{"id":"c81a6aa8-9068-4c0e-b02b-7234ef9a0e9e","arxiv_id":"2411.08416","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Coorbit equivalence of dilation groups is characterized by agreement of essential frequency supports together with quasi-isometry of their full orbit maps, and no irreducible dilation group can realize general anisotropic Besov spaces.","lead":"This mathematics paper provides a geometric criterion for deciding when two different matrix dilation groups define the same wavelet coorbit function spaces. It also proves that anisotropic Besov spaces built from arbitrary expansive matrices cannot be described by an irreducibly acting dilation group.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Connectivity-respecting condition (c2) is left open; if it can fail, the main quasi-isometry classification does not cover all integrably admissible dilation groups.","rationale":"I read the paper in good faith: the core technical results—Theorem 2.7 on uniqueness of the frequency support, Theorems 3.14-3.16 on the orbit quasi-isometry, Theorems 4.3-4.4, and Theorem 5.2—are carefully argued, and I found no internal gap. The proof of Theorem 3.15 is intricate, but the connectivity argument for the set C_x is sound, and the composition arguments in Theorem 4.4 are valid thanks to Lemma 3.11. The main theorems are honest about their hypotheses. The most load-bearing limitation is the open status of (c2): it determines how widely the central classification applies. The authors themselves flag it in Section 3.2, and the reviewing rules require flagging such passages. The reader's weakest_assumption identified exactly this issue, and I agree with that assessment. A proof that O has finitely many connected components—or an explicit counterexample—would settle the matter. Corollary 4.5's omitted proof is a completeness concern but does not change the verdict. I therefore keep the reader's CONDITIONAL verdict unchanged.","tokens_in":35337,"tokens_out":26207,"duration_ms":249896,"concrete_test":"Determine whether the essential frequency support O of an integrably admissible dilation group always has finitely many connected components. If it does, then H0 has finite index in the Lie group H, hence is compactly generated, and (c2) is automatic, removing the concern. If not, construct an integrably admissible H whose frequency support has infinitely many components, for instance using the non-compactly-generated closed matrix group in [46, Appendix A] as a component-group factor, and verify conditions (a1)-(a3). A positive construction would confirm that Theorems 4.4 and 4.7 exclude some integrably admissible dilation groups.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.4 and Corollary 4.7 classify coorbit equivalence only for connectivity-respecting groups. The critical hypothesis is Definition 3.3(c2): the stabilizer H0 of the connected component containing C must be compactly generated. Section 3.2 explicitly states that no example is known of an integrably admissible dilation group failing (c2). If such a group exists, the transition-map characterization, the subgroup criterion, and hence the abstract's claim of a general subgroup characterization would not apply to it. The proof of Theorem 3.16 constructs a word metric from a precompact generating set of H0; without (c2) no such metric is available, so the quasi-isometry between H×C and O is not established. This is a genuine scope limitation, not an internal contradiction. Theorem 5.2 is less exposed because irreducibly admissible groups satisfy (c2) by Example 3.5(2), but the general classification remains conditional. Corollary 4.5 is also stated without proof, though its proof is a routine adaptation of Theorem 4.4, so it is a smaller issue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops coarse-geometric methods for comparing wavelet coorbit spaces associated with different dilation groups H ≤ GL(d, R). It first proves that the essential frequency support of an integrably admissible dilation group is unique (Theorem 2.7), resolving an open point from [30]. It then shows that, under the 'connectivity-respecting' hypotheses of Definition 3.3, the full orbit map p : H × C → O is a quasi-isometry (Theorem 3.16), and uses this to characterize coorbit equivalence of two connectivity-respecting dilation groups in terms of a quasi-isometry between transition maps (Theorem 4.4). Applications include a subgroup criterion (Corollary 4.7), a classification of dilation groups coorbit equivalent to the isotropic group R_+ · Id (Theorem 5.1), and the result that anisotropic Besov spaces defined by general expansive matrices cannot arise from