Composition Operators and Rational Inner Functions II: Boundedness between two different Bergman Spaces

Athanasios Beslikas

REVIEW 4 major objections 3 minor 15 references

Composition Operators and Rational Inner Functions II: Boundedness between two different Bergman Spaces

T0 review · 4 major / 3 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that a composition operator induced by rational inner functions on the bidisc, each with one boundary singularity, is bounded from A²_{β/(2q)−2}(D²) to A²_β(D²) for all β>2q.

desk verdict Plausible theorem, promising strategy, but the proof as written covers only a special case and leaves a real uniformity gap; worth refereeing but not citable yet. read the letter →

arxiv 2509.04366 v1 pith:MBKZDFYZ submitted 2025-09-04 math.CV

Athanasios Beslikas This is my paper · ORCID

classification math.CV MSC 32A3732A4030J10

keywords rationalinnerfunctionscompositionoperatorsweightedBergmanspacesbidiscCarlesonmeasuresLojasiewiczinequalityboundarysingularities

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

The paper establishes a sufficient condition for boundedness of composition operators induced by non-smooth rational inner functions on the bidisc, acting between two different weighted Bergman spaces. The main result, Theorem 2.5, shows that if each coordinate function has the form $\tilde{p}/p$ and each denominator polynomial vanishes at exactly one point of the distinguished boundary $\mathbb{T}^2$, then there exists $q>0$ such that $C_\Phi$ maps $A^2_{\beta/(2q)-2}(\mathbb{D}^2)$ boundedly into $A^2_\beta(\mathbb{D}^2)$ for all $\beta>2q$. This matters because rational inner functions on the bidisc can be non-smooth at boundary singularities, so the classical smooth-symbol methods do not apply. The proof uses the Lojasiewicz inequality to control the rate at which each coordinate approaches its boundary value and then verifies the Carleson measure criterion for the pullback measure.

What carries the argument

The main machinery is the Carleson measure criterion (Lemma 2.1), which turns boundedness of $C_\Phi$ into uniform estimates $V_\beta(\Phi^{-1}(S(\zeta,\delta))) \leq C V_a(S(\zeta,\delta))$ on all Carleson boxes. At each boundary singularity, the Lojasiewicz inequality (a polynomial lower bound $|P(z)| \geq C \operatorname{dist}(z,Z(P))^q$ near a zero) gives a local lower bound for $|\tilde{p}_i - \zeta p_i|$; Lemma 2.2 then ensures that all zeros of $\tilde{p}_i - \zeta p_i$ lie on $\partial \mathbb{D}^2$, so distance to the zero set is comparable to a product of $(1-|z_1|^2)$ and $(1-|z_2|^2)$. Lemma 2.4 computes the $\beta$-weighted volume of the resulting product region, and the proof absorbs the logarithmic factors by setting the domain exponent to $a=\beta/(2q)-2$.

What would settle it

For a concrete RIF pair such as $\Phi=(\phi,\phi)$ with $\phi=(2z_1z_2 - z_1 - z_2)/(2 - z_1 - z_2)$, compute the exact asymptotic of $V_\beta(\Phi^{-1}(S(1,\delta)))$ as $\delta \to 0$. The theorem predicts an upper bound of order $\delta_1^{\beta/4}\delta_2^{\beta/4}$ after absorbing logs, i.e. $q=2$. If the true asymptotic contains an extra logarithmic factor that cannot be absorbed by any finite $q$, or if the pullback volume grows faster than $C \delta_1^{\beta/4}\delta_2^{\beta/4}\sqrt{\log(1/\delta_1)\log(1/\delta_2)}$ for all $C$, then the claimed boundedness for the stated exponent relation would fail.

