{"id":"8be48eb5-7168-44d5-b206-d8592a802286","arxiv_id":"1908.02192","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On a new class of Hartogs domains with parameter b, the Toeplitz operator with symbol K^{-t} is bounded from L^p to L^q exactly under three explicit conditions on p, q, t, n, k and b.","lead":"This mathematics paper finds the exact conditions under which a certain weighted projection operator is bounded between L^p and L^q spaces on a family of Hartogs-type domains with a power parameter b. The result gives a sharp three-region criterion and specializes to known Bergman projection bounds.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the only external dependency is the referenced determinant bound, which is standard under the stated smoothness assumptions.","rationale":"The reader accepted with moderate confidence, and our pass found no fatal gap. We initially worried about the Schur test for q<2 in the middle interval of Theorem 1.3(2), but the integrability lower bound is -(j+1+C_{b,k})/p^*, not -(j-1+C_{b,k})/p^*, so the interval for m_j is nonempty exactly when q exceeds the stated threshold; a concrete check with p=1.1, q=1.8, n=2 gives m_j in (-0.273, -0.202], which is nonempty. We also checked the sign in the exponent inequality of the necessity proof of Theorem 1.3(3); with the intended reading x = 1/l - (2(n+C_{b,k}))/p - (n-1+C_{b,k}), the lower bound is valid and the contradiction follows. The only place where the argument relies on an external result is the uniform determinant bound in Lemma 2.1. That bound is cited from Chen [5, Section 6] and is a standard consequence of boundary regularity of biholomorphic maps between smooth bounded strictly pseudoconvex domains, so I do not regard it as a live objection. The verdict should remain unchanged.","tokens_in":89,"tokens_out":61592,"duration_ms":1234859,"concrete_test":"As a check on the only external input, compute |det Phi'(z)| for a nontrivial admissible example: take n=2, l=1, k=1, b any positive integer, Omega_1=B_1, and phi_1(z)=(z+1/2)/(1+z/2), an automorphism of the unit disk. Then Phi(z)=(phi_1^{-1}(z_1), z_2) maps H^2_{1,b} to H^2_{1,phi_1,b}; verify that |(phi_1^{-1})'(z_1)| is bounded above and below by positive constants on |z_1|<=1. This confirms the reduction Lemma 2.1 in a nontrivial case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 1.3, and the proof rests on two pillars: the reduction Lemma 2.1 to the special domain H^n_{k_j,b}, which uses the uniform bound 0<c<=|det Phi'(z)|<=d inherited from Chen [5, Section 6], and the Schur-test/Green-function estimates on H^n_{k_j,b}. I checked the delicate steps. In the Schur test of Section 3.2, the parameter existence in (3.10)-(3.11) is valid when the lower bound is read as -(j+1+C_{b,k})/p^*, which is exactly the integrability threshold for |zeta|^{m_j p^*+j-1+C} in the complex plane; the L^∞ condition (3.14) is consistent with the detG' powers. In the necessity proof of Theorem 1.3(3), using x = 1/l - (2(n+C_{b,k}))/p - (n-1+C_{b,k}), the exponent inequality r^E >= r^{1/l-1} holds for t <= t_0, and the projection argument in (3.18) correctly leaves the index (0,...,0,k+C_{b,k},...,n-2+C_{b,k},0). I found no internal inconsistency. The determinant bound is a genuine external input, but for bounded smooth Omega_j biholomorphic to the ball, Bell/Fefferman extension theory gives smooth extension to the closure with invertible Jacobian, so the bound is standard rather than a live risk.