irreducibly admissible dilation groups unless the dilation is isotropic (Theorem 5.2). The proofs are detailed and the main hypotheses are stated explicitly, but the central classification is conditional on the connectivity-respecting condition, whose automatic validity is left open in Section 3.2.","tokens_in":35492,"tokens_out":13858,"duration_ms":139410,"significance":"If the main results hold, the paper provides a substantial extension of the coarse-geometric classification of wavelet coorbit spaces from irreducible dilation groups [26] to the full class of integrably admissible dilation groups, including reducible ones. The uniqueness of the essential frequency support is a clean and useful result in itself. The quasi-isometry framework of Section 3 is carefully developed, and Theorem 5.2 gives a sharp negative answer to a natural question about anisotropic Besov spaces. The paper is honest about the unresolved nature of condition (c2) in Definition 3.3, and it gives explicit and well-organized proofs for the main theorems. The significance is tempered by the fact that the abstract and introduction present the subgroup characterization and equivalence criterion without the connectivity-respecting qualification, although the theorems themselves carry that hypothesis.","major_comments":[{"comment":"The abstract and introduction present the subgroup characterization and the coorbit-equivalence criterion as general results, but Theorem 4.4 and Corollary 4.7 are proved only for connectivity-respecting dilation groups, i.e., groups satisfying conditions (c1) and (c2) of Definition 3.3. The paper explicitly states in Section 3.2 that no example is known of an integrably admissible dilation group failing (c2), the compact-generation of the stabilizer H0. If such a group exists, neither the transition-map quasi-isometry characterization nor the subgroup criterion is established for it. This is a genuine scope limitation, not an internal contradiction, but it is load-bearing for the paper's central claims. The authors should either prove that (c2) is automatic for integrably admissible groups, or explicitly qualify the statements in the abstract and introduction and list the validity of (c2) as an open problem.","section":"Abstract; Definition 3.3; Section 3.2"},{"comment":"Corollary 4.5 is stated without proof, with the comment that its proof is similar to the first part of the proof of Theorem 4.4. This corollary is then used in the proofs of Corollary 4.7 and Theorem 5.1(ii). Since it is a necessary-condition tool for the subgroup characterization and for the isotropic classification, the paper should include a complete proof or a precise derivation from Theorem 4.4, rather than leaving this step to the reader.","section":"Corollary 4.5"},{"comment":"The proof of Theorem 5.2 relies on the classification results of [7] for anisotropic Besov spaces and on the identification of coorbit spaces with Besov-type spaces. The argument is coherent, but the step leading to H2 ⊆ SH1 = Z(B) depends on the nontrivial equality SH1 = {C : C^{-1}BC = B}. The authors should make explicit which statements in [7] justify this equality for all C ∈ GL(d, R), since this is the point where the expansive normal-form reduction and the Besov classification are combined.","section":"Theorem 5.2"}],"minor_comments":[{"comment":"In Notation 3.12(A1), W is an open, precompact, symmetric generating set, but the proof of Theorem 3.15 states 'Since W ⊆ H is compact'. This is not literally true; the proof can be repaired by replacing W with its closure (which is compact and still generated by H0) or by using the standard fact that every compact subset of a compactly generated locally compact group lies in some power W^m. The manuscript should clarify this point.","section":"Theorem 3.15, Step 1"},{"comment":"In the displayed formula for the cover induced by P, the symbol C^* appears in 'P = ⋃_{i∈I} h_i^{-T} V^{-T} C^*' but is never defined; it should be B_{\\epsilon/2}(C) (or the corresponding set from the definition of P).","section":"Lemma 3.8, proof"},{"comment":"The citation '[51, Theorem 6.9 1/2]' looks like an incorrect or incomplete reference label; it should be corrected to the precise theorem in [51] that is being invoked.","section":"Theorem 4.3, proof"},{"comment":"The reference to 'Example 3.5(a)' should be 'Example 3.5(1)'.","section":"Example 5.3"},{"comment":"The sentence 'By Lemma A.2, we can choose a compact, connected set C_ξ ⊆ O satisfying {ξ} ∪ C ⊆ C_ξ' is slightly elliptical, since Lemma A.2 applies to an open connected set; the argument should first restrict to the connected component O0 of O containing C and ξ.","section":"Corollary 3.17, proof"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically solid and the main proofs are carefully executed. The recommendation is driven by the gap between the abstract's unqualified claims and the connectivity-respecting hypothesis that is actually required for the central results. If the authors can either close this gap or clearly qualify the scope of the classification, the paper would be well suited for publication. The reliance on prior classification results from [7] in Theorem 5.2 is acceptable, but the relevant hypotheses should be spelled out so that the reader can verify that the reduction to expansive normal form applies."