{"id":"38af16e5-c6ec-4e7b-8203-aae0beb8ab4d","arxiv_id":"2412.16593","paper_version":3,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Rational inner functions with one boundary singularity are claimed to induce unbounded composition operators on the bidisc Bergman space, while stable polynomial symbols give bounded operators between weighted Bergman spaces.","lead":"This paper studies composition operators on the bidisc whose coordinate functions are rational inner functions with boundary singularities. It claims such operators are often unbounded on the Bergman space, but bounded between certain weighted spaces when the defining polynomial is stable.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 2.1 establishes the Carleson-box volume lower bound only for δ∈(1/2,1); it gives no estimate for δ→0, so it cannot yield unboundedness on A^2(D^2).","rationale":"The paper's central claim, Theorem 2.1, is an unboundedness result for composition operators with singular rational inner symbols. The reader's verdict of REJECT is supported: the proof has a real gap that is independent of the distance-identification issue already flagged. Even if Lemma 4.1 is accepted, the chain (4.6)-(4.11) only produces a volume lower bound for boxes of scale δ∈(1/2,1). Unboundedness requires a violation of the Carleson condition at arbitrarily small scales, because boundedness is equivalent to a uniform volume estimate for all boxes. A lower bound on a fixed compact interval of δ is consistent with boundedness, so the argument does not establish the theorem. The reciprocal structure of Lemma 4.1 makes this unavoidable: the bound |φ(z)-1| ≤ Cε/dist^q becomes weak precisely when dist→0, and forcing it below δ pushes the relevant annulus to distance ~δ^{-1/q}, outside the local neighborhood U for small δ. Thus the written proof fails at the point where the theorem needs its quantitative strength. The reader's weakest-assumption is related but distinct; I agree with the overall rejection while locating the load-bearing failure in the unrestricted δ-range required by the Carleson criterion.","tokens_in":11382,"tokens_out":11319,"duration_ms":100964,"concrete_test":"For a concrete symbol satisfying the hypotheses, e.g. p=(2-z1-z2)^2 and φ=\\tilde p/p, which has a single zero τ=(1,1) on T^2 and Lojasiewicz exponent q=2, test the claimed lower bound for δ=0.1, 0.01, 0.001: the annulus in (4.8) has lower endpoint (ε/(2δ))^{1/2} and upper endpoint ε^{1/2}; for δ<1/2 the lower endpoint exceeds the upper endpoint, so the set is empty and the proof's lower bound is unsupported. Independently compute the true volume V_{A^2}({z∈D^2: |φ(z)-1|<δ}) numerically in polar coordinates around (1,1) and compare with Cδ^2 and Cδ^4; this shows which scale the theorem could actually have.","verdict_should_be":"UNCHANGED","load_bearing_attack":"At the core of Theorem 2.1 is the volume estimate (4.10): V(Φ^{-1}(S(e,δ_1))) > C'(ε)δ_1^{4/q}. The set used to justify it is the bi-annulus (ε/(2δ_1))^{1/q} < |z_i-1| < ε^{1/q}, i=1,2. This annulus is nonempty only if (ε/(2δ_1))^{1/q} < ε^{1/q}, i.e. δ_1 > 1/2. Thus the displayed estimate is proved only for δ_1 ∈ (1/2,1). But unboundedness of C_Φ on A^2(D^2) requires inverse images of Carleson boxes of arbitrarily small size δ→0 to have volume not O(δ^4); the finite interval (1/2,1) is compatible with boundedness and gives no violation. The obstruction is not cosmetic: Lemma 4.1 gives |φ-1| ≤ Cε/dist^q, so forcing this bound below δ requires dist ≳ δ^{-1/q}; as δ→0 this region moves away from τ and out of the local neighborhood U, so the bi-annulus construction cannot be rescaled to small δ. Theorem 2.1 is therefore not proven by the argument in §4.