Watch

Extended reading notes

Core claim

The paper's central claim is that boundary singularities of rational inner functions do not, by themselves, prevent bounded composition operators between different weighted Bergman spaces on the bidisc. Specifically, for $\Phi=(\phi,\psi)$ with $\phi=\tilde{p}_1/p_1$ and $\psi=\tilde{p}_2/p_2$, where $p_1$ and $p_2$ are stable polynomials each vanishing at exactly one point of $\mathbb{T}^2$, there is a number $q>0$, coming from the Lojasiewicz exponents of the two local singularities, such that $C_\Phi : A^2_{\beta/(2q)-2}(\mathbb{D}^2) \to A^2_\beta(\mathbb{D}^2)$ is bounded for every $\beta>2q$. The mechanism is that on a neighborhood of the singularity, the Lojasiewicz inequality bounds $|\tilde{p}_i - \zeta p_i|$ from below by a power of the distance to the zero set, and because that zero set lies entire

Load-bearing premise

The proof leans on one uniform rate at which $|\tilde{p}-\zeta p|$ shrinks near the boundary zero, and assumes that rate is valid on a whole fixed neighborhood and turns directly into a product of the two distances to the boundary; if that local rate varies as the neighborhood shrinks, the volume estimate that powers the theorem no longer follows.

Editorial extensions

If this is right

For every pair of rational inner functions of the stated form, the composition operator is bounded from a whole family of weighted Bergman spaces: increasing β shifts the required domain weight a=β/(2q)−2 along with it.
The Lojasiewicz exponent q acts as a quantitative measure of the singularity's severity: the larger q, the more negative the domain weight must be for boundedness.
The theorem recovers the previously studied Knese-function example, where q=2, and extends it to the family Φ_{A,B} with A,B∈T\{1} and |A|+|B|=2, also with q=2.
The proof works only when β>2q, i.e. when the domain weight a=β/(2q)−2 is greater than −1; this is the technical boundary of the Carleson measure method.
The volume estimates are not sharp, so the theorem gives a sufficient but not necessary range of boundedness exponents.

Reading between the lines

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

The single-boundary-zero assumption is likely removable: if p₁ and p₂ have finitely many boundary zeros, the volume estimate should split into finitely many local contributions, with q taken as the maximum of the local Lojasiewicz exponents.
The exponent relation a=β/(2q)−2 suggests a scaling law for RIF symbols: the Lojasiewicz exponent plays the role of an effective order of the boundary zero, analogous to the degree of a Blaschke factor in one variable.
Because the author notes the estimates are not sharp, the optimal domain weight may be less negative than β/(2q)−2; computing exact asymptotics for the Knese function would show how much slack the Lojasiewicz route carries.
One could test the conjecture that the boundedness threshold is governed by the maximum coordinate singularity by computing V_β(Φ^{-1}(S)) explicitly for simple one-sided perturbations such as Φ_{A,B}.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Main take: The paper states a plausible generalization of composition-operator boundedness for pairs of rational inner functions on the bidisc, but the proof as written does not cover the theorem's stated assumptions. The central gap is real, not cosmetic.

What is new: prior work treated C^2-smooth symbols (Bayart, Kosiński) and the author's own earlier paper handled only the diagonal Φ=(φ,φ) for the Knese function. The present statement for arbitrary pairs of RIFs with one boundary singularity is a genuine extension. The strategy is sensible: use the Carleson measure criterion, apply Lojasiewicz near the singularity to control |φ(z)-ζ|, then use the volume estimate from Lemma 2.4. The final exponent comparison works, and the consistency with Example 5.1 in [3] is good evidence the theorem is not vacuous.

Where it breaks down: The proof requires a point η∈T^2 where both p1 and p2 vanish and both renormalized boundary values are 1. The theorem only assumes each p_i has one zero on T^2, not that the zeros coincide. You cannot rotate ψ to put its singularity at the same η without changing the map. So U=U1∩U2 may be empty. Also, the reduction "assume ζ1=1" only works for boxes centered at the singular boundary value (after a rotation of the symbol), not for arbitrary ζ∈T^2. For general ζ, you would need Lojasiewicz estimates for P_ζ=p̃−ζp with an exponent uniform in ζ; the proof supplies none. The phrase "arguing as in Theorem 10 of [10]" does not rescue this, because Kosiński's argument is for C^2-smooth symbols and does not address algebraic zero sets P_ζ, which can even be curves. These are load-bearing gaps: the main theorem is under-proven as written.