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a class of bounded Hartogs domains H^n_{ {k_j, φ_j, b} } in C^n, defined by max_j ‖φ_j(~z_j)‖ < |z_{k+1}|^b < ... < |z_n|^b < 1, where b is a positive integer, and studies the L^p-L^q boundedness of the Toeplitz operator T_{K^{-t}} with symbol K^{-t}, K being the Bergman kernel on the diagonal. The main result, Theorem 1.3, gives a sharp trichotomy for 1 < p ≤ q < ∞: (1) T is unbounded for all t ≥ 0 when q ≥ (2n+2k(b-1))/(n-1+k(b-1)); (2) T is bounded if and only if t ≥ 1/p - 1/q when q lies in a middle interval; (3) T is bounded if and only if t exceeds a p-dependent threshold when q lies in the lower interval. Setting t = 0 and p = q yields Corollary 1.4, a sharp range for L^p boundedness of the Bergman projection, which generalizes results of Chen (b=1) and Beberok. The proof reduces via a biholomorphism to the special domain H^n_{ {k_j,b} } (Lemma 2.1), constructs an explicit orthogonal basis of the Bergman space (Lemma 2.2), and applies Schur's test with carefully chosen weights together with Green-function estimates and explicit test functions for necessity.","tokens_in":13386,"tokens_out":37044,"duration_ms":313800,"significance":"The result is a genuinely wider generalization of earlier work by Chen and Beberok, and it exhibits a new phenomenon: the boundedness range depends on the parameter b through k(b-1), not just the dimension. The criteria are explicit, sharp, and falsifiable, and the proof is essentially self-contained, relying only on standard external results such as Bell's extension theorem for the determinant bound and known pluricomplex Green-function estimates. I checked the delicate parameter ranges in the Schur test (Section 3.2), the exponent estimates in the necessity proofs, and the consistency of the endpoint cases with the Bergman projection corollary; the argument is coherent and the logic is sound. The paper is a solid contribution to the study of special Toeplitz operators on singular Hartogs-type domains.","major_comments":[],"minor_comments":[{"comment":"The subsection heading \"1.2 The orthogonal basis\" appears to be misnumbered; it should be \"2.2\", and the subsequent subsection \"1.3\" should be \"2.3\".","section":"Section 2.2 (heading)"},{"comment":"In the displayed formula for T_{K_1^{-t}}(f_j)(G(η)), the denominator under \\hat{K}_1(η,ζ) should be \\overline{\\det G'(ζ)}, not \\det G'(ζ). The Bergman kernel transformation rule requires the complex conjugate in the second argument; with the printed denominator the angular integration would not select the claimed index β = (0,...,0,k+C_{b,k},...,n-2+C_{b,k},0). The subsequent computation confirms that the conjugate is intended.","section":"Equation (3.18)"},{"comment":"The condition defining the set A in the basis has garbled summation indices; the display uses m both as a summation index and as the lower-bound variable. Please rewrite with unambiguous indices, for example: for m = k+1,...,n, ∑_{s=1}^m α_s + (b-1)∑_{s=1}^k α_s > (1-b)k - m.","section":"Lemma 2.2"},{"comment":"In the proof of necessity in Theorem 1.3(3), the exponent x in h(r) = r^x is defined with an \"l\" that is the index of the annulus (a_{l+1}, a_l]. To avoid confusion, state explicitly that h(r) = r^{x_l} on that annulus, with x_l = 1/l - 2(n+C_{b,k})/p - (n-1+C_{b,k}).","section":"Section 3.3, definition of h"},{"comment":"The lower bound for m_j is written as \"-j+1+C_{b,k} p^*\" in the display; it should be typeset as a fraction, (-j+1+C_{b,k})/p^*, to avoid ambiguity.","section":"Equation (3.10)"},{"comment":"The symbol \"/greaterorsimilar\" in the displayed estimate should be \"≳\".","section":"Proof of Lemma 2.7"}],"recommendation":"minor_revision","confidential_remarks":"The paper is mathematically sound and the main theorem is a clear advance over the cited results. The only issues I found are typographical and notational, including the missing conjugate in (3.18) and the garbled summation indices in Lemma 2.2. These are local and easily fixed. I recommend acceptance after minor revisions; no further technical verification seems necessary."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Know this paper by its main theorem: for the new Hartogs domains H^n_{k_j,phi_j,b}, it gives the sharp L^p-to-L^q range for the Toeplitz operator with symbol K^{-t}, and the bound genuinely depends on b. The dependence is the real news; when b=1 it collapses to Chen's Bergman projection theorem, and when k=n-1, phi=id, it recovers Beberok. The proof is a competent assembly of standard machinery: explicit orthonormal basis via a unitary transform to a product domain, Schur test with carefully chosen weights, and Green-function lower bounds for necessity. I checked the delicate parameter ranges in (3.10)-(3.11) and the exponent comparison in the necessity proof of Theorem 1.3(3); they are consistent. The cited results from Chen, Beberok, Khanh-Liu-Thuc are used as tools, not as hidden inputs of the conclusion. No fitted parameters, no circularity. The citation practice is honest.