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies composition operators C_Φ on weighted Bergman spaces A^2_β(D^2) for symbols Φ=(φ,φ) whose coordinate function φ is a rational inner function (RIF) on the bidisc. The main result, Theorem 2.1, claims that if φ=p̃/p with p stable on D^2, p has a single zero τ on T^2, and |p(z)| ≥ C dist^q(z,Z_p) near τ for some q>1, then C_Φ is unbounded on the unweighted space A^2(D^2). Theorem 2.2 claims boundedness between two different weighted Bergman spaces, A^2_{β/2-2} → A^2_β, for β>4 when p is stable. Theorem 2.3 gives a sufficient volume condition for boundedness on A^2_β(D^n) for Schur-Agler RIFs. The paper also contains three examples: one for the boundedness theorem, one purportedly showing unboundedness, and one for the stable-polynomial boundedness result.","tokens_in":11672,"tokens_out":10723,"duration_ms":87720,"significance":"If Theorem 2.1 were correct, it would be a significant first step: it would give the first unboundedness result for composition operators on A^2(D^2) induced by rational inner symbols with a boundary singularity. The proof strategy—using Carleson-box volume estimates and local estimates of |φ(z)-1| near the singularity—is natural and connects with the existing literature of Bayart and Kosiński. The paper also contains a potentially valid positive result in Theorem 2.2, and the use of established tools (Carleson measure criterion, Lojasiewicz inequality, Rudin's representation, Bickel's stability criterion) is appropriate. However, the main theorem is not established: the proof has a load-bearing gap and the key lemma contains an unjustified geometric comparison. The unconditional significance of the paper therefore rests on a repair that the current argument does not provide.","major_comments":[{"comment":"The volume lower bound V(Φ^{-1}(S(e,δ_1))) > C'(ε)δ_1^{4/q} is proved only for δ_1 ∈ (1/2,1). Indeed, the chosen bi-annulus (ε/(2δ_1))^{1/q} < |z_i-1| < ε^{1/q} is nonempty exactly when δ_1 > 1/2, and the text explicitly restricts to 1 > δ_1 > 1/2. Unboundedness of C_Φ on A^2(D^2) requires a violation of the Carleson estimate for boxes of arbitrarily small side length δ→0; a lower bound on a bounded interval away from zero is compatible with boundedness and gives no contradiction. Moreover, as δ_1→0, the lower radius (ε/(2δ_1))^{1/q} grows without bound, so the bi-annulus eventually leaves the neighborhood U provided by Lemma 4.1; the construction cannot be rescaled to small δ. Thus Theorem 2.1 is not proven by the argument presented.","section":"§4, proof of Theorem 2.1, Eqs. (4.6)–(4.11)"},{"comment":"The proof replaces dist(z, Z_p ∩ T^2) by the Euclidean distance to the point τ, i.e. (|z_1-1|^2+|z_2-1|^2)^{1/2}, and writes dist^q(z, Z_p∩T^2) = (|z_1-1|^2+|z_2-1|^2)^{q/2}. This is unjustified: Z_p is a complex hypersurface through τ, and the distance from z to that hypersurface is generally strictly smaller than the distance from z to the single point τ. Since inequality (4.4) is used as an upper bound on |φ(z)-1|, replacing the larger quantity dist(z,Z_p) in the denominator by the smaller quantity |z-τ| makes the fraction larger, not smaller; the needed comparison is the opposite. This error changes the exponent q in the subsequent volume estimate and is load-bearing for the claimed δ^{4/q} lower bound.","section":"§4, Lemma 4.1, Eqs. (4.4)–(4.5)"},{"comment":"This example repeats the same defect as Theorem 2.1. The annulus constructed after inequality (5.9) requires (1/√2)√(ε/(2δ)) < |z_i-1| < √ε, which is nonempty only when δ > ε/2, and the proof concludes with a 'suitable subinterval of (1/2,1)'. Therefore the displayed lower bound V > C(ε)δ^2 is obtained only for δ in (1/2,1), not for δ→0. Since unboundedness on A^2(D^2) requires boxes of arbitrarily small radius, this example does not demonstrate non-boundedness.","section":"§5, Example 5.2, Eqs. (5.7)–(5.13)"}],"minor_comments":[{"comment":"The text says 'by Lemma 4.3' where the reference should be to Lemma 4.1.","section":"§4, proof of Theorem 2.1"},{"comment":"The phrase 'Inequality (5.10) lets us conclude' appears to refer to inequality (5.9), not (5.10).","section":"§5, Example 5.2"},{"comment":"There are stray '/suppress' commands in the text (e.g., '/suppress Lukasz Kosiński', '/suppress Lojasiewicz') that should be removed; they appear to be typographical artifacts.","section":"Throughout, esp. §1.3, §3.3, §4"},{"comment":"The lemma is stated for a general RIF with one singularity, but the proof tacitly uses that p has a zero at τ and that p̃(τ)=p(τ)=0; this should be stated explicitly for clarity.","section":"Lemma 4.1"}],"recommendation":"reject","confidential_remarks":"The main theorem of the paper, Theorem 2.1, is the advertised 'first unboundedness result' for singular rational inner symbols, and its proof fails for two independent reasons: the volume estimate does not cover arbitrarily small Carleson boxes, and Lemma 4.1 makes an unjustified geometric comparison between distance to a hypersurface and distance to a point. These are not local presentation issues; they strike at the central claim. The positive result in Theorem 2.2 seems plausible and may be salvageable as a separate contribution, but the current manuscript does not establish its headline result. I recommend rejection, while noting that a revised version focusing on the boundedness theorem and a corrected non-boundedness argument could be worth considering."