{"id":"1e062ef9-f455-4351-90df-7bbde8f110d8","arxiv_id":"2411.16010","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Near-maximizers of Cauchy wavelet concentration are quantitatively close to hyperbolic balls and to reproducing kernels, with explicit constants uniform in the wavelet parameter.","lead":"This paper proves sharp, parameter-uniform stability versions of concentration inequalities for Cauchy wavelet transforms: near-maximal concentration forces the set to be nearly a hyperbolic ball and the function to be nearly a reproducing kernel. The same framework yields a new sharp concentration bound for Poisson extensions of Hardy-space functions and recovers the known short-time Fourier transform result as a limit.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.3 Step I relies on a pointwise monotonicity of u*/v* that is asserted to follow from [31] but is not established in the present text; without it the linear-in-deficit estimate does not follow.","rationale":"The reader's weakest assumption identifies exactly the same load-bearing point: Lemma 3.3 inherits a rearrangement comparison from [31] as a black box, and the paper calls the pointwise monotonicity of r(s)=u*/v* an equivalent formulation of [31] without proof or a precise citation. That monotonicity is used in the crucial cases of Lemma 3.3 to convert integral deficit estimates into control of 1-a0^2, and if it fails the central linear-in-deficit bound does not follow. I agree that this is the most load-bearing concern for the main theorem. I also noticed a separate, sketched limsup/liminf exchange in Section 6 around equation (6.4)-(6.7) when passing to the Hardy endpoint; that deserves scrutiny because it concerns the advertised new endpoint result, but it is secondary to the fixed-alpha Theorem 1.2 and can probably be repaired with a compactness argument. The paper has strong independent support in its detailed estimates, explicit constants, and recovery of prior results, so the conditional verdict is appropriate: acceptance should wait until the monotonicity claim is verified against [31] and the limiting arguments are made rigorous.","tokens_in":63,"tokens_out":25143,"duration_ms":488567,"concrete_test":"Pull the exact statement and proof of the relevant result in [31] and check whether it proves, for all normalized f, that s -> u*(s)/v*(s) is nondecreasing, or only that integral_0^s u* <= integral_0^s v* for all s. If only the latter, re-run Lemma 3.3 under only the integral hypothesis: one can construct a decreasing u* with equal total mass, cumulative dominance over v*, and a non-monotone ratio (e.g., piecewise switching between below and above v*), showing the two conditions are not equivalent. Then either locate a missing argument in the present paper or attempt to derive (3.18) from the integral bound alone; if (3.18) cannot be derived, Theorem 1.2 as stated is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Lemma 3.3, Step I (Eqs. (3.18)-(3.22)). The proof that 1-a0^2 is controlled by the deficit uses, twice, the assertion that r(s)=u*(s)/v*(s) is increasing, attributed to [31] as an 'equivalent formulation'. The theorem quoted from [31] is the setwise Faber-Krahn inequality lambda_beta(Omega) <= lambda_beta(Omega*); in rearrangement terms that gives cumulative control integral_0^s u* <= integral_0^s v* for every s, not pointwise monotonicity of the ratio. The displayed estimates in Cases s0>s* and s0<s* require r(s0)>=r(s*) or r(s0)<=r(s*) in a specific direction; if [31] supplies only the integral comparison, the bounds fail because the factor (1-1/r(s0)) can have the wrong sign. The paper neither proves the ratio monotonicity nor cites a theorem number or lemma in [31] that states it. Since this is the only route in the text from the deficit to the lower bound on a0, a gap here removes the linear-in-delta control and with it Theorem 1.2's function stability (1.7).","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a quantitative stability version of the Faber–Krahn inequality for continuous wavelet transforms associated to Cauchy windows. After translating the wavelet problem to Bergman spaces, the main result (Theorem 1.2) asserts that for every α > -1, every s > 0, and every f in B^2_α, the normalized distance from f to the set of normalized Bergman kernels is bounded by an explicit constant times the square root of the Bergman deficit, and the hyperbolic asymmetry of any set Ω with μ(Ω)=s is also bounded by a constant times the square root of the deficit. The authors then recover the Fock-space