That said, none of this suggests the result is false. The gaps look repairable—one might split Carleson boxes depending on whether the center is near the singularity values and prove local uniformity for the Lojasiewicz exponents over compact families of polynomials. But the current manuscript does not do that work.

Audience: researchers in composition operators on polydiscs. The paper is short and readable. It deserves a serious referee: the question is meaningful, the strategy is promising, and a referee could identify whether the missing uniformity can be supplied or whether the statement needs an extra assumption. I would not cite it until the proof is fixed.

Referee Report

4 major / 3 minor

Summary. The paper studies composition operators C_Φ on weighted Bergman spaces of the bidisc, where Φ=(ϕ,ψ) and both coordinate functions are rational inner functions (RIFs) that may have boundary singularities. The main result, Theorem 2.5, asserts that if p_1 and p_2 are stable polynomials each having one zero on T², then there exists q>0 such that C_Φ:A^2_{β/(2q)-2}(D²)→A²_β(D²) is bounded for all β>2q. The proof uses the Carleson measure criterion (Lemma 2.1), a Lojasiewicz lower bound for |p̃_i−ζ p_i| near a boundary zero of the corresponding RIF, and the volume estimate in Lemma 2.4. The paper also gives two examples and an open problem.

Significance. The intended result is a substantial extension of earlier boundedness criteria for composition operators induced by smooth self-maps of the bidisc to the non-smooth setting of rational inner functions. If the theorem were fully proved, it would unify and generalize known examples, including the Knese-type symbols treated in the author's earlier work. The paper does not provide machine-checked proofs or numerical experiments; its value rests on the analytic argument. However, the proof as written has serious gaps in the passage from local Lojasiewicz estimates near a single boundary point to the required global Carleson estimate, and the statement of Theorem 2.5 is broader than what the proof actually establishes.

major comments (4)

[Theorem 2.5, proof, paragraph beginning 'Similarly to [10]'] The proof selects η∈T² with p_1(η)=p_2(η)=0, and then forms U=U_1∩U_2. The theorem only assumes that p_1 and p_2 each have one zero on T², not that they have a common zero. The sentence 'By rotating ψ ... to bring the singularity on the point η' is not a valid reduction: a single rotation of the bidisc moves both domain coordinates by the same unimodular factors and cannot send two distinct boundary zeros to the same point. If the zeros are distinct, U may be empty and the subsequent estimates prove nothing. The theorem either needs the additional assumption p_1(η)=p_2(η)=0 for a common η, or a completely different argument covering disjoint neighborhoods.
[Proof of Theorem 2.5, equation (2.1) and the Lojasiewicz inequality] The Lojasiewicz inequality is applied on U_1∩D², which is not compact unless a closed sub-neighborhood is specified and the inequality is stated with constants uniform there. More importantly, the inequality is proved only for the polynomial P_1=p̃_1−p_1, i.e. for the value ζ=1. The Carleson condition requires an estimate for every ζ∈T². The polynomial P_ζ=p̃_1−ζp_1 varies with ζ, and its zero set Z(P_ζ) can be a curve (e.g. for p=2−z_1−z_2 and ζ=−1, Z(P_ζ) is the union of two lines). The sentence 'Similarly to [10], we can assume ζ_1=1' is not justified by the smooth-symbol argument in [10]. Consequently, the existence of a single uniform exponent q and constant C valid for all box centers is not established.
[Proof of Theorem 2.5, paragraph after equation (2.3)] The sentence 'Arguing as in the proof of Theorem 10 in [10], it is enough to show ...' delegates the global-to-local reduction to a citation. The cited theorem concerns C²-smooth symbols and does not address the algebraic zero sets Z(P_ζ) that arise here. The proof also does not explain how the estimates on the single neighborhood U control Carleson boxes that do not meet U, nor how the logarithmic factors and the choice of δ_0 are made uniform over all ζ and all δ∈(0,2)². This step is load-bearing because the Carleson criterion in Lemma 2.1 requires all boxes.
[Section 3, example Φ_{A,B}] The claim 'After a rotational argument we can assume that the Lojasiewicz exponent for |g̃_{p_{A,B}}−p_{A,B}| will be equal to the one of |p̃−p|' is stated without proof. This is not obvious, since the polynomials depend on parameters A,B and the Lojasiewicz exponent is sensitive to the algebraic structure of the zero set. If this is intended as a concrete illustration of Theorem 2.5, the needed rotational reduction should be spelled out.