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: the paper is a genuinely useful advance in the coarse-geometric approach to wavelet coorbit spaces, and the proof quality is high. The main novelty is that it carries the quasi-isometric classification from irreducible to reducible integrably admissible dilation groups, and it settles an open uniqueness question for the essential frequency support. The negative result about anisotropic Besov spaces (Theorem 5.2) is a nice payoff: it shows you genuinely need reducible representations to cover the general anisotropic case.\n\nThe technical core is the quasi-isometry between the product H×C and the frequency support O for a 'connectivity-respecting' group. The orbit-map estimates in Theorems 3.14–3.16 are carefully argued, and the authors are explicit about standing assumptions. I also appreciate that they prove uniqueness of the frequency support with a clean integrability argument rather than hand-waving. The writing is heavy, but the burden is on the subject, not on them.\n\nWhere I'd push back is on the hypothesis. The classification in Theorem 4.4 and the subgroup criterion in Corollary 4.7 only apply to connectivity-respecting groups, and condition (c2)—that the stabilizer of the relevant connected component is compactly generated—is not known to hold for every integrably admissible dilation group. The authors say so themselves in Section 3.2. That is a real scope limitation, and it means the abstract's claim of a 'general subgroup characterization' is slightly stronger than what is established. If someone later finds an integrably admissible group failing (c2), the main equivalence theorem would not cover it. That doesn't make the paper wrong; it makes it conditional. I'd want the referee to ask the authors to either prove (c2) for all integrably admissible groups, or at least to state the classification with the caveat more prominently.\n\nThe other issue is minor: Corollary 4.5 is stated without proof. It looks like a routine adaptation of Theorem 4.4, but the authors should either prove it or say explicitly why it follows.\n\nOn citations: several load-bearing steps use the authors' own earlier work ([26], [30], [7]). That's not a problem—those are published results, and the dependence is transparent. The Besov classification from [7] is doing real work in Theorem 5.2, but that's a legitimate external machinery, not a loop.\n\nWho is this for? Specialists in wavelet coorbit theory and decomposition spaces. It deserves a serious referee. I'd accept for peer review and let the referee decide whether the connectivity-respecting caveat is acceptable or needs more work.","headline":"A solid, careful extension of the coarse-geometric coorbit classification to the reducible case, with a real open caveat about the connectivity-respecting hypothesis.","tokens_in":36071,"tokens_out":3163,"would_cite":true,"duration_ms":28504,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B35","42C15","42C40","43A65","51F30"],"pacs":[],"model":"deepseek-v4-flash","headline":"Two dilation groups induce identical wavelet coorbit spaces exactly when they share an essential frequency support and the transition map between their parameter spaces is a quasi-isometry.","keywords":["coarse geometry","wavelet coorbit spaces","dilation groups","integrably admissible dilation groups","essential frequency support","anisotropic Besov spaces","quasi-isometry","decomposition spaces"],"falsifier":"A decisive test would be to produce an integrably admissible dilation group that satisfies the first connectivity condition but whose component stabilizer is not generated by finitely many bounded pieces; the authors explicitly state they know no such group, and its existence would show the quasi-isometry classification is not universal. A second test would be to find two connectivity-respecting groups with the same essential frequency support whose transition map is not a quasi-isometry but whose coorbit norms are