{"id":"f7f2b8fc-26d0-4502-85a6-1e1ae1effb50","arxiv_id":"2509.04366","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For pairs of rational inner functions on the bidisc with one boundary singularity each, the induced composition operator is bounded from A^2_{β/(2q)-2} to A^2_β whenever β > 2q, where q is a Lojasiewicz exponent.","lead":"This paper proves a boundedness condition for composition operators on two-variable weighted Bergman spaces when the defining maps are rational inner functions with boundary singularities. The result extends earlier work that only handled smooth maps or a single example map.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof's reduction to a single common singularity and to ζ=(1,1) is not justified; the Lojasiewicz exponent and Carleson estimates are not shown uniform over all box centers.","rationale":"The reader identified the main weak point as the uniformity/global-to-local reduction. My stress test sharpens this: the proof's WLOG normalization requires the two singularities to coincide and the box center to be simultaneously rotated to (1,1), which is not possible for arbitrary ζ when the singular limits differ. This is a concrete, load-bearing gap in the proof of Theorem 2.5. However, the conclusion may still be true and addressable by a more careful parameter-dependent Lojasiewicz argument plus the smooth estimates of Kosiński. I therefore do not change the CONDITIONAL verdict; I only make the requested test more specific.","tokens_in":5525,"tokens_out":23026,"duration_ms":236550,"concrete_test":"Take p1=2−z1−z2 (zero at (1,1)) and p2=2+z1+z2 (zero at (−1,−1)). First verify that no rotation R(z)=(αz1,βz2) maps both zeros to the same point, so U1∩U2 is empty; then directly compute Vβ(Φ^{-1}(S(ζ,δ))) for ζ=(1,1) and ζ=(−1,−1) using explicit p̃1−ζp1 and p̃2−ζp2, and compare the result with δ1^{β/(2q)}δ2^{β/(2q)}. In parallel, for a fixed p, compute the Lojasiewicz exponent q(λ) of p̃−λp near the singularity from the Newton polygon, for λ∈T. If q(λ) is unbounded, the uniform q in Theorem 2.5 fails; if the direct volume estimates fail the required bound, the theorem as stated is false; otherwise the gap is a missing but repairable argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 2.5's proof establishes the Carleson estimate only on U=U1∩U2, a neighborhood of a boundary point η chosen so that p1(η)=p2(η)=0 (and, after normalization, the singular limits are 1). The theorem assumes only that p1 and p2 each have one zero on T²; it does not assume the zeros coincide. A coordinate-wise rotation cannot move two distinct boundary zeros to the same η, so U may be empty in the stated generality. Also, to handle a Carleson box centered at arbitrary ζ∈T² one must rotate the target by ζ^{-1}, which changes the singular boundary values. Hence the normalized situation φ*(η)=ψ*(η)=1 and ζ=(1,1) cannot be achieved simultaneously for all ζ. The polynomial P_ζ = p̃−ζp can in principle be used for every ζ, but the Lojasiewicz exponent q(ζ) may depend on ζ, and the proof supplies no uniformity. The sentence 'Arguing as in the proof of Theorem 10 in [10]' does not fill this gap because [10] treats C²-smooth symbols; it does not address the algebraic zero sets Z(P_ζ), which vary with ζ and can be curves—e.g. for p=2−z1−z2, P_{-1}=2(z1−1)(z2−1). Thus the claimed existence of a single q and the claimed boundedness for all β>2q are not established by the proof as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":5846,"tokens_out":9533,"duration_ms":104207,"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":[{"comment":"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.","section":"Theorem 2.5, proof, paragraph beginning 'Similarly to [10]'"},{"comment":"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.","section":"Proof of Theorem 2.5, equation (2.1) and the Lojasiewicz inequality"},{"comment":"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":"Proof of Theorem 2.5, paragraph after equation (2.3)"},{"comment":"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.","section":"Section 3, example Φ_{A,B}"}],"minor_comments":[{"comment":"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.","section":"Lemma 2.2"},{"comment":"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.","section":"Proof of Theorem 2.5, after equation (2.1)"},{"comment":"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.","section":"Introduction and notation"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a short note whose main theorem is not proved as stated. The proof's reliance on a common boundary zero for p_1 and p_2 is a structural gap: the theorem either needs a strengthened hypothesis or a genuinely new argument. The uniformity issues with the Lojasiewicz exponent and the dependence on ζ are also substantial. I would ask for a revised version that either restricts Theorem 2.5 to the common-zero case or proves the general case, and that gives a self-contained global-to-local reduction. The examples in Section 3 also need supporting arguments. The work is interesting enough to warrant revision, but the present version should not be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nWhere 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.