minor comments (3)

[Lemma 2.2] The statement 'Z(P_ζ) ∩ D² = ∅ and Z(P_ζ) ∩ D² ⊂ ∂D²' appears to contain a typo: the second D² should presumably be the closed bidisc (or the boundary should be described as in the proof, where the more precise decomposition (T×D)∪(D×T)∪T² is given). The notation ∂D² is ambiguous between the topological boundary and the distinguished boundary.
[Proof of Theorem 2.5, after equation (2.1)] The sentence 'Note that U_1∩D² is a small compact set around η' is inaccurate: an open neighborhood of η intersected with the open bidisc is not compact. The authors probably mean a compact sub-neighborhood obtained by taking a closed ball or closure; the distinction matters because the Lojasiewicz inequality is usually stated on compact sets.
[Introduction and notation] The definition of A²_β allows β≥−1, but the main theorem considers β>2q, so β>0. This is consistent, but the abstract says 'β is positive' while the introduction writes 'β≥−1'; the reader would benefit from a uniform notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the exponent q is a Lojasiewicz-geometric quantity, not a fitted parameter, and the proof relies on external Carleson and Lojasiewicz criteria.

full rationale

The central result, Theorem 2.5, derives boundedness from the Carleson measure criterion (Lemma 2.1), Lojasiewicz's inequality (cited as [12]), and an external zero-set lemma (Lemma 2.2, citing [13]). The exponent q is introduced as q = max{q1, q2}, where q1 and q2 are Lojasiewicz exponents obtained from the polynomials p1 and p2; it is not defined in terms of the desired boundedness conclusion, nor is it a parameter fitted to force the theorem. The only self-citation, to the author's previous work [3], appears in Section 3 as a consistency check ('the stated result is consistent with Example 5.1. in the paper [3]') and is not used anywhere in the proof. The proof may have a substantive gap: it assumes a common singularity η for both p1 and p2 and does not justify uniformity of the Lojasiewicz exponent over all Carleson-box centers ζ ∈ T². That is a mathematical-correctness concern, not circularity: no equation is assumed equivalent to the theorem, and no fitted quantity is renamed as a prediction. Accordingly, no circular step is identified.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities. It relies on standard function-theoretic tools (Carleson measures, Lojasiewicz inequality, Rudin's RIF characterization) and a domain assumption on the number of boundary zeros. The only parameter that is not made explicit is the Lojasiewicz exponent q, which is existential.

free parameters (1)

q (Lojasiewicz exponent) = unspecified, q = max(q₁, q₂)
The theorem concludes 'there exists q > 0', so the admissible range of β (β > 2q) is not explicit. q is determined by the polynomials via the Lojasiewicz inequality, but its value is not computed, making the result qualitative.

