{"id":"798d7b8d-efdb-4706-8529-78e2b3158fb2","arxiv_id":"2411.14928","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Bessel-Riesz commutators on R^{n+1}_+, the singular values obey lim t^{1/(n+1)} mu(t,[M_f,R_{lambda,k}]) = C ||f||_{W-dot}^{1,n+1}, and the endpoint weak Schatten norm is equivalent to the same Sobolev norm.","lead":"This paper proves a Weyl-type asymptotic formula for commutators of Bessel-Riesz transforms with multiplication operators, with the asymptotic coefficient equal to a homogeneous Sobolev norm of the symbol. It also characterizes the endpoint weak Schatten norm of these commutators in terms of that Sobolev norm.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Proposition 8.7 depends on Lemma 8.1, whose stated hypothesis omits the vertical commutator l=n+1 although its proof and application require it; the gap is repairable via Lemma 8.4, so the conditional verdict stands.","rationale":"I read the paper in good faith. The central theorem is plausible and well supported by the Schur-multiplier strategy, the Cwikel estimates, and the local-to-global approximation argument. The weakest point is exactly the vertical-index mismatch in Lemma 8.1: the statement assumes horizontal commutators only, while the proof and Proposition 8.7 need the l = n+1 case. I checked that the missing case is already available from Lemma 8.4 for the specific approximants V_j, so the gap is textual and repairable rather than a mathematical obstruction. I also see no independent flaw in the other main ingredients: the boundedness of S_{F_k,l circle H}, S_a, S_b, S_{h_k}, the distance-convergence lemma 8.6, and the Birman-Solomyak approximation lemma 8.9 all appear consistent. Therefore the reader's conditional verdict is appropriate and no verdict change is needed.","tokens_in":36471,"tokens_out":34559,"duration_ms":324842,"concrete_test":"Re-derive Lemma 8.1 with the strengthened hypothesis [M_{x_l},V] in (L^{p,infinity})_0 for all 1 <= l <= n+1, then verify that Proposition 8.7's operators V_j satisfy it for l = n+1: write [M_{x_{n+1}},V_j] = -delta_{k,n+1} E^* M_chi [Delta^{-1/2},M_f] M_chi E - E^* M_chi [partial_k partial_{n+1} Delta^{-3/2},M_f] M_chi E and check that both summands belong to the class L^{(n+1)/2,infinity}(L2(R^{n+1})) given by Lemma 8.4, hence to (L^{n+1,infinity})_0. If the calculation closes, Proposition 8.7 holds and Theorem 1.3 is supported; if not, the spectral asymptotic is not currently established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Proposition 8.7: the Bessel commutator differs from kappa F_{2,0}(0) times the conjugated classical commutator by an element of the separable part (L^{n+1,infinity})_0. The proof transfers this through Lemma 8.2 to Lemma 8.1. Lemma 8.1 is stated only with [M_{x_l},V] in (L^{p,infinity})_0 for 1 <= l <= n, but its proof sums over l = 1,...,n+1 and uses the assumption for l = n+1. Proposition 8.7 applies Lemma 8.1 to V_j = M_{chi_{Q_j}} E^*[R_k,M_f]E M_{chi_{Q_j}} and verifies the horizontal commutators, while the vertical case is not explicitly written. This matters: without separability of [M_{x_{n+1}},V_j], S_H(V_j) need not be separable and the spectral limit in Theorem 1.3 could acquire an extra contribution. The gap is repairable: for these V_j one has [M_{x_{n+1}},R_k] = -delta_{k,n+1} Delta^{-1/2} - partial_k partial_{n+1} Delta^{-3/2}, and Lemma 8.4 with l = n+1 places both resulting terms in L^{(n+1)/2,infinity}, hence in (L^{n+1,infinity})_0. Thus the concern is a genuine proof gap in a critical lemma, not evidence that the asymptotic formula is false.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves two main results for commutators [R_{\\lambda,k}, M_f] of the Bessel–Riesz transforms R_{\\lambda,k} with multiplication operators on the half-space R^{n+1}_+: Theorem 1.2 gives an endpoint weak Schatten class characterization, showing that membership in L^{n+1,\\infty} is equivalent to f belonging to the homogeneous Sobolev space \\dot{W}^{1,n+1}(R^{n+1}_+); Theorem 1.3 establishes a Weyl-type spectral asymptotic formula whose coefficient is an equivalent homogeneous Sobolev seminorm. The proof strategy is to relate the Bessel commutators to classical Riesz commutators through Schur multipliers, prove the necessary Schur multiplier boundedness, and then transfer the known Euclidean spectral asymptotic. A Dixmier trace