\n\nThe only external dependency worth mentioning is the uniform bound 0<c<=|det Phi'(z)|<=d in Lemma 2.1, taken from Chen. If that bound failed, the transfer from the special domain H^n_{k_j,b} to the general one would break. But under the stated smoothness of the Omega_j and biholomorphisms extending smoothly to the boundary, this is standard Bell/Fefferman material, not a live risk. I would not hold the paper hostage to it.\n\nSoft spots are mostly cosmetic. The proof is long and a few estimates are sketched rather than fully written out; there are minor typos (e.g., section numbering jumps to 1.2/2.3, some 'greaterorsimilar' artifacts in the text). The Schur test in Section 3.2 would benefit from one more line explaining why the m_j lower bound is exactly the integrability threshold. But none of this affects the validity of the main theorem as far as I can tell. It was not machine-checked, so a careful referee should redo the exponent bookkeeping, but I expect it to pass.\n\nWho gets value: anyone working on Bergman projections and Toeplitz operators on non-smooth Hartogs-type domains. It is a solid incremental extension with a clean sharp result. If I were the editor, I would send it to peer review. The right referee is someone comfortable with Schur-test estimates on Reinhardt/Hartogs domains; they should ask for expanded details rather than substantive changes.","headline":"Solid sharp Lp-Lq Toeplitz result on a new Hartogs family; the b-dependence is real and the proof checks out.","tokens_in":13982,"tokens_out":2575,"would_cite":true,"duration_ms":26896,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["32A36","32A25","32A07"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a sharp criterion, depending on the exponent t and the domain parameters n, k, b, for when the Toeplitz operator with symbol K^{-t} is bounded between L^p and L^q on a class of bounded Hartogs domains.","keywords":["Hartogs domains","Toeplitz operators","Bergman projection","Bergman kernel","Lp-Lq boundedness","Schur test","pluricomplex Green function","sharp estimates"],"falsifier":"On the special domain $H^n_{k,b}$, take the monomial $f(z)=z_n^{1-n-k(b-1)}$. The proof shows it is in $A^2$ and that $T_{K^{-t}}$ sends it to a nonzero constant multiple of itself; a direct calculation gives $f \\in L^q$ exactly when $q < \\frac{2n+2k(b-1)}{n-1+k(b-1)}$. If one could exhibit any $q$ at or above that threshold for which $T_{K^{-t}}$ is bounded on all of $L^p$, the theorem's first part would be false.","tokens_in":12905,"feed_emoji":"📐","tokens_out":11453,"duration_ms":103217,"temperature":0.7,"pith_summary":"The paper establishes a sharp answer to a \"gain\" question for Toeplitz operators: on a wide class of bounded Hartogs domains, the operator with symbol $K^{{-t}}$ (the t-th power of the reciprocal of the Bergman kernel on the diagonal) is bounded from L^p to L^q exactly when t clears an explicit threshold that depends on p, q, the dimension n, the number k, and the power b appearing in the domain definition. This matters because the answer shows how a boundary singularity, not just smoothness, controls how much integrability a Bergman-type projection can gain. The result unifies and extends the known L^p ranges for the Bergman projection on the classical Hartogs triangle and its fat and power-generalized versions. Setting t=0 and p=q recovers a sharp range for the Bergman projection itself, and the range shrinks toward p=2 as b grows.","feed_headline":"Exact Lp-to-Lq threshold found for Hartogs Toeplitz operators","feed_subtitle":"The