assumptions (6)

standard math Carleson measure criterion for weighted Bergman spaces on the bidisc (Lemma 2.1, from [11])
The boundedness of C_Φ is equivalent to the volume estimate V_β(Φ^{-1}(S(ζ, δ))) ≤ C V_a(S(ζ, δ)) for all Carleson boxes. This is a known theorem and is the central tool.
standard math Rudin's characterization of rational inner functions on the bidisc: φ = z₁^N z₂^M p̃/p (from [14])
Used in the introduction to set the form of φ and ψ. It is a standard result from the cited reference.
standard math Lojasiewicz inequality (cited [12])
Invoked in the proof of Theorem 2.5 to bound |P_ζ(z)| from below by a power of the distance to the zero set. The paper cites the inequality but does not prove it.
standard math Knese's theorem on non-tangential limits of rational inner functions on T² (cited [9])
Used in the introduction to state that RIFs have unimodular non-tangential limits everywhere on T², which underlies the choice of the singular value.
domain assumption Both p₁ and p₂ have exactly one zero on T²
This is the standing assumption of Theorem 2.5. The proof uses the existence of a single singularity η where the Lojasiewicz estimate is applied. It is not ad hoc, but it restricts the class of symbols.
standard math Volume asymptotic V_β({(1−|z₁|²)(1−|z₂|²) ≤ δ}) ≃ δ^{β+1} log(1/δ) (Lemma 2.4)
Proved by a routine change of variables; the paper provides a short proof. It is a background calculation needed for the main estimate.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Composition Operators and Rational Inner Functions II: Boundedness between two different Bergman Spaces." pith.science (2026). https://pith.science/paper/MBKZDFYZ

@misc{pith2026250904366,
  author       = {Pith},
  title        = {Pith review of: Composition Operators and Rational Inner Functions II: Boundedness between two different Bergman Spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBKZDFYZ}},
  note         = {Machine review of arXiv:2509.04366}
}

abstract

In this note we provide a sufficient condition on when the composition operator $C_{\Phi}:A^2_{a}(\mathbb{D}^2)\to A^2_{\beta}(\mathbb{D}^2)$ is bounded, whenever $a\ge-1$ and $\beta$ is positive, with the assumption that $\Phi$ is induced by non-smooth Rational Inner Functions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[3]

Beslikas, Composition Operators and Rational Inner Functions on the bidisc, Proceedings of the American Mathematical Society, Vol

A. Beslikas, Composition Operators and Rational Inner Functions on the bidisc, Proceedings of the American Mathematical Society, Vol. 153, No. 8, August 2025

work page 2025
[10]

Kosi´ nski,Composition operators on the polydisc, Journal of Functional Analysis, Volume 284, Issue 5, 1 March 2023, 109801

L. Kosi´ nski,Composition operators on the polydisc, Journal of Functional Analysis, Volume 284, Issue 5, 1 March 2023, 109801

work page 2023
[1]

F.Bayart, Composition operators on the polydisk induced by affine maps Journal of Functional Analysis Volume 260, Issue 7, 1 April 2011, Pages 1969-2003

work page 2011
[2]

F.Bayart, Composition operators on the Hardy space of the tridisc , to appear in Indiana Uni- versity Mathematics Journal

work page
[4]

Bickel, Joseph A

K. Bickel, Joseph A. Cima, A. Sola, Clark Measures for Rational Inner Functions , Michigan Math. J. 73(5): 1021-1057 (November 2023). DOI: 10.1307/mmj/20216046

work page arXiv 2023
[5]

Bickel, G

K. Bickel, G. Knese, Inner functions on the bidisk and associated Hilbert spaces , Journal of Functional Analysis Volume 265, Issue 11 , 1 December 2013, Pages 2753-2790

work page 2013
[6]

Bickel, J.E

K. Bickel, J.E. Pascoe, A. Sola, Singularities of Rational Functions in higher dimensions, Amer- ican Journal of Mathematics Johns Hopkins University Press Volume 144, Number 4, August 2022

work page 2022
[7]

Bickel, J.E

K. Bickel, J.E. Pascoe, A. Sola, Derivatives of Rational Inner Functions and integrability at the boundary, Proceedings of the London Mathematical Society, Vol. 116, Issue 2, pp.281-329

work page

Show all 15 references

[8]