formula is derived as Corollary 1.4.","tokens_in":36839,"tokens_out":39137,"duration_ms":318738,"significance":"If the gaps identified below are repaired, this is a substantial result: it extends the Euclidean spectral asymptotic of Frank–Sukochev–Zanin to the Bessel setting, answers a question of Rupert Frank, and provides a new instance of the Dixmier trace formula. The paper contains detailed kernel computations, a clear reduction to known Euclidean results, and a novel use of recent Schur multiplier bounds. The endpoint weak Schatten characterization and the spectral asymptotic are natural and important in noncommutative geometry and harmonic analysis.","major_comments":[{"comment":"Lemma 8.1 is stated with the hypothesis that [M_{x_l}, V] \\in (L^{p,\\infty})_0 for 1 \\le l \\le n, but its proof sums the decomposition of S_H(V) over l = 1,\\dots,n+1 and explicitly asserts the hypothesis for l = n+1. Proposition 8.7 verifies condition (ii) of Lemma 8.2 only for l \\le n, and the vertical commutator [M_{x_{n+1}}, V_j] is not established. This is load-bearing: without separability of [M_{x_{n+1}}, V_j], the term S_H(V_j) need not be separable, and the spectral limit in Theorem 1.3 could acquire an extra contribution. The gap appears repairable, since for V_j = M_{\\chi_{Q_j}} E^*[R_k,M_f]E M_{\\chi_{Q_j}} one has [M_{x_{n+1}}, R_k] = -\\delta_{k,n+1}\\Delta^{-1/2} - \\partial_k\\partial_{n+1}\\Delta^{-3/2}, and Lemma 8.4 with l = n+1 places the resulting terms in L^{(n+1)/2,\\infty}, hence in (L^{n+1,\\infty})_0 after compression. The proof should be amended accordingly.","section":"§8, Lemma 8.1 and Proposition 8.7"},{"comment":"The proof of Lemma 8.3 claims that the first assertion follows from Lemma 2.3. However, Lemma 2.3 requires p>2, while the desired conclusion concerns L^{(n+1)/2,\\infty}. For n=1,2,3, the exponent (n+1)/2 is at most 2, so Lemma 2.3 is not applicable. Moreover, even for n\\ge 4, the multiplier g(t)=(1+t^2)^{-1} lies in L^{p,\\infty}(R_+, r^n dr) only for p=(n+1)/2, which is >2 only when n\\ge 3; the endpoint cases require a separate argument (e.g., Birman–Solomyak bounds for pseudo-differential operators of order -2). Since Lemma 8.3 feeds directly into Lemma 8.4 and Proposition 8.7, the proof of Theorem 1.3 for n=1,2,3 is incomplete as written. The authors should provide a valid proof of Lemma 8.3 for all n, either by citing the appropriate endpoint Cwikel/Birman–Solomyak estimates or by a direct singular value estimate.","section":"§8, Lemma 8.3 and Lemma 8.4"},{"comment":"Lemma 2.3 is stated only for n\\ge 2 (ambient dimension at least 3), but it is invoked in the case n=1 (ambient dimension 2) in two places. First, in Lemma 7.4 Step 3, the approximation argument uses the L^{2n+2}-norm and 'By Lemma 2.3'; for n=1, this would require a Cwikel estimate with p=4 for the symbol |\\xi|^{-1}, which is not in L^{4,\\infty}(R^2). The desired conclusion can be proved directly for n=1 by dominating the kernel by C|x-y|^{-1} and applying Lemma 7.2, but that argument is not given. Second, in Lemma 8.5, the n=1 case invokes Lemma 2.3, which is again outside the stated hypothesis; the R^2 estimate with p=4 is true but not covered. These gaps affect the proofs of Theorem 1.2(ii) and Theorem 1.3 for n=1. The authors should either extend Lemma 2.3 to cover dimensions 2 (with appropriate hypotheses) or supply separate arguments.","section":"§7, Lemma 7.4 Step 3, and §8, Lemma 8.5 (n=1)"}],"minor_comments":[{"comment":"The notation in Lemma 2.1 uses the same symbol T for operators on L^2(R^{n+1}_+, m_\\lambda) and L^2(R^{n+1}_+); the equivalence relies on the unitary M_{x_{n+1}}^\\lambda, and it would be clearer to write T_1 and T_2 or to say explicitly that the operator is transported by this unitary.","section":"§2, Lemma 2.1"},{"comment":"In the display after 'It follows from Proposition 7.7', the function f in Proposition 7.7 is assumed to be defined on all of R^{n+1}; in the lower bound proof f is initially on R^{n+1}_+. The application is correct if one replaces f by its zero extension Ef, but this substitution is not stated.","section":"§7, Proof of lower bound in Theorem 1.2"},{"comment":"The arrow notation X \\rightspoonarrow X is introduced in §2.1 but the actual symbol used is 'X \\rightspoonarrow' (a long right arrow with a loop). Please ensure the notation is consistently rendered