{"id":"321d2890-7d80-46d1-bf76-8711d06f213d","arxiv_id":"2606.08939","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"New weighted Gaussian Poincaré inequalities with sharp constants enable complete L2 and Lp stability characterizations for CKN inequalities over the entire known parameter regime.","lead":"The paper develops a new family of weighted Gaussian L2-Poincaré inequalities with explicit sharp constants, optimizers, and stability estimates by using Laguerre polynomial expansions, spherical harmonics, and a Kelvin-type transform. This yields a complete characterization of sharp L2 stability for Caffarelli-Kohn-Nirenberg inequalities across the full parameter range, plus Lp extensions.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption correctly isolates the unverified sufficiency of the new analytic tools; absent the full proofs, no stronger objection can be formulated. The proposed check would directly test whether the transform preserves equality cases across the claimed range.","tokens_in":1754,"tokens_out":210,"duration_ms":14554,"concrete_test":"Extract the explicit form of the Kelvin-type transform and the associated change-of-variables Jacobian from the manuscript; substitute the known CKN optimizers into the transformed inequality and check whether equality is attained for at least one interior and one boundary parameter tuple.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract describes a method (generalized Laguerre expansions + spherical harmonics + Kelvin transform) tailored to singular weights, which directly targets the gap left by prior special-case results. No internal inconsistency or hidden assumption is detectable from the given material; the claimed extension to the full parameter regime is consistent with the stated technical ingredients.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper introduces a new family of weighted Gaussian L²-Poincaré-type inequalities with explicit sharp constants, optimizers, and sharp L²-gradient stability estimates. The approach relies on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform to handle singular weights. As an application, the work completely characterizes the stability of the L²-Caffarelli-Kohn-Nirenberg inequalities throughout the full parameter range (previously limited to special cases) and establishes weighted L^p-Poincaré inequalities for all p>1 together with stability estimates for the L^p-CKN inequalities when p≥2 in the regime where sharp constants and optimizers are known.","tokens_in":1817,"tokens_out":355,"duration_ms":19065,"significance":"If the central claims hold, the manuscript delivers a complete characterization of sharp L² stability for the CKN family across the entire admissible parameter regime, a clear advance over prior special-case results. The explicit sharp constants, optimizers, and the stability-of-stability estimates constitute concrete strengths. The new method tailored to singular weights via orthogonal expansions and the Kelvin transform is a technical contribution that directly addresses a gap in the literature on weighted Poincaré inequalities.","major_comments":[],"minor_comments":[{"comment":"The phrase 'stability of the stability inequality results' in the abstract is concise but may benefit from a brief parenthetical clarification or forward reference to the relevant section when first introduced in the introduction.","section":null},{"comment":"Consider adding a short table or diagram in the introduction that summarizes the admissible parameter ranges for the L² and L^p results (including the previously known special cases) to help readers quickly locate the extension achieved.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive summary and recommendation of minor revision. No major comments were provided in the report.","responses":[],"tokens_in":1295,"tokens_out":42,"duration_ms":9825,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that this work supplies the complete sharp stability picture for L2-CKN inequalities across the full parameter range, plus weighted Gaussian Poincaré inequalities with explicit constants and some Lp extensions for p ≥ 2. Earlier results covered only scattered special cases, so this closes a noticeable gap for tools that show up often in PDE estimates.\n\nThe new method—generalized Laguerre expansions combined with spherical harmonics and a Kelvin transform—targets the singular weights where classical Gaussian techniques break down. That choice makes sense for the problem and appears to avoid circularity by building from orthogonal expansions rather than assuming the target constants.\n\nWhat stands out is the breadth: they get both the stability estimates and the stability of those estimates, plus the Lp versions where sharp constants were already known. If the derivations hold, this would be adopted in applications.\n\nThe soft spot is verification. The abstract states the method works throughout the regime, but without the actual expansions and transform calculations in front of us it is difficult to confirm there are no gaps in the most singular parameter corners or that the constants really stay sharp. Minor issues could appear in the error estimates or in how the spherical harmonic decomposition interacts with the weights.\n\nThis is for people who use CKN or Gaussian-type inequalities in analysis and PDE. A reader working on stability or sharp constants would find the new family and the full-range results directly useful.