{"id":"b8fc8a8d-f67b-4ecc-b0da-455009ccbe9f","arxiv_id":"2607.25859","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The missing endpoint L^2→L^{10/3} bound for Hermite spectral projections in two dimensions, with the optimal λ^{-1/10} decay, is proved.","lead":"This paper proves the sharp size bound for the spectral projector of the Hermite oscillator in two dimensions, resolving the last open endpoint case. Together with earlier work, this completes the optimal eigenfunction estimates for the Hermite operator in every dimension.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The headline theorem rests on the unverified transfer of [11]'s TT* and dimension-independent reductions to d=2; if those do not extend verbatim, Theorem 1.1 is unsupported even if the local estimates are correct.","rationale":"I read the paper in good faith and found the local, multiscale estimate (1.3) internally consistent: the interpolation exponents for q_e=10/3, the dyadic splitting at δ=(μ')^2, and the summation over δ all check out, and I did not find a clear contradiction in the rectangle/orthogonality arguments. The proof is long and technically detailed, and the main body appears to support Proposition 3.1. However, the actual headline Theorem 1.1 is reached only through two quoted transfers from the authors' previous paper [11]: the TT* and annular-summation reduction after Theorem 1.2, and the 'dimension-independent' reduction in Section 2.1. These are not reproduced, and they are genuinely load-bearing: the localized estimate (1.3) is of a different operator type (L^{10/7}→L^{10/3}) than the desired global L^2→L^{10/3} bound, so the TT* mechanism is not a formality. The paper itself explicitly flags this dependence, and no machine-checked or independent verification is offered. This does not mean the result is false; it means the proof as written does not stand alone. The reader's CONDITIONAL verdict with medium risk is exactly calibrated to this situation, so I recommend no change.","tokens_in":56924,"tokens_out":20167,"duration_ms":155594,"concrete_test":"Independently reproduce the reduction of Theorem 1.1 from Theorem 1.2 in d=2: explicitly write the TT* argument and the double dyadic annular sum over μ, μ' for the operator Πλ, using (1.3) and the Koch–Tataru local estimates (1.2), and verify that the sum is bounded by C λ^{-1/10} with no logarithmic factor. Also trace the scaling in Section 2.1 (the λ^{3/5} dilation factor) and the sectorial Whitney decomposition to confirm that (2.2) is exactly equivalent to (1.3) in dimension two. If either step relies on a d≥3 exponent or produces an extra log, the proof of Theorem 1.1 is incomplete as written.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim Theorem 1.1 is not proved self-containedly. Two explicit bridges to [11] are load-bearing: (i) the assertion after Theorem 1.2 that the global L^2→L^{10/3} bound follows from the localized L^{10/7}→L^{10/3} estimate (1.3) by 'the same TT* argument and dyadic annular summation as in [11, pp. 1315–1316 and Section 2.7]'; and (ii) the statement in Section 2.1 that the reduction to the oscillatory integral Theorem 2.1 is 'dimension-independent and apply without change'. Neither is reproduced or verified for d=2. The local estimate (1.3) alone is an off-diagonal L^{q'}→L^q estimate, not an L^2→L^{q_e} estimate; converting the asymmetric gain (μ'/μ)^{1/20} into the global λ^{-1/10} endpoint requires the TT* mechanism to sum over the double annular decomposition without introducing a logarithmic loss or a dimension-dependent exponent mismatch. A failure in either bridge would invalidate Theorem 1.1 even if every estimate in Sections 3–7 is correct. The paper does not provide an independent derivation, a formal check, or a detailed d=2 trace of the cited reductions, so this is the least secure load-bearing premise.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims to prove the endpoint spectral projection estimate for the Hermite operator on R^2: ||Π_λ||_{L^2(R^2)→L^{10/3}(R^2)} ≤ C λ^{-1/10}, λ∈2N_0+2. This would complete the optimal L^2→L^q eigenfunction bounds for Hermite spectral projections in all dimensions, complementing the authors' previous work for d≥3. The proof is built from a reduction to an asymmetric local estimate χ_{A_{λ,μ}} Π_λ χ_{A_{λ,μ'}}: L^{10/7}→L^{10/3}, then to an oscillatory integral operator after rescaling and a sectorial Whitney decomposition. The core of the paper is a multiscale recursive decomposition of the degeneracy set {D(x,y)=0}, with a space–time decomposition, a tangential rectangle covering, an additional input-side tangential refinement, and almost-orthogonality/Cotlar–Stein summation. The authors assert that the final TT* summation from the local