and defined in one place.","section":"§4, Proposition 4.9 and §8, Lemma 8.1"},{"comment":"In the proof of Lemma 8.2, the distance to (L^{p,\\infty})_0 is computed as an infimum over A \\in L^p; this is valid because L^p is dense in (L^{p,\\infty})_0, but the density is not explicitly cited.","section":"§8, Lemma 8.2"}],"recommendation":"major_revision","confidential_remarks":"The paper presents a promising and interesting approach to a natural question, and the main results are likely true. However, the manuscript currently contains several places where a key lemma is invoked outside its stated hypotheses, and these gaps are load-bearing for the main theorem. In particular, Lemma 8.1's proof uses an unverified l=n+1 case, Lemma 8.3 relies on Cwikel estimates that require p>2 while the target exponent is (n+1)/2, and Lemma 2.3 is used in dimension 2 despite being stated only for higher dimensions. These issues are repairable, but they require nontrivial additions or replacements in the proof. I recommend major revision with a careful reworking of Section 8."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper deserves a real referee and very likely a conditional accept. The main result is genuinely new and the proof strategy is sound, but there is a small gap in Lemma 8.1 that the authors need to fix before publication. It is not a sign the theorem is false.\n\nWhat is new: Theorem 1.2 gives the endpoint Schatten–Lorentz characterization for Bessel–Riesz commutators, and Theorem 1.3 gives the Weyl-type asymptotic with the homogeneous Sobolev seminorm as the coefficient. That directly answers Rupert Frank's question and yields a Dixmier trace formula (Corollary 1.4). The reduction of the Bessel problem to the Euclidean Riesz commutator via Schur multipliers is a real technique, not a cosmetic rewrite. The boundedness of those multipliers is imported from Conde-Alonso et al., and the relevant smoothness of the functions F_{k,l} is handled in detail in Appendix A.\n\nThe soft spot: Lemma 8.1 is stated for commutators [M_{x_l}, V] with l ≤ n, but its proof sums over l = 1,...,n+1. Proposition 8.7 applies the lemma to V_j = MχQj E*[R_k,M_f]E MχQj and only verifies the horizontal cases explicitly; this is exactly the load-bearing point where the spectral limit is transferred from Bessel to classical Riesz. If [M_{x_{n+1}}, V_j] were not separable, the asymptotic could pick up an extra term. The good news is that the gap is clearly repairable: a direct computation gives [M_{x_{n+1}}, R_k] = -δ_{k,n+1} Δ^{-1/2} - ∂_k ∂_{n+1} Δ^{-3/2}, and Lemma 8.4 with l=n+1 places both terms in (L^{n+1,∞})_0. So this is a proof gap, not a false theorem. The authors should be asked to expand Lemma 8.1 to cover l=n+1 and spell out the vertical case in Proposition 8.7.\n\nThe paper does lean on several heavy black boxes (the Schur multiplier theorem of Conde-Alonso et al., Cwikel estimates, the Euclidean asymptotic of Frank–Sukochev–Zanin). That is acceptable in a paper like this; the imports are distinct published theorems, and self-citation here is not circularity.\n\nFor whom: functional analysts with taste for singular traces and commutators, and harmonic analysts working on Bessel operators. I would bring it to reading group and would cite it in my own work once the lemma is fixed. The editors should send it to referees; my guess is it will be accepted after a moderate revision.","headline":"Deserves a serious referee: the main asymptotic is new and the proof architecture is sound, with one small and repairable gap in Lemma 8.1 / Proposition 8.7.","tokens_in":37333,"tokens_out":2613,"would_cite":true,"duration_ms":24195,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B10","42B20","43A85"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a Weyl-type spectral asymptotic for Bessel–Riesz transform commutators: the t^{1/(n+1)} singular-value limit exists and equals an f-independent constant times the homogeneous Sobolev seminorm of the symbol.","keywords":["weak Schatten class","Riesz transform commutator","Bessel operator","Sobolev space","spectral asymptotic formula","Dixmier trace","Schur multiplier","singular values"],"falsifier":"For $n=1$, take a smooth compactly supported $f$ on $\\mathbb{R}_+^2$ and compute the singular values of $[R_{\\lambda,1},M_f]$; the theorem predicts $t^{1/2}\\mu(t,[R_{\\lambda,1},M_f])\\to