{"id":"c0611b55-8449-4d5c-9b01-4faf5384b057","arxiv_id":"2506.16958","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For Hermite-type Laguerre expansions, almost everywhere convergence of Bochner-Riesz means holds for all L^p functions when the summability order exceeds λ(p)/2, and the paper attempts to show failure below that threshold.","lead":"This paper studies when Bochner-Riesz means for Hermite-type Laguerre expansions converge to the original function almost everywhere. It proves that the sharp summability exponent is half the classical one, though the claimed sharpness proof has a gap.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.1's monotonicity assertion is backwards on a substantial parameter range: E_k is increasing, not decreasing, so |E_k|≥C0 does not yield |{S^λ_* f=∞}|≥C0. The converse half of Theorem 1.1 is unsupported.","rationale":"The reader's weakest assumption identifies the same load-bearing flaw in Proposition 4.1, and I agree it is decisive. The central theorem has two halves. The sufficiency half via the weighted maximal estimate (Theorem 1.2) is plausible, although Proposition 2.3 and Lemma 2.8 are quoted rather than proved. The converse half, however, is not established: the proof's limiting argument depends on the direction of monotonicity of E_k, and for d=2, d=3, and for d≥4 sufficiently close to the critical index, the threshold decreases instead of increasing. Thus E_k is increasing and the claimed convergence to {S^λ_* f=∞} fails. A finite positive value of the maximal function puts a point in all large E_k without making the maximal function infinite, so the measure bound |E_k|≥C0 cannot be transferred to (1.11). Since the theorem asserts sharp necessity for every λ<λ(p)/2, and the written proof covers at best a complementary parameter range, the manuscript as written does not support its central claim. I am not asserting the theorem is false; the issue is that the written argument has a concrete gap in the necessity proof. This warrants rejection of the paper in its current form, consistent with the reader's verdict.","tokens_in":32531,"tokens_out":12110,"duration_ms":118309,"concrete_test":"Re-run Proposition 4.1 in the concrete case d=2, p=8, λ=0.1 (so λ(p)=1/4 and λ<λ(p)/2). The threshold is c2^{-0.95k}. Verify the monotonicity claim: for any x with 0<S^λ_* f(x)<c, x belongs to all E_k for sufficiently large k, so the sequence is increasing and its limit is {S^λ_* f>0}, not {S^λ_* f=∞}. Then check whether the proof can still conclude |{S^λ_* f=∞}|≥C0 from |E_k|≥C0; without an additional limsup or intersection argument, no such implication holds. This isolates the exact step in Proposition 4.1 that must be repaired.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive problem is in Section 4, Proposition 4.1. After defining E_k={x∈E:S^λ_* f(x)≥c2^{-k} μ_k^{-λ+λ(p)/2}}, the proof asserts that 'since μ_k∼2^{2k} and λ<λ(p)/2, E_k is a decreasing sequence ... which converges to E={S^λ_* f=∞}'. The threshold is c2^{-k}μ_k^{-λ+λ(p)/2}=c2^{k(λ(p)-2λ-1)}. For d=2 and d=3 the exponent is always negative; for d≥4 it is negative on a nonempty interval below λ(p)/2, in particular near the critical index. On this range the threshold decreases to 0, so E_k increases to {S^λ_* f>0}, not to {S^λ_* f=∞}; the claimed intersection/limit argument is invalid. The lower bound |E_k|≥C0 obtained from (4.5) and (4.13) only says that each sublevel set is large. For an increasing family this does not imply positive measure of {S^λ_* f=∞}; a point with 0<S^λ_* f(x)<∞ lies in E_k for all large k. Hence (1.11) and the converse statement in Theorem 1.1 are not established for the stated parameter range. The sufficiency argument may well be sound, and Proposition 2.3 and Lemma 2.8 are also only referenced rather than proved, but the sharpness claim fails as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies almost everywhere convergence of Bochner-Riesz means for Hermite-type Laguerre operators L^α on R_+^d, with α∈[-1/2,∞)^d. The main result, Theorem 1.1, asserts that for 2≤p<∞ and λ≥0, the means S_R^λ(L^α)f converge to f a.e. for every f∈L^p whenever λ>λ(p)/2, and that convergence fails for λ<λ(p)/2 when 2d/(d-1)<p<∞, where