estimate to the global theorem is the same as in their prior paper [11], and that the reduction to the oscillatory integral theorem is dimension-independent.","tokens_in":57270,"tokens_out":9038,"duration_ms":82848,"significance":"If correct, Theorem 1.1 settles the last open endpoint in the Hermite spectral projection problem, a longstanding question following Thangavelu, Karadzhov, Koch–Tataru, and the authors' own d≥3 work. The paper's internal architecture is coherent and does not appear to contain post-hoc fitting: the main new ingredient, an asymmetric input refinement combined with a recursive space–time decomposition, is explicit and yields a concrete (μ'/μ)^{1/20} gain. The local estimates and the almost-orthogonality machinery are presented in substantial detail. However, the global claim is not established self-containedly: two load-bearing transitions are delegated to [11] without a d=2 verification, and the reader cannot check from the present manuscript alone that the TT* summation and the dimension-independent reductions indeed produce the claimed λ^{-1/10} endpoint without loss.","major_comments":[{"comment":"Theorem 1.1 is asserted to follow from Theorem 1.2 by 'the same TT* argument and dyadic annular summation as in [11, pp. 1315–1316 and Section 2.7]'. This is a load-bearing step: (1.3) is an asymmetric L^{10/7}→L^{10/3} bound, not an L^2→L^{10/3} bound, and converting the geometric gain (μ'/μ)^{1/20} into the global λ^{-1/10} requires a double annular summation over μ, μ' without logarithmic loss or dimension-dependent exponent mismatch. No d=2 version of this argument is stated or proved. If the cited summation has a hidden assumption that fails in two dimensions, Theorem 1.1 is unsupported even if all estimates in Sections 3–7 are correct. Please reproduce this step or provide a complete d=2 statement and proof.","section":"Section 1.2, after Theorem 1.2"},{"comment":"The reduction of Theorem 1.2 to the oscillatory-integral Theorem 2.1 is declared 'dimension-independent and apply without change'. This includes the kernel representation and rescaling, the Whitney-type decomposition, the reduction to the asymmetric rectangles B,B', and the treatment of the complementary pieces via [11, (3.3), p. 1329]. The rescaling factor in (2.2) explicitly uses d=2, so the assertion of complete dimension-independence is not self-evident. Since Theorem 2.1 is the main technical target, a gap or a d-dependent constant in any of these reductions would undermine the main theorem. Please include a formal verification or a precise lemma stating the d=2 versions of the cited reductions.","section":"Section 2.1"},{"comment":"The final interpolation and dyadic summation split at δ=(μ')^2 is presented only after Propositions 7.1 and 7.2 are assumed. The claimed endpoint inequality uses the two-scale split and the bound δ*∼λ^{-2/3}μ together with μ'≥C*λ^{-2/3}. This is plausible, but the displayed interpolation is the place where the (μ'/μ)^{1/20} factor must survive the summation over δ. The manuscript does not show the intermediate algebra for the cases (μ')^2≤δ* and δ*≤(μ')^2. Given the stress-test concern from the reader's report, I would ask the authors to expand this deduction and verify explicitly that the constants are uniform in δ and that no logarithmic loss enters in the transition from the local pieces to O_{λ,δ}.","section":"Section 7.1 / Deduction of Proposition 3.1"}],"minor_comments":[{"comment":"The sentence 'Since the support of eψ^ℓ_△(x,y,·) is included in a union of two disjoint intervals where ∂_tP is monotone' is a little terse. A short justification would help, since the monotonicity of ∂_tP is not otherwise stated.","section":"Section 2.2, Eq. (2.22)"},{"comment":"In the display after (2.31), the term |I| is written without absolute values on the left; since y2−w2 can be negative, the inequality should read |I|≥μ^{-1/2}|y2−w2|.","section":"Section 2.4, proof of Lemma 2.6"},{"comment":"The threshold in part (ii), |⟨n'_⊥(x0,y0), y0−y⟩| ≥ C δ μ^{-3/2} (μ')^{1/2}, is dimensionally nonuniform with the threshold in (3.43). This may be a typographical artifact, but it makes the statement hard to verify.","section":"Section 3.2, Lemma 3.9"},{"comment":"The affine transformations L_r, L_s are written with free variables x1,x2 and y1,y2, but the coefficients are not named. It would improve readability to label the coordinates after rescaling explicitly, e.g. u=(u_1,u_2), and to state the Jacobian factor separately.","section":"Section 6.4, Eq. (6.31)"},{"comment":"The paper is long and notation-heavy. A table of the main scales and operators (δ, δ△, δ*, σ, σ⊥, σ′, σ'_⊥, and O^κ_{λ,δ}) would help the reader navigate the recursive definitions. Some references to [11] are used for whole