C_{1,\\lambda}\\|\\nabla f\\|_{L^2}$, so a single $f$ for which this limit exists but differs from that value, or for which the remainder in Proposition 8.7 has positive distance from $(L^{2,\\infty})_0$, would refute the claim.","tokens_in":36310,"feed_emoji":"📐","tokens_out":10013,"duration_ms":86807,"temperature":0.7,"pith_summary":"This paper establishes a Weyl-type spectral asymptotic for commutators of Bessel–Riesz transforms on the half-space $\\mathbb{R}_+^{n+1}$, answering a question posed to the authors. The main theorem states that for every bounded symbol $f$ whose gradient lies in $L^{n+1}$, the singular values of the commutator satisfy $\\lim_{t\\to\\infty} t^{1/(n+1)} \\mu(t,[R_{\\lambda,k},M_f]) = C_{n,\\lambda}\\|f\\|^{(k)}_{\\dot{W}^{1,n+1}(\\mathbb{R}_+^{n+1})}$, with $C_{n,\\lambda}$ independent of $f$. The asymptotic coefficient is exactly an equivalent homogeneous Sobolev seminorm, meaning the leading spectral decay records only the gradient of $f$. As consequences, the paper derives a Dixmier trace formula for $|[R_{\\lambda,k},M_f]|^{n+1}$ and a rigidity statement: if the limit is zero, $f$ must be constant.","feed_headline":"Commutator spectra obey a Weyl law with Sobolev coefficient","feed_subtitle":"The leading decay of the commutator's singular values equals a constant times the L^{n+1} norm of the gradient of f.","key_machinery":"The central mechanism is the reduction of the Bessel–Riesz commutator to the classical Riesz commutator. Proposition 3.6 expresses $[R_{\\lambda,k},M_{E^*f}]$ as a combination of Schur multipliers — operations that multiply an operator's integral kernel by a bounded function of the two variables — applied to the conjugated classical commutators $M_{x_{n+1}^{-\\lambda}}E^*[R_l,M_f]EM_{x_{n+1}^{\\lambda}}$. The relevant multiplier symbols are built from $F_{k,l}\\circ H$, $a$, $b$, $h_m$ with $H(x,y)=|x-y|(x_{n+1}y_{n+1})^{-1/2}$; their $L^p$ boundedness is proved in Propositions 4.8–4.9 via a recent Schur-multiplier criterion and transference. Proposition 8.7 then shows that, modulo the ideal $(L^{n+1,\\infty})_0$ of operators whose $t^{1/(n+1)}\\mu(t)$ tends to zero, the Bessel commutator differs from $\\kappa^{(3)}_{n,\\lambda}F_{2,0}(0)$ times the conjugated classical commutator only by negligible terms. The Birman–Solomyak approximation lemma (Lemma 8.9) turns this equivalence into the exact limit formula.","core_discovery":"The central discovery is that the Bessel–Riesz commutator is asymptotically equivalent, at the level of the separable part of the weak Schatten ideal, to a constant multiple of the classical Riesz commutator conjugated by the weight $M_{x_{n+1}^{\\pm\\lambda}}$ and restricted to the half-space. Through this identification, the endpoint weak Schatten characterization of Theorem 1.2 — membership in $L^{n+1,\\infty}$ if and only if $f\\in\\dot{W}^{1,n+1}$ — upgrades to the Weyl-type asymptotic of Theorem 1.3. Consequently, Corollary 1.4 states that for any normalised continuous trace $\\varphi$ on $L^{1,\\infty}$, one has $\\varphi(|[R_{\\lambda,k},M_f]|^{n+1}) = C_{n,\\lambda}\\|f\\|^{(k)}_{\\dot{W}^{1,n+1}(\\mathbb{R}_+^{n+1})}$.","pith_inferences":["The paper leaves $C_{n,\\lambda}$ as an unspecified positive constant; combining Proposition 8.7 with the classical asymptotic identifies it as $\\kappa^{(3)}_{n,\\lambda}F_{2,0}(0)$ times the classical coefficient, a direct byproduct of the proof.","The same transfer through Schur multipliers should produce Weyl-type asymptotics for commutators of Riesz transforms associated with other Bessel-type degenerate operators, provided the kernel symbols satisfy the smoothness conditions of Theorem 4.1.","The rigidity statement for zero limit suggests a quantitative stability version: if the singular-value limit is small, $f$ should be close to a constant in $\\dot{W}^{1,n+1}$; this is not stated in the paper."],"forward_implications":["The limit in Theorem 1.3 exists for every bounded $f$ with gradient in $L^{n+1}$, so the asymptotic spectrum is completely determined by the homogeneous Sobolev seminorm; no further information about $f$ enters.","Corollary 1.4 upgrades the asymptotic to a trace formula: on $|[R_{\\lambda,k},M_f]|^{n+1}$, every normalised continuous trace evaluates to the same constant times $\\|f\\|^{(k)}_{\\dot{W}^{1,n+1}}$.","Corollary 1.5 says that if the singular-value limit