λ(p)=max{d(1/2-1/p)-1/2,0}. The sufficiency is derived from a weighted L^2 maximal estimate (Theorem 1.2), which is proved through a square-function estimate (Proposition 3.1) using heat-kernel bounds, negative-power estimates, and a generalized trace lemma (Lemma 2.8). The necessity is approached by constructing functions from Laguerre generating functions, following the strategy of [21], and showing that the maximal operator is infinite on a set of positive measure.","tokens_in":32829,"tokens_out":7673,"duration_ms":75169,"significance":"If the result were correct, it would be a meaningful extension of the Hermite-operator results of [5] and the twisted-Laplacian results of [21] to the full Laguerre parameter range, showing that the critical summability index for a.e. convergence is λ(p)/2 and that inverse-square potential perturbations do not alter it. The paper contains useful technical contributions, particularly the detailed kernel estimates for negative powers of L^α via the heat semigroup and the derivation of the critical index without free parameters. However, the necessity proof contains a monotonicity error that invalidates the converse statement of Theorem 1.1 as written, and the sufficiency argument rests on several substantial lemmas whose proofs are omitted. The advertised sharpness claim is therefore not established.","major_comments":[{"comment":"The assertion that the sets E_k are decreasing and converge to E={S_*^λ f=∞} is incorrect. With μ_k∼2^{2k}, the threshold defining E_k is c 2^{-k} μ_k^{-λ+λ(p)/2}=c 2^{k(λ(p)-2λ-1)}. For d=2 and d=3 the exponent λ(p)-2λ-1 is always negative when λ<λ(p)/2; for d≥4 it is negative for λ near λ(p)/2 but positive for small λ (e.g., d=4, p=∞, λ=0 gives exponent 1/2). When the exponent is negative, the threshold decreases with k, so E_k is increasing, not decreasing, and the union of the E_k is contained in {S_*^λ f>0}, not in {S_*^λ f=∞}. Consequently the estimate |E_k|≥C_0 obtained from (4.5) and the inclusion (4.13) does not imply the positive-measure divergence set (1.11). This invalidates the necessity half of Theorem 1.1 for the full stated range.","section":"Section 4, Proof of Proposition 4.1"},{"comment":"The quantity A^ε_{β,d}(δ) is defined only for 0≤β≤1 when d=1 and for 1<β<d when d≥2, yet Proposition 3.1 and Theorem 1.2 are stated for all 0≤β<d. In particular, for d≥2 and 0≤β≤1, the case needed for the embedding L^p↪L^2((1+|x|)^{-β}) when p is close to 2, the definition gives no value. The proof mentions that the range is extended by interpolation, but no definition of A^ε_{β,d}(δ) on the missing interval is supplied, leaving a central statement incomplete.","section":"Section 3.1, Eq. (3.4)"},{"comment":"Lemma 2.8 (the generalized trace lemma) is stated with its proof omitted entirely, with only the remark that the proof in [5] extends to this setting; Proposition 2.3 (the Littlewood-Paley inequality) is likewise only referenced. These results are load-bearing: Lemma 2.8 is used to prove the square-function estimates (3.13) and (3.14), and Proposition 2.3 is used to close (3.41). The adaptation to the Laguerre operator with inverse-square potentials and general α∈[-1/2,∞)^d is not a purely notational change, so the sufficiency argument currently rests on unverified assertions.","section":"Section 2, Lemma 2.8 and Proposition 2.3"}],"minor_comments":[{"comment":"In the formula for L^α, the index j is used in (α_j^2-1/4) while the denominator is x_i^2; the index should be i.","section":"Abstract and Section 1"},{"comment":"The notation |α|_1 for the coordinate sum Σ α_i, which may be negative, is nonstandard and easily confused with an L^1 norm; a different symbol would avoid ambiguity.","section":"Section 1.2"},{"comment":"The sentence 'we thus deduce from (3.31), (3.37) and (3.37)' should refer to (3.31), (3.36), and (3.37).","section":"Section 3.4, after (3.37)"},{"comment":"The exponent in (4.6) writes |α| without a subscript; it should be |α|_1.","section":"Lemma 4.2, Eq. (4.6)"},{"comment":"The reference '[35, 41, Chapter]' is incomplete; a precise chapter or theorem number is needed.","section":"Section 2, Eq. (2.4)"}],"recommendation":"reject","confidential_remarks":"The necessity proof's monotonicity