stability result of [14] as α → ∞ and obtain a new Hardy-space concentration inequality with stability as α → -1 (Theorem 1.4). Sharpness is discussed in Section 4 via a variational second-variation analysis and explicit test functions.","tokens_in":1581,"tokens_out":1665,"duration_ms":111823,"significance":"If the technical gaps identified below are repaired, this is a substantial contribution. The paper gives explicit, parameter-uniform constants rather than qualitative compactness arguments, and it genuinely unifies the Bergman, Fock, and Hardy regimes. The derivation of a new quantitative Hardy-space concentration result for Poisson extensions is a notable novelty, and the recovery of the known Fock stability theorem as a limiting case is a useful external check. The proof of the main function-stability estimate is detailed and does not rely on numerical fitting or hidden parameters. However, the central rearrangement step in Lemma 3.3 and the deferred lemmas in Section 4 are load-bearing for the advertised claims, so the result cannot be accepted as fully established in the present form.","major_comments":[{"comment":"The proof of (3.18) hinges on the assertion that r(s)=u*(s)/v*(s) is increasing on [0,∞), which is called 'an equivalent formulation of the main result from [31]'. The theorem quoted from [31] at the beginning of the introduction, however, is the setwise Faber–Krahn inequality λβ(Ω) ≤ λβ(Ω*); in rearrangement terms that gives the cumulative comparison ∫_0^s u* ≤ ∫_0^s v* for every s. Cumulative comparison does not imply pointwise monotonicity of the ratio u*/v*. The estimates in the cases s0 > s* and s0 < s* use the sign of (1 - 1/r(s0)) and (1 - r(s0)) in an essential way, and without monotonicity those inequalities can fail. Since no proof and no exact lemma or theorem number in [31] is supplied, this is a genuine gap. Please provide a proof of the monotonicity or a precise citation to a statement in [31] that contains it. This step is load-bearing for (3.21), (3.22), and ultimately for the linear-in-δ estimate in Theorem 1.2.","section":"§3, Lemma 3.3, Step I (Eq. (3.20))"},{"comment":"The sharpness analysis in Section 4 depends on three technical results that are not proved in the text. Lemma 4.1 is described as a 'standard adaptation' of [14, Lemma 3.2]; Lemma 4.3 is deferred by saying that the proof is purely technical and referring to 'the techniques' in [14, Appendix A], without a precise statement; Lemma 4.5 is imported from [6] and [14]. Because Theorem 4.2 and Corollary 4.9 need quantitative lower bounds with explicit constants in the Bergman setting, the transfer from the Fock-space arguments in [14] is not automatic. The stated modulus of continuity and the ε0(s,α), C(s,α) dependence require either full proofs or exact theorem statements with the required parameter dependence. As written, Section 4 does not establish the sharpness claims (i)–(iii) of Corollary 4.9.","section":"§4.1, Lemmas 4.1, 4.3, 4.5"},{"comment":"The argument for recovering the Fock stability result from [14] is not sufficient. The text says that 'the left-hand side of (1.7) is simply the ℓ2-norm of f ∈ B^2_α' and concludes that this quantity is independent of α. This is inaccurate: the left-hand side of (1.7) is an infimum over normalized reproducing kernels fω, and both the kernels and the coefficient normalization depend on α. The infimum over ω can be strictly smaller than the full norm, so the asserted independence does not follow. A limiting argument for the distance itself, analogous to the one used for Theorem 1.4 in Section 6, is required. Without it, Theorem 1.3 is not derived from Theorem 1.2 in the present text.","section":"§5.2, recovery of Theorem 1.3"}],"minor_comments":[{"comment":"The definition of u* via the condition μ({u>u*(s)})=s should specify the convention at discontinuities or plateaus of the distribution function; the standard generalized inverse would make the following integral manipulations unambiguous.","section":"§2.3, Eq. (2.3)–(2.4)"},{"comment":"The display (3.1) has mismatched parentheses in the interval for t; it should read t ∈ ((α+1)/π c0, (α+1)/π a0^2). Please correct the typesetting.","section":"§3, Lemma 3.1"},{"comment":"There is a typo in the phrase 'the results are aslo sharp' in the paragraph introducing Section 4.2; it should read 'also sharp'.","section":"§4.2, Corollary 4.9"},{"comment":"References [23] and [24] appear to point to the same arXiv preprint (arXiv:2212.14008) under slightly different titles; please verify and merge or disambiguate them.