is zero, then $f$ is constant; the asymptotic coefficient therefore detects all non-trivial symbols.","Theorem 1.2 gives two-sided endpoint bounds: $[R_{\\lambda,k},M_f]\\in L^{n+1,\\infty}$ if and only if $f\\in\\dot{W}^{1,n+1}$, with norms equivalent up to an $f$-independent constant."],"supporting_citations":[{"why":"Supplies the classical singular-value asymptotic for Riesz commutators that Lemma 8.8 and the final approximation step adapt to the half-space.","marker":"[19]"},{"why":"Provides the Euclidean endpoint characterization and the approximation scheme used in the proof of the upper bound and in Corollary 1.4.","marker":"[28]"},{"why":"Gives the L^p Schur multiplier criterion that Theorem 4.1 and the boundedness of the F_{k,l}∘H multipliers rely on.","marker":"[6]"},{"why":"Establishes the preceding endpoint weak Schatten result for Bessel–Riesz commutators and the kernel formula used in Section 3.","marker":"[15]"},{"why":"Supplies the spectral approximation lemma that appears as Lemma 8.9 in the final step.","marker":"[4]"},{"why":"Provides the singular-value and trace machinery used to control the separable part of the weak Schatten ideal.","marker":"[29]"},{"why":"Gives the Dixmier trace formula that Corollary 1.4 extends to the Bessel half-space setting.","marker":"[7]"}],"fun_headline_variants":["Bessel-Riesz commutator follows a Weyl law","Commutator singular values scale with Sobolev norm","Weyl asymptotics for Bessel-Riesz commutator spectra","Sobolev coefficient sets commutator spectral decay","Endpoint weak Schatten implies Weyl law for Bessel-Riesz"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The entire asymptotic transfers from the classical Riesz commutator to the Bessel one through Proposition 8.7, and that transfer leans on Lemma 8.1, whose stated hypothesis covers commutators with the horizontal coordinates only, although its proof sums over all $n+1$ coordinates; if the vertical-coordinate estimate failed, the remainder would not lie in the ideal of faster-decaying singular values and the limit in Theorem 1.3 could pick up an extra term.","fun_headline_variants_meta":{"raw":{"variants":["Bessel-Riesz commutator follows a Weyl law","Commutator singular values scale with Sobolev norm","Weyl asymptotics for Bessel-Riesz commutator spectra","Sobolev coefficient sets commutator spectral decay","Endpoint weak Schatten implies Weyl law for Bessel-Riesz"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000328,"raw_usage":{"total_tokens":1823,"prompt_tokens":929,"completion_tokens":894,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":545,"completion_tokens_details":{"reasoning_tokens":810}},"tokens_in":545,"tokens_out":894,"duration_ms":8283,"temperature":1.0,"reasoning_tokens":810,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:44:36.473748+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For $n=1$, take a smooth compactly supported $f$ on $\\mathbb{R}_+^2$ and compute the singular values of $[R_{\\lambda,1},M_f]$; the theorem predicts $t^{1/2}\\mu(t,[R_{\\lambda,1},M_f])\\to C_{1,\\lambda}\\|\\nabla f\\|_{L^2}$, so a single $f$ for which this limit exists but differs from that value, or for which the remainder in Proposition 8.7 has positive distance from $(L^{2,\\infty})_0$, would refute the claim.","supporting_citations":[{"cited_title":"Frank, F","cited_arxiv_id":null,"evidence_quote":"Supplies the classical singular-value asymptotic for Riesz commutators that Lemma 8.8 and the final approximation step adapt to the half-space."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Euclidean endpoint characterization and the approximation scheme used in the proof of the upper bound and in Corollary 1.4."},{"cited_title":"Conde-Alonso, A.M","cited_arxiv_id":null,"evidence_quote":"Gives the L^p Schur multiplier criterion that Theorem 4.1 and the boundedness of the F_{k,l}∘H multipliers rely on."},{"cited_title":"Birman and M.Z","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral approximation lemma that appears as Lemma 8.9 in the final step."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the singular-value and trace machinery used to control the separable part of the weak Schatten ideal."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the Dixmier trace formula that Corollary 1.4 extends to the Bessel half-space setting."}],"review_version":1}