{"id":"4802d3e9-dce4-46d3-a204-4260b438bbad","arxiv_id":"2607.09585","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For the inverse-square Schrödinger operator, the two-power-weight bound for H_a^{-s/2} holds for 1<p,q<∞ precisely under ordering, scaling, sum, and strict origin conditions, with Lorentz replacements at the origin-critical boundaries.","lead":"The paper finds the exact range of power weights for which a fractional integral of the inverse-square Schrödinger operator maps one weighted L^p space into another. Specialists in harmonic analysis of singular potentials can use the sharp conditions and Lorentz endpoint replacements without re-deriving them.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates the sole external dependency (kernel comparison + uniqueness of extension) and correctly judges the remainder of the paper to be elementary and checkable line-by-line. My re-examination of the necessity tests, the three-region bounds, the Lorentz pairing, and the appendix multi-block construction finds no additional soft spot that would move the verdict. The concrete verification step above simply reconfirms the internal arithmetic of the Stein–Weiss reduction; it is expected to pass. Consequently the ACCEPT verdict with high confidence stands unchanged.","tokens_in":8930,"tokens_out":461,"duration_ms":5535,"concrete_test":"Independently verify that the three Stein–Weiss applications in the sufficiency proof of Theorem 2.1 (with parameters (s,α,β), (s+σ,α,β+σ), (s+σ,α+σ,β)) each satisfy the classical conditions (4.8) whenever (1.6)–(1.9) hold, and that the resulting operator sum is dominated by the model kernel via (4.7); if any parameter triple violates A+B≥0 or the individual upper bounds, the decomposition argument fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorems 2.1–2.3) is elementary real analysis once the Killip–Miao–Visan–Zhang–Zheng two-sided kernel comparison is granted. Necessity follows from standard scaling, origin-concentration, far-field translation, and multi-block tests (Sections 3 and A); sufficiency is a three-region decomposition into classical Stein–Weiss pieces (Lemma 4.1 + (4.7)–(4.10)). The only external black box is precisely the one the Reader flags: transfer of the model-kernel range to the spectral operator via the known pointwise comparison plus uniqueness of the weighted extension on a spectral core. That step is standard, explicitly stated, and does not introduce an internal inconsistency or hidden assumption that would collapse the argument if the comparison holds. No further load-bearing gap appears in the necessity/sufficiency logic, the Lorentz pairing, or the Hardy-critical specialization.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper determines the complete strong two-power-weight mapping range of the fractional integral operator associated with the Friedrichs extension Ha = −Δ + a|x|−2 in the attractive Hardy range. Starting from the known two-sided pointwise kernel comparison of Killip–Miao–Visan–Zhang–Zheng, it proves that || |x|−β Ha−s/2 f ||_Lq ≲ || |x|α f ||_Lp holds for 1 < p, q < ∞ if and only if p ≤ q, the scaling relation 1/q = 1/p + (α + β − s)/d, the sum condition α + β ≥ 0, and the strict origin conditions α + σ < d/p′, β + σ < d/q (Theorems 2.1–2.2). At either origin-critical equality the strong and weighted weak-type estimates fail, while the Lorentz replacement L^{p,1} \to L^{q,∞} holds (Theorem 2.3). Weighted Sobolev consequences and the Hardy-critical Friedrichs case are recorded separately.","tokens_in":9068,"tokens_out":971,"duration_ms":7087,"significance":"The result supplies an explicit, sharp five-condition characterization for a scale-invariant operator whose kernel carries simultaneous near-diagonal and origin singularities. Necessity is obtained from four concrete test-function constructions plus a multi-block argument for p ≤ q; sufficiency reduces the model kernel to three classical Stein–Weiss pieces via a transparent three-region decomposition. The Lorentz endpoint analysis and the distinction between the full-space sum condition and the weaker radial sum condition of Nowak–Stempak are clean and useful. The paper makes no exaggerated claims about heat kernels or spectral multipliers and correctly isolates the transfer from the model kernel to the spectral operator as the sole external black box. Within the literature on