{"id":"e9aae1f9-0000-490e-b7e2-cf1c33aaaa43","arxiv_id":"2505.02136","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Generalized matrix-weighted Besov-Triebel-Lizorkin spaces with growth functions are fully characterized (phi-transform, maximal functions, molecules, wavelets), and p-invariance holds iff the matrix weight is eigenvalue-comparable to a scalar.","lead":"This paper builds a complete real-variable theory for a new family of matrix-weighted Besov-Triebel-Lizorkin spaces, using growth functions to capture many previously separate spaces in one framework. It also answers an open question about invariance on the integrable index, characterising it by a comparability condition on the matrix weight's eigenvalues.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Full-range F-space characterizations rest on the unproved matrix-weighted Fefferman-Stein inequality for A_{p,∞} from [15]; if that estimate fails for p<1 or q≤1, Theorem 2.5 and its consequences lose their stated scope.","rationale":"I read the paper in good faith and found the internal proofs detailed and coherent conditional on the cited results from [15]. The reader's weakest_assumption correctly identifies the external dependence on matrix A_{p,∞} theory, specifically Lemma 3.9(ii) and Lemma 3.21. My stress-test confirms that these are load-bearing: they are used at multiple central junctions, and the paper does not reprove them or justify the full index range p∈(0,∞), q∈(0,∞] for the Fefferman-Stein-type inequality. This is a genuine risk rather than a discovered error. Because reliance on prior work is standard and the cited companion paper is by the same group, I do not see grounds to change the ACCEPT verdict; however, the verification step I propose would settle whether the concern lands. If the cited results are found not to extend to p<1 or q≤1, the paper's claims would need to be restricted accordingly, and the verdict should then be reconsidered.","tokens_in":108584,"tokens_out":24934,"duration_ms":290185,"concrete_test":"Obtain the proofs of [15, Corollary 5.8] and [15, Lemma 6.8]. For Corollary 5.8, check whether the argument uses the unweighted Fefferman-Stein vector-valued maximal inequality for M on L^p(l^q) with p,q>1, and whether a convexification step supplies the case p<1 or q<1. Then test Lemma 3.9(ii) at the endpoint: take m=2, W(x)=diag(|x|^α, |x|^β) with α<β≤0 (as in Remark 6.5, W∈A_{r,∞} for all r), p=q=1/2, and a sequence t_Q supported on cubes of a single scale; compare the two sides of (3.9). If a counterexample emerges, the full-range theorem must be restricted. For Lemma 6.8, verify that the upper/lower dimensions in (3.32)-(3.33) are finite for every W∈A_{p,∞} and that the estimate (3.34) holds with constants independent of W in the stated class.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central machinery for the Triebel-Lizorkin-type spaces depends on Lemma 3.9(ii), a matrix-weighted vector-valued inequality for W∈A_{p,∞} covering p∈(0,∞), q∈(0,∞]. The proof is entirely delegated to [15, Corollary 5.8], with no verification of the endpoint cases p<1 or q≤1. This lemma is used at crucial steps: (3.16) in the sequence-space coincidence, (3.40) in the averaging-space coincidence, and (4.5)-(4.6) and (4.12) in the Peetre-maximal and Littlewood-Paley characterizations. If [15, Cor 5.8] requires p,q>1 — as the classical scalar Fefferman-Stein vector-valued inequality does — then the φ-transform characterization (Theorem 2.5), and with it the molecular and wavelet theorems, are unsupported for the full range claimed in the abstract. A second external pillar, Lemma 3.21 (reducing-operator estimates from [15, Lemma 6.8]), is needed to obtain the strongly doubling condition used in Theorem 3.27 and to define the almost-diagonal thresholds in Theorem 5.6. If the dimension parameters (3.32)-(3.33) are not finite for all W∈A_{p,∞}, the molecule and wavelet results also overreach. These two citations are the single most load-bearing dependency in the paper.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces homogeneous generalized matrix-weighted Besov-Triebel-Lizorkin-type spaces ˙A^{s,υ}_{p,q}(W) associated with matrix A_{p,∞} weights and growth functions υ defined on dyadic cubes. It establishes a full real-variable theory for these spaces: the φ-transform characterization (Theorem 2.5), Peetre-type maximal and Littlewood-Paley characterizations (Theorems 4.4 and 4.7), and molecular and wavelet characterizations (Theorems 5.17 and 5.20). The boundedness of almost diagonal operators on the corresponding sequence spaces is proved (Theorem 5.6), and the growth-function condition is shown to be optimal in the sense that, for almost increasing υ, boundedness of almost diagonal operators forces υ to satisfy the growth condition (Proposition 5.9). As applications, the paper gives necessary and sufficient conditions for p-invariance of the Triebel-Lizorkin-type spaces, namely W ∼ E_W I_m (Theorems 6.4 and 6.6), and for Sobolev-type embeddings (Theorem 6.11). The authors emphasize that the framework is new even in the scalar-valued setting.","tokens_in":108847,"tokens_out":10902,"duration_ms":119980,"significance":"If the results are correct, this is a substantial contribution: it unifies and extends the known matrix-weighted BTL spaces, supplies an optimal growth-function scale, and provides sharp necessity results rather than only sufficient conditions. The p-invariance criterion and the optimality of the growth condition are particularly strong and conceptually novel claims, and the authors correctly point out that several consequences are new even when the weight