nevertheless equivalent for all $1\\le p,q\\le\\infty$, which Theorem 4.4 says cannot happen.","tokens_in":35090,"feed_emoji":"📐","tokens_out":10980,"duration_ms":102089,"temperature":0.7,"pith_summary":"This paper asks when two different matrix dilation groups produce the same wavelet coorbit spaces, the Banach function spaces defined by imposing norm conditions on wavelet coefficients. Its answer, for a broad class of admissible dilation groups, is geometric: two groups are coorbit equivalent exactly when they have the same essential frequency support and the map between their parameter spaces is a quasi-isometry. This turns a norm-by-norm comparison of function spaces into a checkable statement about large-scale geometry, and it also settles the open question of uniqueness of the essential frequency support. The same machinery proves that anisotropic Besov spaces for general expansive matrices require reducible dilation groups: they cannot be coorbit spaces of an irreducibly admissible dilation group unless the dilation is isotropic.","feed_headline":"Coorbit equivalence is a quasi-isometry test","feed_subtitle":"A quasi-isometry between parameter spaces decides when two dilation groups give identical wavelet coorbit spaces.","key_machinery":"The load-bearing object is the essential frequency support $O$, the open full-measure set with $O=H^{T}C$ for compact $C$; it is shown unique and serves as the common space on which coorbit norms live. The full orbit map $p:H\\times C\\to O$, $(h,\\xi)\\mapsto h^{-T}\\xi$, is the carrier of the classification: under the connectivity-respecting conditions it is a surjective quasi-isometry from the product of a word metric on $H$ and a bounded metric on $C$ to the cover-induced chain metric on $O$. The connectivity-respecting conditions require $C$ to be compact connected and the stabilizer of the connected component containing $C$ to be compactly generated. The proof route goes through induced covers and their weak equivalence, the condition that each set of one cover meets only boundedly many sets of the other and conversely, because coorbit spaces are Besov-type decomposition spaces determined by such covers.","core_discovery":"The paper's central result (Theorem 4.4) is that for connectivity-respecting integrably admissible dilation groups $H_1,H_2$ with essential frequency supports $O_1=H_1^{T}C_1$, $O_2=H_2^{T}C_2$, coorbit equivalence is equivalent to $O_1=O_2$ and the transition map $p_2^{*}\\circ p_1:(H_1\\times C_1,d_{H_1\\times C_1})\\to(H_2\\times C_2,d_{H_2\\times C_2})$ being a quasi-isometry. To get there, the paper proves that every integrably admissible dilation group has a unique essential frequency support (Theorem 2.7) and that the full orbit map $p:H\\times C\\to O$, $(h,\\xi)\\mapsto h^{-T}\\xi$, is a surjective quasi-isometry under the connectivity-respecting assumptions (Theorem 3.16). The applications include a subgroup criterion: for $H_1\\subseteq H_2$, coorbit equivalence holds exactly when $H_2/H_1$ is compact, and an anisotropic Besov space for an expansive matrix admits an irreducible coorbit description only when the dilation is a scalar multiple of the identity.","pith_inferences":["Inference: the uniqueness of the essential frequency support gives a fast necessary condition for coorbit equivalence, so new dilation groups can first be checked for support agreement before any quasi-isometry is computed.","Inference: the theorem's scope depends on the connectivity-respecting condition (c2), so a natural search is for an integrably admissible dilation group whose component stabilizer is not compactly generated; the authors state they do not know such an example.","Inference: the support-based picture suggests that coorbit spaces are ultimately determined by the shape of the frequency support together with the large-scale structure of the group action, so the same method could be adapted to dilation groups acting on cones or sectors rather than all of $\\mathbb{R}^d\\setminus\\{0\\}$."],"forward_implications":["The essential frequency support of an integrably admissible dilation group is unique, so a single well-defined frequency set is attached to each such group.","Coorbit equivalence for connectivity-respecting groups can be decided by one quasi-isometry check between explicit metric spaces, eliminating norm-by-norm verification.","For nested dilation groups $H_1\\subseteq H_2$, the coorbit spaces coincide exactly when the quotient $H_2/H_1$ is compact.","A one-parameter group $\\exp(\\mathbb{R}X)$ is coorbit equivalent to the isotropic $\\mathbb{R}_+\\cdot I$ exactly when $X=sI+Y$ with $s\\neq 0$ and $\\exp(\\mathbb{R}Y)$ relatively