Bergman-kernel power t that turns on bounded Toeplitz operators is now known exactly.","key_machinery":"The proof is carried by four pieces. First, a biholomorphism $\\Phi$ sends the general domain $H^n_{k_j,\\varphi_j,b}$ to the special domain $H^n_{k_j,b}$ where all $\\varphi_j$ are identity maps, and a uniform two-sided bound on $|\\det \\Phi'|$ transfers $L^p$-$L^q$ boundedness of the Toeplitz operators between the two domains. Second, an explicit biholomorphism $\\Psi$ to a product of unit balls and punctured disks normalizes the Bergman kernel on the diagonal to a product formula involving $1-\\|\\eta_j\\|^2$ and $1-|\\eta_j|^2$, so the kernel's singular structure is visible. Third, a generalized Schur test with power weights $h_1$, $h_2$, $f$ turns the desired boundedness into two integral estimates on balls and disks. Fourth, necessity is forced by testing against monomials and by lower bounds from the pluricomplex Green function, which show that if $t$ is below the stated threshold the operator cannot be bounded.","core_discovery":"On the domain $H^n_{k_j,\\varphi_j,b}$, defined by $\\max_j \\|\\varphi_j(\\tilde z_j)\\| < |z_{k+1}|^b < \\cdots < |z_n|^b < 1$, the paper proves that $T_{K^{-t}}: L^p \\to L^q$ is bounded if and only if one of three mutually exclusive regimes holds. For $q \\geq \\frac{2n+2k(b-1)}{n-1+k(b-1)}$, the operator is unbounded for every $t \\geq 0$. For $q$ strictly between $\\frac{2(n-1)+2k(b-1)}{n+1+k(b-1)-2/p}$ and the upper threshold, boundedness holds exactly when $t \\geq \\frac{1}{p} - \\frac{1}{q}$. For $q$ from $p$ up to that middle interval, boundedness holds exactly when $t > \\frac{1}{2p} + \\frac{1-p}{2p}\\cdot\\frac{n+1+k(b-1)}{n-1+k(b-1)}$. Thus the exponent $t$ is the only adjustable knob: once it clears the stated threshold, the operator gains the full $L^p$-to-$L^q$ integrability that the geometry permits.","pith_inferences":["A natural testable extension is to let $b$ be a positive real number instead of an integer; the biholomorphism $\\Psi$ would no longer be single-valued, so the paper's method does not directly apply, and the sharp threshold may change or require a different normalization.","Because the proof isolates the role of the parameter $C_{b,k}=k(b-1)$ as an effective exponent shift, one can expect analogous $L^p$-$L^q$ thresholds for other radial symbols, such as powers of the distance to the boundary, with $C_{b,k}$ replaced by a suitable boundary-order parameter.","The necessity argument via the pluricomplex Green function could also yield two-sided estimates of the operator norm near the critical exponents, a quantitative question the paper does not address."],"forward_implications":["The Bergman projection on $H^n_{k_j,\\varphi_j,b}$ is $L^p$-bounded exactly for $p$ in $\\left(\\frac{2n+2k(b-1)}{n+1+k(b-1)}, \\frac{2n+2k(b-1)}{n-1+k(b-1)}\\right)$, so the admissible $p$-range shrinks as $b$ increases and collapses to $\\{2\\}$ as $b \\to \\infty$.","For any fixed $p,q$ in the middle regime, taking $t \\geq \\frac{1}{p} - \\frac{1}{q}$ makes $T_{K^{-t}}$ bounded, so a sufficiently negative power of the Bergman kernel always buys the full possible gain in integrability.","The boundedness criterion is unchanged when the biholomorphisms $\\varphi_j$ are replaced by any other biholomorphisms from the same smooth domains onto unit balls, because only the uniform determinant bound matters.","Corollary 1.4 recovers the known dimension-only interval for $b=1$ and the known power-generalized Hartogs triangle range when $l=1$, $k=n-1$, $\\varphi_1$ is the identity; in that sense the new statement contains the earlier sharp results as endpoints.","The sharp ranges in the theorem apply to the special Toeplitz operator with symbol $K^{-t}$ on every domain in this class, not just to the Bergman projection, so the result gives a complete $L^p$-$L^q$ picture for these operators."],"supporting_citations":[{"why":"Establishes the uniform determinant bound for the biholomorphism between the special and general domains, which Lemma 2.1 