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper depends heavily on [14] and [31], and several authors of the present paper are also authors of those works. This is not by itself problematic, but the unproved 'equivalent formulation' in Lemma 3.3 needs to be checked carefully against the actual content of [31]. If the authors can supply a proof or an exact citation for the monotonicity of u*/v*, and provide the missing details in Section 4, the paper would be a strong contribution. If the monotonicity assertion is false, the main stability estimate would be unsupported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is the bottom line. The paper contains a genuinely new package: a uniform-in-α stability theorem for the Cauchy wavelet/Bergman concentration, a new Hardy-space concentration inequality as α→-1, and a recovery of the Fock-space stability result [14] as α→∞. The main machinery in Section 3, especially the level-set and asymmetry estimates, looks like serious, honest work. But there is a load-bearing step in Lemma 3.3 that I cannot accept as written. The proof claims the ratio r(s)=u*(s)/v*(s) is increasing on [0,∞), calling this 'an equivalent formulation of the main result from [31].' The theorem from [31] quoted in the paper is the setwise Faber-Krahn inequality, which in rearrangement terms gives cumulative control ∫_0^s u* ≤ ∫_0^s v* for every s. That does not imply pointwise monotonicity of the ratio. Both cases in Step I of Lemma 3.3 use the monotonicity in a specific direction to get the linear-in-deficit bound on 1-a0^2. If that monotonicity fails, (3.18) fails, and with it the function-stability estimate (1.7) and everything downstream, including the Hardy endpoint. So the gap is not cosmetic. It may be that [31] actually contains the needed statement; if so, the authors should cite the exact lemma. If not, they need a direct proof. The sharpness section is also not self-contained: Lemma 4.1, Lemma 4.3, and Lemma 4.5 are deferred to [14] and [6]. For an advertised sharpness claim that is thin—but it is secondary to the stability theorem. The limiting arguments in Sections 5 and 6 involve sketched interchanges of limits; they look plausible but need filling in. My overall read: the results are significant if the gap can be closed, and the paper deserves a serious referee. I would not cite it for the stability theorem until the monotonicity claim is resolved. But I would bring it to a reading group to go through Lemma 3.3 alone.","headline":"A likely-fixable gap in Lemma 3.3 stands between this paper and its advertised stability theorem.","tokens_in":97,"tokens_out":7441,"would_cite":false,"duration_ms":127768,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42C40","30H20","30H10","47A75","49K21","49R05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Near-maximal wavelet concentration forces functions and sets close to extremal shapes, uniformly in the Cauchy-wavelet parameter.","keywords":["wavelet transform","Bergman space","Hardy space","Fock space","Faber–Krahn inequality","concentration inequality","stability estimate","Poisson kernel"],"falsifier":"Test the monotonicity of $r(s)=u_*(s)/v_*(s)$ numerically or analytically on a dense grid of $s$ for a near-extremal function $f\\in B^2_\\alpha(\\mathbb{D})$; any violation would invalidate Step I of Lemma 3.3 and hence the claimed bound $1-a_0^2\\le M_\\alpha(s)\\delta(f;\\Omega,\\alpha)$. Equivalently, compute the quotient in Proposition 4.8 for $f_\\varepsilon(z)=a_0+a_2 z^2$ at fixed $\\alpha$ and small $a_2/a_0$: if the distance-to-kernels over the square root of the deficit ever fails to match the stated constant behavior, the square-root stability rate fails.","tokens_in":42333,"feed_emoji":"📐","tokens_out":6967,"duration_ms":61637,"temperature":0.7,"pith_summary":"The paper establishes a sharp quantitative version of the Faber–Krahn inequality for continuous wavelet transforms with Cauchy windows, stated equivalently as Theorem 1.2 for Bergman spaces $B^2_\\alpha$ of the disk. For every $\\alpha>-1$, any set of hyperbolic measure $s$, and any normalized $f$, the distance from $f$ to the set of normalized Bergman kernels is at most a computable constant times $\\delta(f;\\Omega,\\alpha)^{1/2}$, and the hyperbolic asymmetry of $\\Omega$ satisfies the same square-root bound. The interest is that these constants stay controlled as $\\alpha$ varies across its full range: letting $\\alpha\\to\\infty$ recovers the known Fock-space stability theorem, while letting $\\alpha\\to-1$ yields a new concentration