Knese, Rational Inner Functions in the Schur-Agler class of the polydisc , Publicacions Matem` atiques, Vol

G. Knese, Rational Inner Functions in the Schur-Agler class of the polydisc , Publicacions Matem` atiques, Vol. 55, No. 2 (2011), pp. 343-357

2011
[9]

Knese, Integrability and regularity of Rational Inner Functions, Proceedings of the London Mathematical Society, 2015, Vol 111, Issue 6, p1261

G. Knese, Integrability and regularity of Rational Inner Functions, Proceedings of the London Mathematical Society, 2015, Vol 111, Issue 6, p1261

2015
[11]

H. Koo, M. Stessin, K. Zhu, Composition operators on the polydisc induced by smooth symbols , Journal of Functional Analysis 254 (2008) 2911–2925

2008
[12]

Lojasiewicz, Introduction to Complex Analytic Geometry ,Transl

S. Lojasiewicz, Introduction to Complex Analytic Geometry ,Transl. from the Polish by Maciej Klimek.- Basel: Boston: Berlin: Birkh¨ auscr, 1991

1991
[13]

Bickel, K., Eldred Pascoe, J., and Sola, A. (2020). Level curve portraits of rational inner func- tions. ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE, 21(special, 1-2), 449-494. https://doi.org/10.2422/2036-2145.201804.025

2020
[14]

Rudin, Function Theory in polydisks , W

W. Rudin, Function Theory in polydisks , W. A. Benjamin, Inc., New York Amsterdam, 1969

1969
[15]

J.H.Shapiro The essential norm of a composition operator, Ann. Math. 12 (1987), 375-404. Athanasios Beslikas, Doctoral School of Exact and Natural Studies Institute of Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University Lojasiewicza 6 PL30348, Cra...

1987

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [3]

Beslikas, Composition Operators and Rational Inner Functions on the bidisc, Proceedings of the American Mathematical Society, Vol

A. Beslikas, Composition Operators and Rational Inner Functions on the bidisc, Proceedings of the American Mathematical Society, Vol. 153, No. 8, August 2025

work page 2025

[2] [10]

Kosi´ nski,Composition operators on the polydisc, Journal of Functional Analysis, Volume 284, Issue 5, 1 March 2023, 109801

L. Kosi´ nski,Composition operators on the polydisc, Journal of Functional Analysis, Volume 284, Issue 5, 1 March 2023, 109801

work page 2023

[3] [1]

F.Bayart, Composition operators on the polydisk induced by affine maps Journal of Functional Analysis Volume 260, Issue 7, 1 April 2011, Pages 1969-2003

work page 2011

[4] [2]

F.Bayart, Composition operators on the Hardy space of the tridisc , to appear in Indiana Uni- versity Mathematics Journal

work page

[5] [4]

Bickel, Joseph A

K. Bickel, Joseph A. Cima, A. Sola, Clark Measures for Rational Inner Functions , Michigan Math. J. 73(5): 1021-1057 (November 2023). DOI: 10.1307/mmj/20216046

work page arXiv 2023

[6] [5]

Bickel, G

K. Bickel, G. Knese, Inner functions on the bidisk and associated Hilbert spaces , Journal of Functional Analysis Volume 265, Issue 11 , 1 December 2013, Pages 2753-2790

work page 2013

[7] [6]

Bickel, J.E

K. Bickel, J.E. Pascoe, A. Sola, Singularities of Rational Functions in higher dimensions, Amer- ican Journal of Mathematics Johns Hopkins University Press Volume 144, Number 4, August 2022

work page 2022

[8] [7]

Bickel, J.E

K. Bickel, J.E. Pascoe, A. Sola, Derivatives of Rational Inner Functions and integrability at the boundary, Proceedings of the London Mathematical Society, Vol. 116, Issue 2, pp.281-329

work page