error is not a minor patch: the set-inclusion argument points in the wrong direction for the entire parameter range d=2,3 and for a substantial range in d≥4, so the converse of Theorem 1.1 is unsupported. The sufficiency may be salvageable, but it depends on omitted proofs of key lemmas. I recommend rejection, though the kernel estimates in Section 2 may be useful for a future revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThe thing to know: this paper has a genuine new sufficiency result, but its sharpness claim is not supported. The sufficiency half (Theorem 1.2, the weighted maximal estimate) adapts the Chen–Duong–He–Lee–Yan machinery to the full parameter range α∈[−1/2,∞)^d by proving kernel estimates for (1+L^α)^{−β} through the heat semigroup. That is real work and a real extension: previous Laguerre results were one-dimensional and non-sharp, and the inverse-square perturbation range is handled without the Hardy-inequality or oscillatory-integral constraints. The conclusion that the discrete spectrum cuts the summability index in half, matching the Hermite and twisted-Laplacian cases, is a genuine contribution.\n\nThe problem is Section 4. In Proposition 4.1, E_k is defined with threshold c·2^{−k}·μ_k^{−λ+λ(p)/2}. With μ_k∼2^{2k} and λ<λ(p)/2, the threshold is c·2^{k(λ(p)−2λ−1)}. The paper asserts E_k is decreasing to {S_*^λ f = ∞}. That is only true when the exponent is positive. For d=2, and for λ near λ(p)/2 in higher dimensions, the exponent is negative; the threshold decays, E_k is increasing, and the limit is {S_*^λ f > 0}, not {S_*^λ f = ∞}. The lower bound |E_k|≥C0 then does not imply divergence on a positive-measure set. So the converse half of Theorem 1.1 is not established for a substantial parameter range. This is not a minor gap; it is the entire necessity argument.\n\nThere are also two technical lemmas (Proposition 2.3 and Lemma 2.8) stated without proof and deferred to [5]. Since the sufficiency argument leans on them, a referee will need to verify they really transfer. The definition of A^ε_{β,d}(δ) in (3.4) is also incomplete in overlapping cases. The citation pattern is fine — the paper builds on [5] and [21] with proper attribution, and self-citation is not an issue.\n\nBottom line: the sufficiency half looks plausible and is worth taking seriously, but as written the paper overclaims. I would send it to a serious referee, with the clear expectation that the necessity part either be fixed or removed, and the omitted proofs supplied. If the converse is dropped or corrected, the sufficiency result alone is publishable. For now, the sharp index claim should not be quoted.","headline":"Plausible new sufficiency result for a.e. convergence of Laguerre Bochner-Riesz means, but the claimed sharp converse rests on a monotonicity error and needs major revision before the sharpness claim is credible.","tokens_in":33415,"tokens_out":3850,"would_cite":true,"duration_ms":36361,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B25","42B15","42C10","35P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Hermite-type Laguerre expansions, Bochner-Riesz means converge almost everywhere precisely when the summability exponent exceeds λ(p)/2 — half the classical L^p index.","keywords":["Bochner-Riesz means","almost everywhere convergence","Laguerre expansions","Hermite type Laguerre operator","maximal operator","square function estimates","critical summability index","inverse-square potential"],"falsifier":"Compute T_k=$c2^{{-k}}$$μ_k^{{-λ+λ(p)/2}}$ with μ_k∼$2^{{2k}}$: T_k=$c2^{{-k(λ(p)-2λ+1)}}$→0, so the sets E_k grow with k rather than shrink; if a direct evaluation of sup_R |S_R^λ f| on E for the constructed f shows it is finite a.e. for some λ<λ(p)/2, the sharpness claim is false.","tokens_in":32236,"feed_emoji":"🎯","tokens_out":8609,"duration_ms":84713,"temperature":0.7,"pith_summary":"The paper aims to determine the exact Bochner-Riesz summability exponent at which Laguerre expansions of Hermite type converge to a function pointwise almost everywhere. Its main claim is that for 2≤p<∞ and every f∈L^p(ℝ_+^d), the means S_R^λ(L^α)f converge a.e. whenever λ>λ(p)/2, and that convergence can fail when λ<λ(p)/2 for 2d/(d−1)<p<∞. The critical