inequality for Hardy-space functions, whose extremals are Poisson kernels, together with its sharp stability version. A sympathetic reader should care because this unifies and extends the geometry of near-extremals in three classical function-space settings with one parameter-free mechanism.","feed_headline":"Wavelet concentration tied to Bergman kernels with uniform stability","feed_subtitle":"Near-maximal mass forces functions and sets close to extremals, recovering Fock and new Hardy cases.","key_machinery":"The load-bearing object is the one-dimensional rearrangement comparison between $u_*(s)$, the inverse distribution function of $u(z)=|f(z)|^2(1-|z|^2)^{\\alpha+2}$, and the explicit comparison function $v_*(s)=\\frac{\\alpha+1}{\\pi}(1+s/\\pi)^{-(\\alpha+2)}$. The argument uses that the ratio $r(s)=u_*(s)/v_*(s)$ is increasing, an equivalent formulation of the base Faber–Krahn inequality, to convert integral estimates of $\\int_0^{s_*}(v_*-u_*)\\,ds$ into a bound on $1-a_0^2$, where $a_0^2$ measures the largest value of $u$ against the kernel. Around this core, Lemma 3.1 bounds the distribution function by a sharp level-set estimate obtained through a Taylor expansion of the level radius, Lemma 3.2 gives the integral lower bound, and Lemma 3.3 assembles these into the linear-in-deficit control $1-a_0^2\\le M_\\alpha(s)\\delta(f;\\Omega,\\alpha)$. A reproducing-kernel argument (Lemma 3.4) then converts this coefficient control into the actual Bergman-norm distance to the normalized kernels.","core_discovery":"The paper's central discovery is Theorem 1.2: for every $\\alpha>-1$, every $s>0$, and every $f\\in B^2_\\alpha(\\mathbb{D})$ with $\\mu(\\Omega)=s$, the normalized distance from $f$ to the set of normalized Bergman kernels satisfies $\\inf_{|c|=\\|f\\|,\\omega\\in\\mathbb{D}} \\|f-cf_\\omega\\|_{B^2_\\alpha}/\\|f\\|_{B^2_\\alpha} \\le C(1+\\frac{\\alpha+2}{\\alpha+1}[(1+s/\\pi)^{\\alpha+1}-1])^{1/2}\\delta(f;\\Omega,\\alpha)^{1/2}$, and the hyperbolic asymmetry satisfies $A_\\mathbb{D}(\\Omega)\\le K(s,\\alpha)\\delta(f;\\Omega,\\alpha)^{1/2}$, with explicit computable constants. Both estimates are sharp in the exponent $1/2$ and in their dependence on $\\alpha$ (Corollary 4.9). Translated back to the upper half plane this is the wavelet formulation Theorem 1.1. The limiting cases belong to the same discovery: as $\\alpha\\to\\infty$ the inequality converges to the known Fock-space Faber–Krahn stability result, and as $\\alpha\\to-1$ it produces Theorem 1.4, a concentration inequality for Poisson extensions in $H^2(\\mathbb{D})$ whose extremals are Poisson kernels, with a stability version whose constant is continuous in the set measure.","pith_inferences":["The stability mechanism interpolates between Hardy and Fock geometries, so one testable conjecture the paper leaves implicit is that the optimal constants $M_\\alpha(s)$ and $K(s,\\alpha)$ are monotone in $\\alpha$ for fixed $s$; this could be probed by computing the same second variation along the $z^2$ perturbations at interior values of $\\alpha$.","The authors leave open whether the constant threshold $c\\ge(\\alpha+2)^{-1/(\\alpha+1)}$ is optimal for each fixed $\\alpha$; refining the level-set comparison behind (3.25) until this threshold is attained would sharpen the Hardy-space stability bound.","If the rearrangement comparison could be run without the log-subharmonicity of $|f|^2$, the Hardy-space inequality would extend from analytic $H^2$ functions to general $L^2$ boundary data, a direction the paper explicitly identifies as needing new ideas.","The variational machinery of Section 4 suggests a numerical route to search for non-radial near-extremal shapes: compute the Hessian of $K_\\alpha$ at the ball in directions orthogonal to the kernel and look for directions where the second variation is less negative than the universal lower bound."],"forward_implications":["If the main theorem is correct, then for every fixed $\\alpha$ the exponent $1/2$ in the stability estimates cannot be improved: Corollary 4.9 exhibits near-extremal perturbations for which the distance is of the order of the square root of the deficit, and not higher.","The Fock-space stability theorem for the short-time Fourier transform follows as the $\\alpha\\to\\infty$ limit of the same inequality, so the Bergman result is a genuine one-parameter umbrella for the Gaussian case.","The $\\alpha\\to-1$ limit gives a new concentration inequality for Hardy spaces $H^2(\\mathbb{D})$: the mass of a Poisson extension is maximized by the Poisson kernel, with a sharp