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nYou should know two things about this paper. First, it's the first attempt I've seen to treat composition operators on the bidisc whose symbols are rational inner functions with boundary singularities. That's a natural gap in the Bayart–Kosinski program, and Theorem 2.2 (boundedness for stable polynomials between different Bergman spaces) looks like a real, citable result. Second, the advertised unboundedness result, Theorem 2.1, is not proved. The paper deserves a serious referee, but the core idea is worth keeping on the radar.\n\nWhat's new: the question itself, and the observation that Lojasiewicz-type control on the denominator can in principle be fed into Carleson-measure estimates. Theorem 2.2 seems to work; the proof via Bickel's inequality gives the right box-counting. Examples 5.1 and 5.3 are useful sanity checks.\n\nThe soft spots are at the center. The proof of Theorem 2.1 estimates the volume of Φ^{-1}(S(e,(δ,δ))) by a bi-annulus that exists only for δ∈(1/2,1). The Carleson criterion requires control for all δ arbitrarily small; a finite interval of scales can always be absorbed into the constant. So the argument never touches the scales that matter. Lemma 4.1, which is supposed to provide the local bound |φ−1| ≤ Cε/dist(z,τ)^q, replaces dist(z,Zp) by dist(z,τ) even though Zp is a hypersurface through τ. Those two distances are not comparable for points near the hypersurface; inside D^2 the ratio can be arbitrarily small. This is the central estimate, and it is not justified. Example 5.2 inherits both problems and adds a sign error: it estimates |φ+1| where the relevant quantity is |φ−1|. So the paper's main claim, as written, is unsupported.\n\nWho is this for? Specialists in composition operators on polydiscs or rational inner functions. If you work in that area, you'd want to know about the question and the partial result, but you cannot rely on the unboundedness theorem. A serious referee could probably help the author fix the technical steps, but the revision would be substantial. I'd send it to review rather than desk-reject, because the question is genuine and Theorem 2.2 is solid. Just don't expect the central theorem to survive in its current form.\n\nBest,\n[Your name]","headline":"Good question and a solid boundedness result for stable polynomials, but the main unboundedness theorem is not proven—the volume estimates only cover δ>1/2 and the key lemma misidentifies distances.","tokens_in":12168,"tokens_out":11375,"would_cite":false,"duration_ms":91413,"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":"One boundary singularity forces an unbounded composition operator","keywords":["composition operators","rational inner functions","bidisc","weighted Bergman spaces","boundary singularities","algebraic distance estimates","Schur-Agler class","polydisc"],"falsifier":"Test the comparison in Lemma 4.1 for $p=2-z_1-z_2$: near $(1,1)$, $Z_p$ is the line $z_1+z_2=2$, so the distance from $z$ to $Z_p$ is comparable to $|p(z)|$, whereas the distance from $z$ to the point $(1,1)$ is not; the claimed bound on $|\\phi(z)-1|$ in terms of the point distance therefore does not follow from the hypothesis. A direct computation of the volume $V(\\{z:|\\phi(z)+1|<\\delta\\})$ in Example 5.2 would show which exponent of $\\delta$ is actually correct.","tokens_in":11162,"feed_emoji":"📐","tokens_out":18752,"duration_ms":142212,"temperature":0.7,"pith_summary":"The paper asks whether a rational inner self-map of the bidisc can induce an unbounded composition operator on the Bergman space when the map has a boundary singularity. Its main theorem answers yes: for a symbol whose two coordinate functions are the same rational inner function $\\phi=\\widetilde p/p$ with a denominator having exactly one zero on the distinguished boundary and satisfying a mild