inverse-square Schrödinger operators this is a solid, self-contained contribution of permanent reference value.","major_comments":[],"minor_comments":[{"comment":"In the abstract and title the operator is written with a straight double quote (Schr\"odinger); replace by the proper umlaut or LaTeX \\\"o throughout for consistency with the body text.","section":"Abstract / title"},{"comment":"Section 5.1, display (5.4): the identification |x|−β−σ ∫ |y|^{s+σ−d} |f(y)| dy = |x|−d/q ∫ |y|−d/p′ g(y) dy relies on the scaling relation (5.3); a one-line reminder that α = s + σ − d/p would make the equality immediate for the reader.","section":"Section 5.1"},{"comment":"Appendix A, after (A.7): the phrase “Because rj = c Rj, the expression … is bounded below by a positive constant independent of j” is correct, but inserting the explicit factor c^{s−d/p+d/q} would make the independence of j completely transparent.","section":"Appendix A"},{"comment":"References [9] (Sun–Wang) is listed as appearing in 2026; if the paper is still only on arXiv, a note “arXiv preprint” would avoid a future citation mismatch.","section":"References"}],"recommendation":"accept","confidential_remarks":"The manuscript is short, carefully written, and free of load-bearing gaps once the Killip et al. kernel comparison is granted. It is a natural fit for a classical-analysis or harmonic-analysis journal; I see no reason to request further external review."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"This is a short, careful harmonic-analysis note that finishes a concrete mapping question. Starting from the Killip–Miao–Visan–Zhang–Zheng two-sided kernel comparison, the authors give the complete strong two-power-weight range for H_a^{-s/2} (and the model kernel) under five explicit conditions: p≤q, the usual scaling, α+β≥0, and the two strict origin restrictions involving σ. At either origin boundary they show strong and weighted weak-type fail while L^{p,1}\to L^{q,∞} holds. That is the actual new content.\n\nWhat they do well is keep the argument elementary and self-contained. Necessity is four standard tests (scaling, input/output origin concentration, far-field translation) plus a multi-block construction for p≤q in the appendix. Sufficiency is a three-region split that reduces to three classical Stein–Weiss pieces. The full-space sum condition α+β≥0, forced by translated balls and invisible to radial tests, is the clearest improvement over the Nowak–Stempak Hankel result they cite. The Lorentz endpoints and the short Hardy-critical discussion are cleanly handled. They flag the black-box transfer from model kernel to spectral operator and claim no new heat-kernel or multiplier theory.\n\nSoft spots are minor and proportional. Everything rides on the Killip et al. comparison plus uniqueness of the weighted extension on a spectral core; that is standard and explicitly stated, not hidden. Novelty is moderate: the pieces are classical, the range is the contribution. Significance stays inside weighted inequalities for inverse-square operators. No free parameters, no circularity, citations look honest.\n\nThis is for people who already work with singular potentials or two-weight fractional integrals. A serious referee should see it; the math checks line-by-line. I would cite the range statement when I need the precise conditions, and I would accept it for peer review without hesitation.","headline":"Clean, elementary sharp-range note for a known inverse-square kernel; solid and citeable within the subfield, nothing more.","tokens_in":9713,"tokens_out":527,"would_cite":true,"duration_ms":4752,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J10","42B35","35P05","46E30"],"pacs":[],"model":"grok-4.5","headline":"The paper gives the exact range of exponents for which two power-weighted fractional integrals of the inverse-square Schrödinger operator are bounded, and shows what replaces the bound at the origin-critical endpoints.","keywords":["inverse-square Schrödinger operator","fractional integral","Hardy–Littlewood–Sobolev inequality","Stein–Weiss inequality","power weights","Lorentz spaces","two-weight estimates"],"falsifier":"Construct a test function that saturates one of the four necessary conditions (for example a truncated power near the origin that forces α + σ = d/p′) and check whether the weighted Lq norm of the fractional integral stays bounded by a constant