compact.","Homogeneous anisotropic Besov spaces for general expansive matrices cannot be realized as coorbit spaces of irreducibly admissible dilation groups unless the dilation is isotropic."],"supporting_citations":[{"why":"supplies the coorbit-space framework and Besov-type norm equivalence for integrably admissible dilation groups.","marker":"[30]"},{"why":"supplies the precursor coarse-geometric classification for irreducible dilation groups, including the cover-metric equivalence theorem reused here.","marker":"[26]"},{"why":"supplies the decomposition-space classification showing coorbit space equality is equivalent to weak equivalence of frequency covers.","marker":"[51]"},{"why":"supplies the notion of integrably admissible dilation groups and the admissible vectors with compactly supported Fourier transform.","marker":"[11]"},{"why":"supplies the identification of wavelet coorbit spaces with decomposition spaces for irreducible admissible dilation groups.","marker":"[31]"},{"why":"supplies the classification of anisotropic Besov spaces for expansive matrices that the irreducibility application relies on.","marker":"[7]"},{"why":"defines the anisotropic Besov spaces that the paper classifies within coorbit theory.","marker":"[3]"}],"fun_headline_variants":["One quasi-isometry decides all coorbit spaces","Dilation groups meet via coarse geometry","Coorbit spaces: same if orbits are quasi-isometric","Reducible reps unify Besov and coorbit theory","Quasi-isometry criterion for dilation group equivalence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification rests on the assumption that each dilation group's frequency support is generated by one compact connected piece and that the group elements sending that piece's connected component to itself can be generated by finitely many bounded pieces; the authors know of no integrably admissible group that violates the second part.","fun_headline_variants_meta":{"raw":{"variants":["One quasi-isometry decides all coorbit spaces","Dilation groups meet via coarse geometry","Coorbit spaces: same if orbits are quasi-isometric","Reducible reps unify Besov and coorbit theory","Quasi-isometry criterion for dilation group equivalence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000639,"raw_usage":{"total_tokens":2921,"prompt_tokens":900,"completion_tokens":2021,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":516,"completion_tokens_details":{"reasoning_tokens":1946}},"tokens_in":516,"tokens_out":2021,"duration_ms":14977,"temperature":1.0,"reasoning_tokens":1946,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T21:38:40.872374+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive test would be to produce an integrably admissible dilation group that satisfies the first connectivity condition but whose component stabilizer is not generated by finitely many bounded pieces; the authors explicitly state they know no such group, and its existence would show the quasi-isometry classification is not universal. A second test would be to find two connectivity-respecting groups with the same essential frequency support whose transition map is not a quasi-isometry but whose coorbit norms are nevertheless equivalent for all $1\\le p,q\\le\\infty$, which Theorem 4.4 says cannot happen.","supporting_citations":[{"cited_title":"F¨ uhr and J","cited_arxiv_id":null,"evidence_quote":"supplies the coorbit-space framework and Besov-type norm equivalence for integrably admissible dilation groups."},{"cited_title":"F¨ uhr and R","cited_arxiv_id":null,"evidence_quote":"supplies the precursor coarse-geometric classification for irreducible dilation groups, including the cover-metric equivalence theorem reused here."},{"cited_title":"Voigtlaender","cited_arxiv_id":null,"evidence_quote":"supplies the decomposition-space classification showing coorbit space equality is equivalent to weak equivalence of frequency covers."},{"cited_title":"Currey, H","cited_arxiv_id":null,"evidence_quote":"supplies the notion of integrably admissible dilation groups and the admissible vectors with compactly supported Fourier transform."},{"cited_title":"F¨ uhr and F","cited_arxiv_id":null,"evidence_quote":"supplies the identification of wavelet coorbit spaces with decomposition spaces for irreducible admissible dilation groups."},{"cited_title":"Cheshmavar and H","cited_arxiv_id":null,"evidence_quote":"supplies the classification of anisotropic Besov spaces for expansive matrices that the irreducibility application relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines the anisotropic Besov spaces that the paper classifies within coorbit theory."}],"review_version":1}