lower-bound condition, $C_\\Phi$ is unbounded on $A^2(\\mathbb{D}^2)$. A contrasting theorem shows that when the denominator is stable on $\\mathbb{D}^2$, the same kind of symbol induces a bounded operator between appropriate weighted Bergman spaces. Together the results say that boundary singularities of rational inner functions are not automatically harmless, and the transition between boundedness and unboundedness is controlled by the zero set of the denominator.","feed_headline":"One boundary singularity forces an unbounded composition operator","feed_subtitle":"On the bidisc, one boundary zero makes the composition operator unbounded; stable denominators stay bounded.","key_machinery":"The load-bearing object is a rational inner function $\\phi=\\widetilde p/p$ on the bidisc, where $\\widetilde p$ is the reflection of $p$; such functions have modulus one on the distinguished boundary but may have singularities where $p$ vanishes there. Near a single boundary zero $\\tau$, Lemma 4.1 converts the assumed lower bound on $|p(z)|$ into an upper bound on $|\\phi(z)-1|$, showing that the sublevel set $\\{z:|\\phi(z)-1|<\\delta\\}$ contains a bi-annulus of Euclidean width comparable to $\\delta^{1/q}$. Its Bergman volume is at least $C\\delta^{4/q}$. The boundedness criterion on weighted Bergman spaces says $C_\\Phi$ is bounded only if pulled-back volumes of boundary boxes grow no faster than the boxes themselves, which on $A^2(\\mathbb{D}^2)$ is $\\delta^4$; the $4/q<4$ growth rate is the contradiction that yields unboundedness.","core_discovery":"The paper's central claim is Theorem 2.1: if $\\Phi=(\\phi,\\phi)$ is a holomorphic self-map of the bidisc and $\\phi=\\widetilde p/p$ is a rational inner function whose denominator polynomial $p$ is stable on $\\mathbb{D}^2$, has exactly one zero $\\tau$ on $\\mathbb{T}^2$, and obeys $|p(z)|\\ge C\\,\\mathrm{dist}^{q}(z,Z_p)$ near $\\tau$ for some $q>1$, then $C_\\Phi$ is unbounded on the unweighted Bergman space $A^2(\\mathbb{D}^2)$. The proof shows that the inverse image of a boundary box around the singular value contains a thin bi-annulus whose volume is bounded below by $C\\delta^{4/q}$, while boundedness on $A^2(\\mathbb{D}^2)$ would force that volume to be at most $C\\delta^4$; since $q>1$, the lower bound is the larger one as $\\delta\\to0$. The paper also proves Theorem 2.2, that for stable $p$ the operator $C_\\Phi$ is bounded from $A^2_{\\beta/2-2}(\\mathbb{D}^2)$ to $A^2_\\beta(\\mathbb{D}^2)$ for all $\\beta>4$, and Theorem 2.3, a sufficient volume condition in the Schur-Agler class on the polydisc.","pith_inferences":["A direct comparison of the distance to the zero set with the distance to the singular point would decide whether the exponent in Theorem 2.1 is optimal; for $p=2-z_1-z_2$, whose zero set near $(1,1)$ is a line, $\\mathrm{dist}(z,Z_p)$ is comparable to $|p(z)|$, so the natural sharp exponent there is $1$ rather than $2$.","The proof treats only diagonal symbols $\\Phi=(\\phi,\\phi)$; a natural next step is to test whether one singular coordinate is already enough when paired with a smooth second coordinate, since the volume mechanism would then see only one shrinking direction.","If the volume bound in the singular example is sharp at $\\delta^2$, the corresponding operator should remain unbounded on every weighted space $A^2_\\beta$ with $\\beta<4$; computing this threshold would turn the paper's boundedness/unboundedness gap into a sharp phase transition."],"forward_implications":["Every rational inner function satisfying the single-zero lower-bound hypothesis with $q>1$ induces a composition operator that is unbounded on $A^2(\\mathbb{D}^2)$.","For a stable denominator $p$ on $\\mathbb{D}^2$, the same diagonal symbol is bounded from $A^2_{\\beta/2-2}(\\mathbb{D}^2)$ into $A^2_\\beta(\\mathbb{D}^2)$ for every $\\beta>4$, with a polydisc analogue holding for $\\beta>2n$.","For Schur-Agler rational inner functions on the polydisc, a volume decay condition of the form $V_\\beta(\\{z:\\sum_j(1-|z_j|^2)\\mathrm{SOS}_{ij}\\le\\delta_i M\\})\\le C\\delta_i^{n(\\beta+2)}$ is sufficient for boundedness of $C_\\Phi$ on $A^2_\\beta(\\mathbb{D}^n)$.","The examples show that the same denominator $2-z_1-z_2$ can produce a bounded operator between weighted spaces for $\\beta\\ge8$ and an unbounded operator on the unweighted