multiple of the weighted Lp norm; any counter-example that remains bounded would refute the claimed necessity.","tokens_in":9788,"feed_emoji":"📐","tokens_out":818,"duration_ms":6885,"temperature":0.7,"pith_summary":"The paper classifies when the fractional integral of the inverse-square Schrödinger operator maps one power-weighted Lebesgue space into another. Starting from a known two-sided pointwise formula for the integral kernel, it proves that the strong estimate holds for 1 < p, q < ∞ if and only if four conditions are satisfied: the output integrability cannot be better than the input, the usual scaling relation that balances the order of the integral against the weight exponents, a non-negative sum of the two weight exponents, and two strict origin conditions that keep the weights from being too singular at zero. At either origin-critical equality the strong bound and the ordinary weak-type bound both fail, but a Lorentz-space replacement still holds. The same range is transferred from a model kernel to the actual spectral operator, and weighted Sobolev inequalities are recorded as immediate consequences. The result is of interest because the inverse-square potential is scale-invariant and already produces singular factors at the origin, so the classical Stein–Weiss theory must be adjusted by three separate pieces of the kernel.","feed_headline":"Exact power-weight range for inverse-square fractional integrals","feed_subtitle":"Four conditions decide the strong map; Lorentz spaces save the origin-critical endpoints.","key_machinery":"The model kernel Ks,σ, which is the right-hand side of the known two-sided comparison for the integral kernel of Ha−s/2. It is split into three disjoint geometric regions (near-diagonal, output-origin, input-origin); each piece is then controlled by a classical Stein–Weiss inequality, and the same tests that prove necessity for the model transfer to the spectral operator by the lower comparison.","core_discovery":"For the Friedrichs operator Ha = −∆ + a|x|−2 in the attractive Hardy range, the two-weight estimate || |x|−β Ha−s/2 f ||_Lq ≲ || |x|α f ||_Lp holds for 1 < p, q < ∞ precisely when p ≤ q, 1/q = 1/p + (α + β − s)/d, α + β ≥ 0, α + σ < d/p′ and β + σ < d/q. At either origin-critical boundary the strong and weighted weak-type estimates fail while the Lorentz estimate L^{p,1} → L^{q,∞} remains true.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Sharp two-weight range for inverse-square fractional integrals of Ha","Four conditions set strong maps for Ha^{-s/2} power weights","Origin-critical endpoints fail strong but hold in Lorentz spaces","Exact p-q range for two-weight estimates of inverse-square Schrödinger","Weighted Sobolev consequences from Ha fractional integral bounds"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The entire range for the spectral operator is transferred from a model kernel by a known two-sided pointwise kernel comparison; if that comparison fails for some admissible a or s, the operator statement collapses while the model-kernel statement remains.","fun_headline_variants_meta":{"raw":{"variants":["Sharp two-weight range for inverse-square fractional integrals of Ha","Four conditions set strong maps for Ha^{-s/2} power weights","Origin-critical endpoints fail strong but hold in Lorentz spaces","Exact p-q range for two-weight estimates of inverse-square Schrödinger","Weighted Sobolev consequences from Ha fractional integral bounds"]},"model":"grok-4.5","effort":"low","cost_usd":0.005276,"raw_usage":{"total_tokens":1532,"prompt_tokens":883,"num_sources_used":0,"completion_tokens":71,"cost_in_usd_ticks":52760000,"prompt_tokens_details":{"text_tokens":883,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":578,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":883,"tokens_out":71,"duration_ms":8846,"temperature":1.0,"reasoning_tokens":578,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T01:56:44.284650+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct a test function that saturates one of the four necessary conditions (for example a truncated power near the origin that forces α + σ = d/p′) and check whether the weighted Lq norm of the fractional integral stays bounded by a constant multiple of the weighted Lp norm; any counter-example that remains bounded would refute the claimed necessity.","supporting_citations":[],"review_version":1}