\n\nThat 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.\n\nAudience: 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.","headline":"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.","tokens_in":6361,"tokens_out":9947,"would_cite":false,"duration_ms":97010,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32A37","32A40","30J10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["rational inner functions","composition operators","weighted Bergman spaces","bidisc","Carleson measures","Lojasiewicz inequality","boundary singularities"],"falsifier":"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.","tokens_in":5393,"feed_emoji":"📐","tokens_out":8102,"duration_ms":77734,"temperature":0.7,"texified_at":"2026-08-05T20:25:21.048813+00:00","pith_summary":"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.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":8668,"prompt_tokens":961,"completion_tokens":7707,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":961,"completion_tokens_details":{"reasoning_tokens":6693}},"feed_headline":"Boundary singularity need not break Bergman boundedness","feed_subtitle":"For rational inner functions with one boundary zero per coordinate, a negative-weight domain keeps the composition operator bounded.","key_machinery":"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 $a=\\beta/(2q)-2$.","core_discovery":"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","pith_inferences":["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}."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Rudin's theorem giving the representation of rational inner functions on the bidisc as p̃/p, which is the form used throughout.","marker":"[14]"},{"why":"States the Carleson box volume criterion (Lemma 2.1) that reduces boundedness of the composition operator to pullback measure estimates.","marker":"[11]"},{"why":"Provides the Lojasiewicz inequality used to get a polynomial lower bound near the boundary singularity.","marker":"[12]"},{"why":"Supplies the global reduction and the method for choosing ζ₁=1 and absorbing logarithmic factors over Carleson boxes.","marker":"[10]"},{"why":"Proves that the zeros of p̃−ζp lie on the boundary of the bidisc, which is used to turn distance into products of (1−|z_i|²) factors.","marker":"[13]"},{"why":"Contains the earlier Knese-function example with q=2 that the theorem recovers and extends.","marker":"[3]"}],"fun_headline_variants":["RIF boundary zeros don't preclude Bergman boundedness","Weighted Bergman bound survives rational inner singularities","Lojasiewicz exponent controls composition operator bound","Bidisc composition bounded despite one boundary zero","Negative-weight Bergman space tames RIF singularity"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["RIF boundary zeros don't preclude Bergman boundedness","Weighted Bergman bound survives rational inner singularities","Lojasiewicz exponent controls composition operator bound","Bidisc composition bounded despite one boundary zero","Negative-weight Bergman space tames RIF singularity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000152,"raw_usage":{"total_tokens":979,"prompt_tokens":622,"completion_tokens":357,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":366,"completion_tokens_details":{"reasoning_tokens":283}},"tokens_in":366,"tokens_out":357,"duration_ms":3933,"temperature":1.0,"reasoning_tokens":283,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T10:13:08.331990+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Rudin, Function Theory in polydisks , W","cited_arxiv_id":null,"evidence_quote":"Supplies Rudin's theorem giving the representation of rational inner functions on the bidisc as p̃/p, which is the form used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the Carleson box volume criterion (Lemma 2.1) that reduces boundedness of the composition operator to pullback measure estimates."},{"cited_title":"Lojasiewicz, Introduction to Complex Analytic Geometry ,Transl","cited_arxiv_id":null,"evidence_quote":"Provides the Lojasiewicz inequality used to get a polynomial lower bound near the boundary singularity."},{"cited_title":"Kosi´ nski,Composition operators on the polydisc, Journal of Functional Analysis, Volume 284, Issue 5, 1 March 2023, 109801","cited_arxiv_id":null,"evidence_quote":"Supplies the global reduction and the method for choosing ζ₁=1 and absorbing logarithmic factors over Carleson boxes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves that the zeros of p̃−ζp lie on the boundary of the bidisc, which is used to turn distance into products of (1−|z_i|²) factors."},{"cited_title":"Beslikas, Composition Operators and Rational Inner Functions on the bidisc, Proceedings of the American Mathematical Society, Vol","cited_arxiv_id":null,"evidence_quote":"Contains the earlier Knese-function example with q=2 that the theorem recovers and extends."}],"review_version":1}