained under a weaker one-sided or on-diagonal heat-kernel bound. The scope and conditional character of the main theorem therefore remain incompletely specified from the material under review.","section":"Abstract (standing assumption)"}],"minor_comments":[{"comment":"The abstract is clearly written and lists the main tools, but the notation for the broken-power weights w_{α_0,α_∞} and the precise meaning of “one-sided weight equalities” and “same-side power/scale corner” are not expanded; a short parenthetical definition would help readers who encounter the result only via the abstract.","section":"Abstract"},{"comment":"The claimed range 0 < s < min{d, d−2σ_0, d−2σ_∞} is stated as maximal for the clean two-sided kernel estimate; it would be useful already in the abstract to indicate briefly whether the kernel bound fails (or merely loses cleanliness) at the endpoints s = d−2σ_0 or s = d−2σ_∞.","section":"Abstract"}],"recommendation":"uncertain","confidential_remarks":"This is an abstract-only review: the full manuscript was not available. The recommendation is therefore “uncertain” rather than a substantive accept/revise/reject decision. Once the full text is supplied, the load-bearing steps to re-examine are (i) the passage from the Ishige–Kabeya–Ouhabaz heat-kernel bound to the claimed maximal open s-interval for the clean kernel estimate, and (ii) the necessity arguments for every equality case in the Lorentz classification, especially scale equality when q < p and the simultaneous-corner restriction (u,v)=(1,∞). If those arguments are complete and correct, the result appears to be of high quality for a serious journal in harmonic analysis / mathematical physics."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"Punchline: under the Ishige–Kabeya–Ouhabaz two-sided ground-state heat-kernel bound, they pin the maximal open interval where the fractional kernel of a radial Schrödinger operator with inverse-square asymptotics stays clean, then give a full necessary-and-sufficient list for the broken-power Lorentz estimate, including signed ground-state exponents, q < p, one-sided weight equalities, both scale equalities, and the simultaneous corners.\n\nWhat is actually new is the package, not any single gadget. The clean-kernel range 0 < s < min{d, d−2σ₀, d−2σ∞} is stated as maximal, and the Lorentz side covers every equality case they list (u ≤ v even when q < p; input/output power endpoints force u=1 or v=∞; same-side corner only (1,∞)). That is more than a routine extension. The named tools—local Lorentz–HLS, rank-one endpoints, annular sequence spaces, triangular matrix theorem, nine-block decomposition—are the right ones for this program, and the abstract is precise about what is assumed and what is claimed.\n\nSoft spot, in proportion: we only have the abstract. Every sharpness statement rides on the external IKO hypothesis, and we cannot check whether the nine-block argument really closes the gaps when signs of σ₀, σ∞ and the relation q ≶ p vary, or whether necessity of every listed equality is proved inside that s-interval. That is not a manufactured flaw; it is simply the limit of an abstract-only read. The circularity burden is low—the range is not fitted to free parameters of the paper itself.\n\nThis is for people who already work on weighted inequalities for Schrödinger operators and fractional kernels with singular potentials. A serious referee in that circle should see the full text. I would send it to peer review rather than desk-reject; the claims are clear, the scope is complete within the subfield, and the standing assumption is standard. If the proofs hold, it is a useful reference classification. If they do not, the referee will find it. Either way it deserves the look.","headline":"Abstract-only: a complete N&S Lorentz classification under the IKO heat-kernel hypothesis, with a clean maximal s-range claim that looks like solid specialist work but cannot be audited yet.","tokens_in":3269,"tokens_out":549,"would_cite":false,"duration_ms":12041,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J10","42B20","46E30","47G40"],"pacs":[],"model":"grok-4.5","headline":"Fractional powers of radial Schrödinger operators with inverse-square asymptotics admit a clean two-sided kernel estimate and sharp broken-power Lorentz bounds on a maximal open range of exponents.","keywords":["Schrödinger operators","fractional powers","inverse-square potentials","Lorentz spaces","broken-power weights","heat kernel estimates","Hardy-Littlewood-Sobolev inequalities","radial potentials"],"falsifier":"Produce a radial potential V for which the Ishige–Kabeya–Ouhabaz heat-kernel bound holds, yet either the claimed two-sided kernel comparison for H^{−s/2} fails for some s inside the stated open interval, or the Lorentz estimate holds while one of the listed necessary-and-sufficient exponent conditions fails (or fails while all listed conditions hold).","tokens_in":3225,"feed_emoji":"⚡","tokens_out":1267,"duration_ms":20077,"temperature":0.7,"pith_summary":"The paper studies fractional powers of radial Schrödinger operators H = −Δ + V(|x|) on Euclidean space of dimension at least 2, where the positive harmonic function U of H behaves like one power near the origin and another power at infinity. Assuming a known two-sided bound on the ground-state heat kernel, the authors identify the largest open interval of fractional orders s for which the integral kernel of H^{−s/2} is comparable to an explicit expression built from U and the distance |x − y|. Within that range they supply a complete necessary-and-sufficient list of conditions on the Lorentz and power-weight exponents under which the operator maps one weighted Lorentz space into another. The classification covers signed ground-state exponents, the reverse-order range q < p, all one-sided weight equalities, both scale equalities, and simultaneous endpoint corners. A sympathetic reader cares because the result gives the precise mapping properties of these nonlocal operators under the most general broken-power weights compatible with the two-scale geometry of the potential.","feed_headline":"Sharp Lorentz bounds for fractional inverse-square Schrödinger powers","feed_subtitle":"A clean kernel formula yields necessary and sufficient conditions covering reverse order and all endpoint corners.","key_machinery":"The clean two-sided kernel estimate for H^{−s/2} written in terms of the ground-state harmonic function U; once available, this formula reduces the operator bounds to local Lorentz–Hardy–Littlewood–Sobolev estimates, rank-one endpoint arguments, geometric annular sequence spaces, a triangular matrix theorem, and a nine-block decomposition of the plane.","core_discovery":"Under the two-sided ground-state heat-kernel estimate of Ishige–Kabeya–Ouhabaz, the kernel of H^{−s/2} satisfies the clean two-sided estimate K_s^H(x,y) ≃ |x−y|^{s−d} U(|x|)U(|y|)/[U(|x|+|x−y|)U(|y|+|x−y|)] precisely when 0 < s < min{d, d−2σ₀, d−2σ∞}. In that same range the broken-power Lorentz estimate ‖w_{−β₀,−β∞} H^{−s/2} f‖_{L^{q,v}} ≲ ‖w_{α₀,α∞} f‖_{L^{p,u}} holds if and only if a complete list of necessary-and-sufficient conditions on the exponents is satisfied; the list includes all one-sided power equalities, both scale equalities (governed by u ≤ v even when q < p), and the simultaneous endpoint corners where the only admissible pair is (u,v) = (1,∞).","pith_inferences":["The nine-block decomposition and annular sequence spaces may transfer to other operators whose kernels obey similar two-scale asymptotic formulas, such as fractional Laplacians with potentials or Bessel operators.","Once the heat-kernel hypothesis is verified for a concrete potential class, the Lorentz estimates become unconditional, suggesting a program of checking the Ishige–Kabeya–Ouhabaz bound for larger families of inverse-square-type potentials.","The necessity of (u,v) = (1,∞) at simultaneous corners indicates critical sensitivity to logarithmic divergences at those endpoints, which could be tested numerically on model operators such as the pure inverse-square potential.","The clean-kernel range 0 < s < min{d, d−2σ₀, d−2σ∞} may also delimit related estimates such as weighted Sobolev embeddings or Riesz-potential inequalities associated with H."],"forward_implications":["The mapping properties of H^{−s/2} between weighted Lorentz spaces are completely classified for all admissible broken-power weights.","Scale equality is controlled solely by the relation u ≤ v between the Lorentz second indices, independently of whether q is larger or smaller than p.","An input power endpoint forces u = 1 and an output power endpoint forces v = ∞; at a same-side power/scale corner only the pair (1, ∞) is admissible.","The same kernel formula and exponent conditions apply to any radial Schrödinger operator whose heat kernel satisfies the Ishige–Kabeya–Ouhabaz bound.","Signed ground-state exponents σ₀, σ∞ inside (−d/2, d/2) are fully covered by the classification."],"fun_headline_variants":["Clean kernel formula gives sharp broken-power Lorentz bounds for H^{-s/2}","Necessary and sufficient Lorentz estimates for fractional inverse-square Schrödinger","Full range of broken-power Lorentz bounds under Ishige-Kabeya-Ouhabaz heat kernels","Endpoint-complete classification for Lorentz norms of radial Schrödinger fractional powers","Sharp s-range and (u,v) conditions for weighted Lorentz estimates of H^{-s/2}"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire argument takes as given the external two-sided ground-state heat-kernel estimate of Ishige, Kabeya and Ouhabaz for the operator H.","fun_headline_variants_meta":{"raw":{"variants":["Clean kernel formula gives sharp broken-power Lorentz bounds for H^{-s/2}","Necessary and sufficient Lorentz estimates for fractional inverse-square Schrödinger","Full range of broken-power Lorentz bounds under Ishige-Kabeya-Ouhabaz heat kernels","Endpoint-complete classification for Lorentz norms of radial Schrödinger fractional powers","Sharp s-range and (u,v) conditions for weighted Lorentz estimates of H^{-s/2}"]},"model":"grok-4.5","effort":"low","cost_usd":0.007288,"raw_usage":{"total_tokens":2042,"prompt_tokens":1145,"num_sources_used":0,"completion_tokens":110,"cost_in_usd_ticks":72880000,"prompt_tokens_details":{"text_tokens":1145,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":787,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":1145,"tokens_out":110,"duration_ms":5697,"temperature":1.0,"reasoning_tokens":787,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T02:16:13.266848+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Produce a radial potential V for which the Ishige–Kabeya–Ouhabaz heat-kernel bound holds, yet either the claimed two-sided kernel comparison for H^{−s/2} fails for some s inside the stated open interval, or the Lorentz estimate holds while one of the listed necessary-and-sufficient exponent conditions fails (or fails while all listed conditions hold).","supporting_citations":[],"review_version":1}