value λ(p)/2 is half of the classical L^p-critical index λ(p), a gain that the paper attributes to the discrete spectrum of L^α being bounded away from zero. If correct, the result also shows the sharp index is stable under inverse-square potential perturbations of the harmonic oscillator.","feed_headline":"Half the usual summability index suffices for Laguerre a.e. convergence","feed_subtitle":"For Hermite-Laguerre expansions, pointwise convergence holds above λ(p)/2 — half the L^p summability threshold.","key_machinery":"The engine of the proof is the maximal Bochner-Riesz operator S_*^λ(L^α)f(x)=sup_{R>0}|S_R^λ(L^α)f(x)|, bounded on the weighted space $L^{2}$(ℝ_+^d,(1+|x|)^{−β}) when λ>max{(β−1)/4,0}. To prove that bound, the paper decomposes the means into dyadic frequency pieces and reduces the maximal operator to a square function; two auxiliary estimates carry the argument, a trace-type bound on the spectral projections P_n^α with weight (1+|x|)^{−β}, and a weighted bound for the negative powers (1+L^α)^{−β} obtained from the heat kernel of L^α. The lower bound uses functions built from the Laguerre generating function so that their spectral projections are concentrated on a fixed annulus, mimicking eigenfunctions of L^α there.","core_discovery":"The central discovery is a sharp threshold phenomenon for almost everywhere convergence of Bochner-Riesz means associated with the Hermite-Laguerre operator L^α=−Δ+|x|^2+∑($α_j^{2}$−1/4)/$x_j^{2}$ on the half-space ℝ_+^d. Theorem 1.1 asserts that for α∈[−1/2,∞)^d and 2≤p<∞, lim_{R→∞} S_R^λ(L^α)f=f a.e. for every f∈L^p whenever λ>λ(p)/2, while for p>2d/(d−1) and λ<λ(p)/2 there are functions whose maximal means are infinite on a set of positive measure. The sufficiency is proved through a weighted $L^{2}$ maximal estimate (Theorem 1.2), and the paper argues the index λ(p)/2 is sharp by constructing counterexamples from the generating function of Laguerre polynomials. The paper's framing is that the inverse-square potential is a small perturbation of the harmonic oscillator in the spectral sense, leaving the critical a.e. summability index unchanged.","pith_inferences":["The proof's necessity part, as printed, relies on the sets E_k being decreasing; with the stated threshold and μ_k∼2^{2k} they are increasing, so the negative-index conclusion needs an additional argument beyond what is written.","A repaired necessity argument might replace the decreasing-family claim by a limsup of increasing sets, which could still prove divergence if the union has positive measure.","If the same half-index mechanism holds for other Schrödinger operators with discrete spectra bounded away from zero, the trace-and-square-function route may transfer to more general confining potentials; this extrapolation is not in the paper.","The endpoint λ=λ(p)/2 is left open: the stated results concern strict inequality and do not decide convergence at the critical index itself."],"forward_implications":["For d=1 the theorem gives a.e. convergence for every λ>0 and every p≥2, with no gap between sufficiency and necessity since λ(p)=0.","For all α∈[−1/2,∞)^d and p≥2, the summability index for a.e. convergence is λ(p)/2, exactly half the index needed for L^p convergence of the same means.","Taking α_j=±1/2 recovers the known Hermite-operator result and shows the index is insensitive to turning on inverse-square potentials.","The weighted maximal estimate yields a.e. convergence for weighted L^2, then by embedding for all L^p with p≥2.","When λ<λ(p)/2 and p>2d/(d−1), there exist L^p functions for which the maximal means blow up on a set of positive measure, so no smaller index can work."],"supporting_citations":[{"why":"Supplies the maximal-operator and square-function framework, the trace lemma for spectral projections, and the weighted estimates adapted to the Hermite-Laguerre setting.","marker":"[5]"},{"why":"Establishes the classical sharp a.e. summability exponent and the high-frequency square-function technique reused in Lemma 3.2.","marker":"[10]"},{"why":"Provides the counterexample construction via Schrödinger propagator kernels and the inequality connecting spectral projections to maximal means.","marker":"[21]"},{"why":"Gives the heat-kernel