reduction steps; even when the argument is said to be identical, a precise pointer to theorem/lemma numbers would improve verifiability.","section":"General presentation"}],"recommendation":"major_revision","confidential_remarks":"The two bridges to [11] are the weakest point. The authors should be asked whether the TT* and annular summation, and the 'dimension-independent' reduction, can be reproduced in an appendix or stated as separate lemmas with full hypotheses. My own reading suggests that the central local estimates are very likely correct, but the present manuscript does not alone support Theorem 1.1 as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague —\n\nThis is the genuinely new result: it closes the d=2 endpoint q_e=10/3 in sharp L^2–L^q bounds for Hermite spectral projections, finishing a problem left open by Thangavelu, Karadzhov, and Koch–Tataru. Combined with the authors' d≥3 paper, the result now holds in all dimensions. That matters. The proof is also clearly the product of serious work: the multiscale space–time decomposition, the asymmetric refinement on the input side, and the nearly orthogonal summation are new and are developed with real care in Sections 3–7. The geometry is worked out in detail, the constants are fixed up front, and there is no sign of post-hoc selection. The endpoint estimate comes from interpolation of genuine L^2 and L^1→L^∞ bounds, so the architecture is coherent.\n\nWhat I trust: the local oscillatory integral estimates (Proposition 6.1, the stationary-phase reduction, the off-diagonal decay in Section 5) look honest and are presented in enough detail to referee. The paper correctly isolates the 2D difficulty in the gain (µ'/µ)^{1/20}; the logic from there to the global result is the standard one.\n\nThe soft spot is exactly what the stress-test note says: Theorem 1.1 is not proved self-containedly. The global bound is said to follow from (1.3) by 'the same TT* argument and dyadic annular summation as in [11, pp. 1315–1316 and Section 2.7]', and the reduction to the oscillatory integral theorem is called dimension-independent without a d=2 argument. Both are load-bearing, and neither is reproduced. If the TT* summation turns out to have a dimension-dependent exponent, the paper as written does not support Theorem 1.1 even if all the local estimates are correct. I do not think that is the most likely outcome — [11] is peer-reviewed and the logic is plausible — but this is a genuine premise that the authors should pin down, not a stylistic quibble.\n\nMinor notes: Lemma 3.9 is stated with proof omitted, and the bottom-scale analogues are compressed; a few other estimates are stated without full derivation. None of these looks like a real gap at my reading.\n\nThis paper is for harmonic analysts working on eigenfunction bounds, Bochner–Riesz means, and related restriction theory. It deserves a serious referee, and my recommendation is to send it out with the explicit instruction that the referee verify the two quoted reductions from [11] in d=2.","headline":"A serious, largely self-contained proof of the last open 2D Hermite endpoint — the main soft spot is that two load-bearing reduction steps are quoted from the authors' prior d≥3 paper without a d=2 trace.","tokens_in":57726,"tokens_out":2981,"would_cite":true,"duration_ms":26021,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B99","42C10"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves the sharp L^2→L^{10/3} bound for Hermite spectral projections in two dimensions, completing the optimal eigenfunction bounds in all dimensions.","keywords":["Hermite operator","spectral projection","eigenfunction bounds","endpoint estimate","oscillatory integrals","almost orthogonality","degenerate phase","two dimensions"],"falsifier":"Construct an L²-normalized sequence f_λ supported in an annulus of width λ^{−2/3} near √λ S¹ and compute ‖Π_λ f_λ‖_{L^{10/3}}; if this exceeds Cλ^{−1/10} for arbitrarily large λ, the endpoint estimate is false.","tokens_in":56834,"feed_emoji":"📐","tokens_out":4444,"duration_ms":39864,"temperature":0.7,"pith_summary":"The paper establishes the missing endpoint estimate for Hermite spectral projections in two dimensions: for an eigenvalue λ, the projection operator Π_λ maps L^2(ℝ²) to L^{10/3}(ℝ²) with norm at most Cλ^{−1/10}. This closes the only remaining gap in a long line of sharp L^2→L^q eigenfunction bounds, matching the known lower bound and removing the logarithmic loss that had persisted at q_e = 10/3. The proof localizes near the sphere √λ S¹, decomposes the operator in space and time according to the size of a discriminant D, and harvests an extra factor (μ′/μ)^{1/20} from an asymmetric refinement of the input annulus. A sympathetic reader would