uses to transfer boundedness.","marker":"[5]"},{"why":"Supplies the generalized Schur test with weights h1, h2, f used to prove the sufficiency of the thresholds.","marker":"[11]"},{"why":"Provides the fat Hartogs triangle boundedness result and the monomial and Green-function testing technique adapted in the necessity proofs.","marker":"[12]"},{"why":"Gives the earlier L^p range for the power-generalized Hartogs triangle, recovered here as a special case in Corollary 1.4.","marker":"[2]"},{"why":"Supplies the pluricomplex Green function lower bound used in the necessity proof of part (2).","marker":"[9]"},{"why":"Supplies the same Green-function estimate on bounded pseudoconvex domains, used in the same necessity argument.","marker":"[3]"}],"fun_headline_variants":["Sharp t-threshold for Lp-Lq Toeplitz maps on Hartogs domains","Exact Bergman-kernel exponent t for bounded Toeplitz operators","Hartogs Toeplitz boundedness pinned to one power t","Three regimes settle Lp-Lq Toeplitz on Hartogs","Criteria for Toeplitz Lp-Lq: t must clear sharp bound"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole transfer from the general domains to the special ones rests on the uniform bound $0 < c \\leq |\\det \\Phi'(z)| \\leq d$ for the biholomorphism $\\Phi$; if that bound fails, the equivalence of boundedness between the two families of domains could break, and the computations on the special domain would no longer say anything about the general one.","fun_headline_variants_meta":{"raw":{"variants":["Sharp t-threshold for Lp-Lq Toeplitz maps on Hartogs domains","Exact Bergman-kernel exponent t for bounded Toeplitz operators","Hartogs Toeplitz boundedness pinned to one power t","Three regimes settle Lp-Lq Toeplitz on Hartogs","Criteria for Toeplitz Lp-Lq: t must clear sharp bound"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1380,"prompt_tokens":899,"completion_tokens":481,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":379}},"tokens_in":515,"tokens_out":481,"duration_ms":4969,"temperature":1.0,"reasoning_tokens":379,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:55:39.962439+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the special domain $H^n_{k,b}$, take the monomial $f(z)=z_n^{1-n-k(b-1)}$. The proof shows it is in $A^2$ and that $T_{K^{-t}}$ sends it to a nonzero constant multiple of itself; a direct calculation gives $f \\in L^q$ exactly when $q < \\frac{2n+2k(b-1)}{n-1+k(b-1)}$. If one could exhibit any $q$ at or above that threshold for which $T_{K^{-t}}$ is bounded on all of $L^p$, the theorem's first part would be false.","supporting_citations":[{"cited_title":"Chen, The Lp boundedness of the Bergman projection for a class of boundedHartogs domains, J","cited_arxiv_id":null,"evidence_quote":"Establishes the uniform determinant bound for the biholomorphism between the special and general domains, which Lemma 2.1 uses to transfer boundedness."},{"cited_title":"Khanh, J","cited_arxiv_id":null,"evidence_quote":"Supplies the generalized Schur test with weights h1, h2, f used to prove the sufficiency of the thresholds."},{"cited_title":"Khanh, J","cited_arxiv_id":null,"evidence_quote":"Provides the fat Hartogs triangle boundedness result and the monomial and Green-function testing technique adapted in the necessity proofs."},{"cited_title":"Beberok, Lp boundedness of the Bergman projection on some generalized H artogs triangles, Bull","cited_arxiv_id":null,"evidence_quote":"Gives the earlier L^p range for the power-generalized Hartogs triangle, recovered here as a special case in Corollary 1.4."},{"cited_title":"Herbort, The Bergman metric on hyperconvex domains","cited_arxiv_id":null,"evidence_quote":"Supplies the pluricomplex Green function lower bound used in the necessity proof of part (2)."},{"cited_title":"B/suppress locki, The Bergman metric and the pluricomplex Green function, Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the same Green-function estimate on bounded pseudoconvex domains, used in the same necessity argument."}],"review_version":1}