\n\nIt deserves peer review. The gap is real and the approach is tailored to it; a referee can check the technical steps without the paper being obviously flawed on its face.","headline":"The paper claims the first full sharp L2 stability for CKN inequalities over all parameters via a new Laguerre-Kelvin method, but the details need checking.","tokens_in":2300,"tokens_out":396,"would_cite":false,"duration_ms":18000,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Weighted Gaussian Poincaré inequalities with singular weights yield sharp stability for the full range of Caffarelli-Kohn-Nirenberg inequalities.","keywords":["Caffarelli-Kohn-Nirenberg inequalities","weighted Gaussian Poincaré inequalities","sharp stability estimates","Laguerre polynomial expansions","Kelvin-type transform","spherical harmonic decompositions","singular weights"],"falsifier":"A concrete counterexample consisting of a specific admissible parameter tuple together with a function for which the derived stability constant is strictly larger than the one claimed, or for which no optimizer attains the constant.","tokens_in":2657,"feed_emoji":"","tokens_out":777,"duration_ms":20532,"temperature":0.7,"pith_summary":"The paper constructs a new family of weighted Gaussian L²-Poincaré inequalities that admit explicit sharp constants, optimizers, and gradient stability estimates even when the weights are singular. These inequalities extend the classical Gaussian Poincaré inequality and serve as the vehicle for a complete characterization of stability in the L²-Caffarelli-Kohn-Nirenberg inequalities across every admissible parameter tuple, including the stability of the stability statements themselves. The same framework produces weighted L^p-Poincaré inequalities for all p greater than 1 and stability estimates for the L^p-CKN inequalities when p is at least 2, in every regime where sharp constants are already known. Earlier results had covered only isolated special cases of the parameter space.","feed_headline":"Full stability of CKN inequalities now characterized with sharp estimates","feed_subtitle":"New weighted Gaussian Poincaré family supplies explicit constants and covers the entire parameter range for both L2 and Lp versions.","key_machinery":"The method of generalized Laguerre polynomial expansions combined with spherical harmonic decompositions and a Kelvin-type transform, which produces the sharp constants, optimizers, and stability estimates for the singular weights in every admissible parameter regime.","core_discovery":"The authors introduce a new family of weighted Gaussian L²-Poincaré-type inequalities with explicit sharp constants, optimizers, and corresponding sharp L²-gradient stability estimates. This family substantially extends the classical Gaussian Poincaré inequality. Owing to the singular nature of the weights involved, standard approaches to classical Gaussian Poincaré inequalities do not apply. To overcome this difficulty, they develop a new method based on generalized Laguerre polynomial expansions, spherical harmonic decompositions, and a Kelvin-type transform. As an application, they completely characterize the stability of the L²-Caffarelli-Kohn-Nirenberg inequalities by establishing sharp","pith_inferences":["The same expansion technique may apply to other families of inequalities with radial singular weights that currently lack stability results.","Explicit knowledge of the optimizers could be used to test numerical approximations of the inequalities in high-dimensional or non-radial settings.","The full-parameter stability may allow quantitative control of the deficit in related Sobolev-type embeddings that rely on CKN as an intermediate step."],"forward_implications":["Sharp L²-gradient stability estimates hold for the CKN inequalities in every parameter regime.","The stability statements themselves satisfy their own stability estimates.","Weighted L^p-Poincaré inequalities exist for every p greater than 1.","Stability estimates for the L^p-CKN inequalities hold for all p at least 2 whenever sharp constants and optimizers are known."],"fun_headline_variants":["Sharp stability of CKN inequalities characterized for all parameters","Weighted Gaussian Poincare inequalities provide CKN sharp stability","CKN L2 stability complete with new family of weighted inequalities","Lp stability extensions for CKN inequalities now fully sharp"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The combination of generalized Laguerre polynomial expansions, spherical harmonic decompositions, and Kelvin-type transform is sufficient to produce sharp constants, optimizers, and stability estimates for the singular weights and for every admissible parameter tuple in the CKN family.","fun_headline_variants_meta":{"raw":{"variants":["Sharp stability of CKN inequalities characterized for all parameters","Weighted Gaussian Poincare inequalities provide CKN sharp stability","CKN L2 stability complete with new family of weighted inequalities","Lp stability extensions for CKN inequalities now fully sharp"]},"model":"grok-4.3","cost_usd":0.005669,"raw_usage":{"total_tokens":2741,"prompt_tokens":732,"num_sources_used":0,"completion_tokens":63,"cost_in_usd_ticks":56687000,"prompt_tokens_details":{"text_tokens":732,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1946,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":732,"tokens_out":63,"duration_ms":12236,"temperature":1.0,"reasoning_tokens":1946,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T16:04:25.678422+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A concrete counterexample consisting of a specific admissible parameter tuple together with a function for which the derived stability constant is strictly larger than the one claimed, or for which no optimizer attains the constant.","supporting_citations":[],"review_version":1}