Gaussian bound and the integral representation of the heat semigroup used throughout the paper.","marker":"[33]"},{"why":"Supplies kernel estimates for negative powers (1+L^α)^{−β} that yield the weighted estimate in Lemma 2.4.","marker":"[34]"},{"why":"Provides the pointwise asymptotic for Laguerre functions (Lemma 2.7) and the Hermite/Laguerre transplantation background.","marker":"[47]"},{"why":"Gives the Schrödinger propagator kernel formula for L^α, motivating the generating-function counterexample construction.","marker":"[48]"},{"why":"Justifies reducing maximal means to square functions via the inequality used at the start of the proof of Theorem 1.2.","marker":"[36]"}],"fun_headline_variants":["Sharp a.e. convergence threshold for Hermite-Laguerre means","Bochner-Riesz means for Laguerre: a.e. convergence at half index","Laguerre expansions: a.e. convergence at λ(p)/2 sharp","Hermite-Laguerre Bochner-Riesz: pointwise convergence at half index","A.e. convergence for Laguerre means at critical index λ(p)/2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The necessity proof assumes the sets E_k = {x∈E: S_*^λ f(x) ≥ c $2^{{-k}}$ $μ_k^{{-λ+λ(p)/2}}$} form a decreasing family shrinking to {S_*^λ f = ∞}; with μ_k∼$2^{{2k}}$ and λ<λ(p)/2 this threshold actually decreases, so the sets increase and the intersection argument as written does not yield divergence.","fun_headline_variants_meta":{"raw":{"variants":["Sharp a.e. convergence threshold for Hermite-Laguerre means","Bochner-Riesz means for Laguerre: a.e. convergence at half index","Laguerre expansions: a.e. convergence at λ(p)/2 sharp","Hermite-Laguerre Bochner-Riesz: pointwise convergence at half index","A.e. convergence for Laguerre means at critical index λ(p)/2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000285,"raw_usage":{"total_tokens":1781,"prompt_tokens":1151,"completion_tokens":630,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":767,"completion_tokens_details":{"reasoning_tokens":525}},"tokens_in":767,"tokens_out":630,"duration_ms":6033,"temperature":1.0,"reasoning_tokens":525,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:18:28.087371+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute T_k=$c2^{{-k}}$$μ_k^{{-λ+λ(p)/2}}$ with μ_k∼$2^{{2k}}$: T_k=$c2^{{-k(λ(p)-2λ+1)}}$→0, so the sets E_k grow with k rather than shrink; if a direct evaluation of sup_R |S_R^λ f| on E for the constructed f shows it is finite a.e. for some λ<λ(p)/2, the sharpness claim is false.","supporting_citations":[{"cited_title":"T., He D., Lee S., Yan L.: Almost everywhere convergence of Bochner–Riesz means for the Hermite operators","cited_arxiv_id":null,"evidence_quote":"Supplies the maximal-operator and square-function framework, the trace lemma for spectral projections, and the weighted estimates adapted to the Hermite-Laguerre setting."},{"cited_title":"L., Vega, L.: Almost everywhere summability of Fourier integrals","cited_arxiv_id":null,"evidence_quote":"Establishes the classical sharp a.e. summability exponent and the high-frequency square-function technique reused in Lemma 3.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the counterexample construction via Schrödinger propagator kernels and the inequality connecting spectral projections to maximal means."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the heat-kernel Gaussian bound and the integral representation of the heat semigroup used throughout the paper."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies kernel estimates for negative powers (1+L^α)^{−β} that yield the weighted estimate in Lemma 2.4."},{"cited_title":"Princeton Univ","cited_arxiv_id":null,"evidence_quote":"Provides the pointwise asymptotic for Laguerre functions (Lemma 2.7) and the Hermite/Laguerre transplantation background."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Schrödinger propagator kernel formula for L^α, motivating the generating-function counterexample construction."},{"cited_title":"M., Weiss G.: Introduction to Fourier Analysis on Euclidean Spaces","cited_arxiv_id":null,"evidence_quote":"Justifies reducing maximal means to square functions via the inequality used at the start of the proof of Theorem 1.2."}],"review_version":1}