care because the result settles the question of optimal eigenfunction bounds in every dimension and reveals a mechanism—weak interaction between unequal annuli—that may be reusable elsewhere.","feed_headline":"Hermite endpoint bound in 2D hits sharp exponent λ^{−1/10}","feed_subtitle":"The missing L²→L^{10/3} estimate closes optimal eigenfunction bounds for Hermite spectral projections in every dimension.","key_machinery":"The load-bearing object is the discriminant D(x,y) = 1 + ⟨x,y⟩² − |x|² − |y|², which controls the stationary-point equation for the phase P in the Hermite–Schrördinger representation: D > 0 gives two nondegenerate critical times, D < 0 gives none, and D = 0 produces a cubic degeneracy. The proof recursively decomposes the operator over dyadic scales of |D| from δ△ ∼ μμ′ down to δ∗ ∼ λ^{−2/3}μ, covers the spatial regions by tangential rectangles adapted to the level sets of D, and then subdivides each input rectangle tangentially by an additional factor (μ′/μ)^{1/2}. Interpolation between L² diagonal bounds and L¹→L^∞ kernel bounds yields the asymmetric gain (μ′/μ)^{1/20}, and Cotlar–Stein al","core_discovery":"The central discovery is Theorem 1.1: for λ ∈ 2ℕ₀+2 and q_e = 10/3, the spectral projection Π_λ satisfies ‖Π_λ‖_{L²(ℝ²)→L^{10/3}(ℝ²)} ≤ Cλ^{−1/10}. Together with the previously known non-endpoint bounds, this gives the optimal exponent across the full range 2 ≤ q ≤ ∞ in two dimensions, and combined with the authors' earlier d ≥ 3 results, it completes the optimal L²–L^q eigenfunction bounds for the Hermite operator in all dimensions. The engine of the proof is a localized asymmetric estimate: operators χ_{A_{λ,μ}} Π_λ χ_{A_{λ,μ′}} with μ′ ≪ μ gain a factor (μ′/μ)^{1/20} relative to the earlier bound, and this geometric decay is exactly what makes the dyadic summation over annuli converge.","pith_inferences":["The same multiscale space–time decomposition may transfer to other spectral projections whose phase has a quadratic degeneracy surface, such as twisted or magnetic Laplacians, where endpoint bounds still carry logarithmic losses.","A natural stress test is whether the asymmetric gain (μ′/μ)^{1/20} persists all the way to the bottom scale μ′ ∼ λ^{−2/3}; the summation split at δ = (μ′)² in Section 7.1 suggests this is the sharp borderline.","The proof supports the heuristic that, unlike in dimension one, no equidistributed eigenfunction can concentrate near √λ S¹, which is why the two-dimensional endpoint is log-free while the one-dimensional endpoint has a logarithm."],"forward_implications":["The endpoint bound at q_e = 10/3, combined with known estimates, yields the optimal λ^{β(q)} rate for every 2 ≤ q ≤ ∞ in dimension two, with no logarithmic loss.","Together with the d ≥ 3 endpoint results, the problem of optimal L²–L^q eigenfunction bounds for Hermite spectral projections is now closed in all dimensions.","The localized asymmetric estimate (1.3) implies the global bound through the TT* argument and dyadic annular summation, so the gain for μ′ ≪ μ is the decisive quantitative fact.","Applications include sharpened L^p convergence of Hermite Bochner–Riesz means and strong unique continuation for parabolic inequalities, where such spectral projection bounds are used."],"fun_headline_variants":["2D Hermite endpoint bound resolved: L²→L^{10/3} sharp","Hermite eigenfunction bound complete in all dimensions","Sharp L^{10/3} estimate for Hermite projections in 2D","Optimal Hermite bounds: missing 2D case now proven"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof inherits, without reproducing the details, the TT* reduction, dyadic annular summation, and dimension-independent oscillatory-integral reductions from the authors' earlier paper; if those reductions do not extend verbatim to d = 2, Theorem 1.1 as proved here is not self-contained.","fun_headline_variants_meta":{"raw":{"variants":["2D Hermite endpoint bound resolved: L²→L^{10/3} sharp","Hermite eigenfunction bound complete in all dimensions","Sharp L^{10/3} estimate for Hermite projections in 2D","Optimal Hermite bounds: missing 2D case now proven"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00078,"raw_usage":{"total_tokens":3310,"prompt_tokens":801,"completion_tokens":2509,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":2430}},"tokens_in":545,"tokens_out":2509,"duration_ms":17331,"temperature":1.0,"reasoning_tokens":2430,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T01:16:39.590753+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an L²-normalized sequence f_λ supported in an annulus of width λ^{−2/3} near √λ S¹ and compute ‖Π_λ f_λ‖_{L^{10/3}}; if this exceeds Cλ^{−1/10} for arbitrarily large λ, the endpoint estimate is false.","supporting_citations":[],"review_version":1}