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Spectral asymptotic formula of Bessel--Riesz commutator

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.14928 v1 pith:R5NEHLTF submitted 2024-11-22 math.FA

Zhijie Fan , Ji Li , Fedor Sukochev , Dmitriy Zanin This is my paper · ORCID
classification math.FA MSC 47B1042B2043A85
keywords weakSchattenclassRiesztransformcommutatorBesseloperatorSobolevspacespectralasymptoticformulaDixmiertraceSchurmultipliersingularvalues
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $\varphi(|[R_{\lambda,k},M_f]|^{n+1}) = C_{n,\lambda}\|f\|^{(k)}_{\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})}$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [§8, Lemma 8.1 and Proposition 8.7] 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.
  2. [§8, Lemma 8.3 and Lemma 8.4] 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.
  3. [§7, Lemma 7.4 Step 3, and §8, Lemma 8.5 (n=1)] 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.
minor comments (4)
  1. [§2, Lemma 2.1] 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.
  2. [§7, Proof of lower bound in Theorem 1.2] 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.
  3. [§4, Proposition 4.9 and §8, Lemma 8.1] 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.
  4. [§8, Lemma 8.2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Bessel–Riesz asymptotic is derived by reduction to independent classical results, not by defining the output in terms of its inputs.

full rationale

The paper's main result (Theorem 1.3) is a genuine asymptotic statement: it identifies lim_{t→∞} t^{1/(n+1)} μ(t,[R_{λ,k},M_f]) with C_{n,λ}‖f‖^{(k)}_{Ẇ^{1,n+1}}, where ‖·‖^{(k)} is an equivalent seminorm. The left-hand side is a singular-value limit, not a norm, so the equality is not definitional. The load-bearing reduction is Proposition 8.7, which approximates the Bessel–Riesz commutator by κ F_{2,0}(0) times a conjugated classical Riesz commutator modulo the separable ideal (L^{n+1,∞})_0. The classical spectral asymptotic is then imported from [19, Proposition 8.6] (Frank–Sukochev–Zanin, Trans. AMS 2023) and Lemma 8.8 repeats that argument mutatis mutandis. These are separate published theorems with independent proofs, so citing them is evidence, not circularity, even though two of the authors (Sukochev and Zanin) are co-authors of [19]. Similarly, Theorem 1.2 relies on the endpoint characterisation [28, Theorem 1], again an independent external theorem, and the upper bound is further supported by the alternative Cwikel-estimate proof in Section 6. The constant C_{n,λ} is f-independent and arises from the kernel comparison constant F_{2,0}(0); it is not fitted to the data being predicted. No equation in the paper defines the target quantity in terms of itself, and no fitted parameter is renamed as a prediction. The reviewer-flagged gap in Lemma 8.1 (hypothesis stated for l≤n while the proof sums l=1,…,n+1) is a proof-correctness concern about verifying the vertical commutator, not a circularity: it concerns whether the remainder is separable, and Lemma 8.4 provides the ingredients to repair it. Therefore the derivation chain is self-contained against external benchmarks and the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No numerical parameters are fitted to data; n and lambda are inputs of the problem. The proof imports several published theorems as black boxes; these are standard mathematical assumptions rather than ad hoc choices. No new physical or geometric entities are introduced.

assumptions (6)
  • domain assumption The Bessel operator Delta_lambda has the self-adjoint extension defined via the Fourier-Bessel transform in Section 2.3.
    Used throughout to define R_{lambda,k} and its functional calculus; it is a standard construction for the Bessel operator.
  • standard math The Schur multiplier criterion of Conde-Alonso, Gonzalez-Perez, Parcet, and Tablate [6, Theorem A] applies to the symbols considered here.
    Imported as a black box in Theorem 4.2 and used for boundedness of S_{F_{k,l} circ H}.
  • standard math The Euclidean endpoint weak Schatten characterization [28, Theorem 1] is valid.
    Used in Lemma 5.1 for the upper bound and in Lemma 7.6 for the lower bound.
  • standard math The Euclidean spectral asymptotic formula [19, Proposition 8.6] is valid.
    Used in Lemma 8.8 as the known Euclidean benchmark for the asymptotic coefficient.
  • standard math Abstract Cwikel estimates from [26] and [29] apply in the Bessel and Euclidean settings.
    Used in Theorem 6.2 for the alternative upper bound and in Lemma 8.4 for separability of local commutators.
  • standard math Macaev's triangular truncation theorem holds on L_p of B(L2(R^{n+1}_+)).
    Used in Proposition 4.9 to prove boundedness of the Schur multiplier S_b.

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Pith. "Pith review of Spectral asymptotic formula of Bessel--Riesz commutator." pith.science (2026). https://pith.science/paper/R5NEHLTF

@misc{pith2026241114928,
  author       = {Pith},
  title        = {Pith review of: Spectral asymptotic formula of Bessel--Riesz commutator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/R5NEHLTF}},
  note         = {Machine review of arXiv:2411.14928}
}
abstract

Let $R_{\lambda,j}$ be the $j$-th Bessel--Riesz transform, where $n\geq 1$, $\lambda>0$, and $j=1,\ldots,n+1$. In this article, we establish a Weyl type asymptotic for $[M_f,R_{\lambda,j}]$, the commutator of $R_{\lambda,j}$ with multiplication operator $M_f$, based on building a preliminary result that the endpoint weak Schatten norm of $[M_f,R_{\lambda,j}]$ can be characterised via homogeneous Sobolev norm $\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})$ of the symbol $f$. Specifically, the asymptotic coefficient is equivalent to $\|f\|_{\dot{W}^{1,n+1}(\mathbb{R}_+^{n+1})}.$ Our main strategy is to relate Bessel--Riesz commutator to classical Riesz commutator via Schur multipliers, and then to establish the boundedness of Schur multipliers.

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