stability version, and the qualitative inequality is itself new.","The set-stability estimate transfers by the paper's limiting argument to Euclidean asymmetry in the Fock case and hyperbolic asymmetry in the Hardy case, so near-extremal sets are quantitatively close to balls in all three settings.","Because Theorem 1.2 is stated on the disk with hyperbolic measure, pulling it back by biholomorphisms yields the same concentration and stability statements in any simply connected domain, as noted in Remark 5."],"supporting_citations":[{"why":"Supplies the base Faber–Krahn inequality for the wavelet transform and the monotonicity of the rearrangement ratio used in Lemma 3.3.","marker":"[31]"},{"why":"Provides the stability framework for the short-time Fourier transform that this paper adapts and refines, and which is recovered as the $\\alpha\\to\\infty$ limit.","marker":"[14]"},{"why":"Establishes the original sharp concentration inequality for the short-time Fourier transform that the Fock-space limit recovers.","marker":"[28]"},{"why":"Used for the measure-preserving transport map in the set-stability proof of (1.8).","marker":"[12]"},{"why":"Supplies the incomplete Beta and Gamma function identities used in the second-variation and sharpness computations.","marker":"[11]"},{"why":"Connects the wavelet transform to Bergman spaces and explicitly posed the concentration problem that the base inequality answers.","marker":"[2]"}],"fun_headline_variants":["Sharp stability for wavelet concentration inequalities","Uniform sharp bounds for wavelet concentration","Stable Faber-Krahn for wavelets and Hardy spaces","Near-extremal wavelet concentration forces kernel shapes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof takes as given the base Faber–Krahn rearrangement theorem, specifically that the ratio $u_*(s)/v_*(s)$ of rearranged level functions is increasing, and all of the linear-in-deficit control in Lemma 3.3 collapses if that external comparison, or its quantitative strength, fails.","fun_headline_variants_meta":{"raw":{"variants":["Sharp stability for wavelet concentration inequalities","Uniform sharp bounds for wavelet concentration","Stable Faber-Krahn for wavelets and Hardy spaces","Near-extremal wavelet concentration forces kernel shapes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000209,"raw_usage":{"total_tokens":1443,"prompt_tokens":1016,"completion_tokens":427,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":632,"completion_tokens_details":{"reasoning_tokens":371}},"tokens_in":632,"tokens_out":427,"duration_ms":4434,"temperature":1.0,"reasoning_tokens":371,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:39:16.002957+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Test the monotonicity of $r(s)=u_*(s)/v_*(s)$ numerically or analytically on a dense grid of $s$ for a near-extremal function $f\\in B^2_\\alpha(\\mathbb{D})$; any violation would invalidate Step I of Lemma 3.3 and hence the claimed bound $1-a_0^2\\le M_\\alpha(s)\\delta(f;\\Omega,\\alpha)$. Equivalently, compute the quotient in Proposition 4.8 for $f_\\varepsilon(z)=a_0+a_2 z^2$ at fixed $\\alpha$ and small $a_2/a_0$: if the distance-to-kernels over the square root of the deficit ever fails to match the stated constant behavior, the square-root stability rate fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the base Faber–Krahn inequality for the wavelet transform and the monotonicity of the rearrangement ratio used in Lemma 3.3."},{"cited_title":"G´ omez, A","cited_arxiv_id":null,"evidence_quote":"Provides the stability framework for the short-time Fourier transform that this paper adapts and refines, and which is recovered as the $\\alpha\\to\\infty$ limit."},{"cited_title":"Nicola and P","cited_arxiv_id":null,"evidence_quote":"Establishes the original sharp concentration inequality for the short-time Fourier transform that the Fock-space limit recovers."},{"cited_title":"Figalli and F","cited_arxiv_id":null,"evidence_quote":"Used for the measure-preserving transport map in the set-stability proof of (1.8)."},{"cited_title":"https://dlmf.nist.gov/, Release 1.1.12 of 2023-12-15","cited_arxiv_id":null,"evidence_quote":"Supplies the incomplete Beta and Gamma function identities used in the second-variation and sharpness computations."},{"cited_title":"An Inverse Problem for Localization Operators","cited_arxiv_id":"1202.5841","evidence_quote":"Connects the wavelet transform to Bergman spaces and explicitly posed the concentration problem that the base inequality answers."}],"review_version":1}