space, so the boundedness range is sensitive to the weights and to the numerator."],"supporting_citations":[{"why":"supplies the real-analytic lower bound on |p| near its zero set used to turn the hypothesis into a pointwise estimate on |phi-1|.","marker":"[16]"},{"why":"gives the Carleson-box volume criterion that converts inverse-image volume bounds into boundedness of the composition operator.","marker":"[2]"},{"why":"provides the polydisc version of the volume criterion and the first-order-condition framework being extended to singular symbols.","marker":"[13]"},{"why":"gives the representation of rational inner functions as reflected quotients that fixes the form of the symbol.","marker":"[18]"},{"why":"provides the stability criterion |tilde p|^2-|p|^2 >= C(1-|z1|^2)(1-|z2|^2) used in the boundedness theorem for stable denominators.","marker":"[5]"},{"why":"supplies the exponent q=2 for the polynomial 2-z1-z2 used in the concrete unbounded example.","marker":"[4]"},{"why":"defines the Schur-Agler class and the sum-of-squares decompositions behind the volume condition in Theorem 2.3.","marker":"[10]"},{"why":"establishes the existence of non-tangential limits at singular points, which allows normalizing the boundary value at the singularity to 1.","marker":"[11]"}],"fun_headline_variants":["One boundary zero forces unbounded composition on bidisc","Boundary singularity makes operator unbounded","One torus point dooms bounded composition","Rational inner maps: one boundary zero unbounded"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The unboundedness conclusion rests on the claim that, near the single boundary zero, the denominator shrinks like a power of the distance from $z$ to that point; if the zero set is a curve rather than a point, that comparison can fail and the exponent used in the volume estimate changes.","fun_headline_variants_meta":{"raw":{"variants":["One boundary zero forces unbounded composition on bidisc","Boundary singularity makes operator unbounded","One torus point dooms bounded composition","Rational inner maps: one boundary zero unbounded"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000882,"raw_usage":{"total_tokens":3799,"prompt_tokens":919,"completion_tokens":2880,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":535,"completion_tokens_details":{"reasoning_tokens":2823}},"tokens_in":535,"tokens_out":2880,"duration_ms":19720,"temperature":1.0,"reasoning_tokens":2823,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:29:50.817231+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the comparison in Lemma 4.1 for $p=2-z_1-z_2$: near $(1,1)$, $Z_p$ is the line $z_1+z_2=2$, so the distance from $z$ to $Z_p$ is comparable to $|p(z)|$, whereas the distance from $z$ to the point $(1,1)$ is not; the claimed bound on $|\\phi(z)-1|$ in terms of the point distance therefore does not follow from the hypothesis. A direct computation of the volume $V(\\{z:|\\phi(z)+1|<\\delta\\})$ in Example 5.2 would show which exponent of $\\delta$ is actually correct.","supporting_citations":[{"cited_title":"Lojasiewicz, Introduction to Complex Analytic Geometry ,Transl","cited_arxiv_id":null,"evidence_quote":"supplies the real-analytic lower bound on |p| near its zero set used to turn the hypothesis into a pointwise estimate on |phi-1|."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the Carleson-box volume criterion that converts inverse-image volume bounds into boundedness of the composition operator."},{"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":"provides the polydisc version of the volume criterion and the first-order-condition framework being extended to singular symbols."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the stability criterion |tilde p|^2-|p|^2 >= C(1-|z1|^2)(1-|z2|^2) used in the boundedness theorem for stable denominators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the exponent q=2 for the polynomial 2-z1-z2 used in the concrete unbounded example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the Schur-Agler class and the sum-of-squares decompositions behind the volume condition in Theorem 2.3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes the existence of non-tangential limits at singular points, which allows normalizing the boundary value at the singularity to 1."}],"review_version":1}