is scalar or trivial. The proof structure is systematic and mostly detailed, and the paper is careful in attributing external estimates to prior work. However, the central real-variable characterizations depend on a matrix-weighted Fefferman-Stein inequality and on reducing-operator estimates imported from [15] without reproving or even stating their precise hypotheses; the confidence in the full claimed range therefore rests on those external results.","major_comments":[{"comment":"The matrix-weighted Fefferman-Stein inequality stated for W ∈ A_{p,∞} and p ∈ (0,∞), q ∈ (0,∞] is the single most load-bearing external input in the paper: it is used at (3.16), (3.40), (4.5)-(4.6), and (4.12), and through those estimates it underpins Theorems 2.5, 4.4, 4.7, 5.17, and 5.20. The proof of Lemma 3.9(ii) consists only of an invocation of [15, Corollary 5.8], with no statement of the range of p and q for which that result is proved and no discussion of the endpoint cases p < 1 or q ≤ 1. Since the classical scalar Fefferman-Stein vector-valued inequality is restricted to p, q > 1, the full-range claims in this paper are contingent on an unverified external estimate. Please either state the precise theorem from [15] and verify that it covers p ∈ (0,∞), q ∈ (0,∞], or give a proof for the endpoint cases.","section":"§3.1, Lemma 3.9(ii)"},{"comment":"In the proof of Theorem 5.6, the estimate labeled (5.17) is central to the subcritical case δ2 < 1/p and is introduced with the phrase 'Applying some arguments similar to those used in the proof of [16, Theorem 4.19]', after which the estimate is asserted without proof. As written, this is not a complete proof of the boundedness of almost diagonal operators on ˙as,υ_{p,q}(W), which is in turn used for the molecular and wavelet characterizations. Please supply the full derivation of (5.17), or give a precise lemma-level reference in [16] whose hypotheses are explicitly checked here.","section":"§5.1, Eq. (5.17)"}],"minor_comments":[{"comment":"The statement says that the functions υ_{1/q,W} and υ_{1/p,W} defined in (2.18) 'are growth functions', but the standing assumption in Theorem 6.4 is only that E_W is a scalar doubling weight, which does not imply E_W ∈ A∞ and hence does not imply that these functions belong to any class G(δ1,δ2;ω). The proof in fact uses only the monotonicity of Q ↦ ∫_Q E_W, which always holds. Please adjust the wording or add the additional assumption E_W ∈ A∞.","section":"§6.1, Theorem 6.4"},{"comment":"Several technical lemmas and corollaries are stated with 'we omit the details' (for example Lemmas 2.8, 2.9, 2.10, 3.16, 3.17, 3.24, and Corollaries 3.33 and 3.34). Some of these are routine, but for a paper of this length, a remark indicating which omitted proofs are completely standard and which are direct adaptations of proofs in [15] or [16] would significantly aid the reader.","section":"§2.2, Lemma 2.8 and elsewhere"},{"comment":"In the proof of Lemma 3.30, the choice of N uses the notation max{...} ∩ N; if the displayed maximum is not an integer, this is fine, but the formula would be clearer if written with an explicit ceiling, for example N ∈ (max{...}, ∞) ∩ N.","section":"§3.2, Lemma 3.30"},{"comment":"The notation ˙b^s_{∞,∞}(C^m) is said to be as in Remark 3.4, but Remark 3.4 defines averaging sequence spaces with a general growth function υ and does not specify that υ ≡ 1 for this classical space. Please clarify the normalization of ˙b^s_{∞,∞}(C^m), since it is used as an unweighted classical space in the proof.","section":"§6.1, Theorem 6.4(ii)"}],"recommendation":"major_revision","confidential_remarks":"The paper is well within the scope of the journal and the results are potentially significant. My main concern is not the architecture of the proofs but the heavy reliance on the matrix-weighted Fefferman-Stein inequality from [15] for the full range p ∈ (0,∞), q ∈ (0,∞]. If the authors can confirm that [15, Corollary 5.8] indeed covers p < 1 and q ≤ 1, or provide a proof for those endpoints, I would be inclined to accept after minor revision. The incomplete derivation of (5.17) should also be addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This is a serious, substantial paper. It introduces generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces with growth functions and matrix A_{p,∞} weights, then proves the full real-variable package: φ-transform, Peetre maximal, Littlewood–Paley, molecular, and wavelet characterizations. Two applications stand out: a sharp p-invariance criterion (iff W ~ E_W I_m) and a Sobolev-type embedding characterized at the sequence level. The growth condition on υ is shown necessary as well as sufficient for almost diagonal boundedness, which is a genuinely nice optimality result. The proofs are detailed and the organization is strong; the averaging-space reduction via reducing operators is an effective and natural device.\n\nThe soft spots are real but localized. The biggest is Lemma 3.9(ii), the matrix-weighted vector-valued inequality used for all Triebel–Lizorkin-type results. It is stated for p ∈ (0,∞), q ∈ (0,∞], but the proof is entirely delegated to [15, Corollary 5.8]. Neither the paper nor the stress-test note confirms that the cited result covers p < 1 or q ≤ 1; the classical scalar Fefferman–Stein inequality needs p,q > 1, so this is not an idle worry. If [15, Cor 5.8] fails or needs p,q > 1, then Theorem 2.5, and with it the maximal, Littlewood–Paley, molecular, and wavelet theorems, lose their advertised range. Lemma 3.21 (reducing-operator norm estimates from [15, Lemma 6.8]) does similar load-bearing work for strongly doubling and the almost-diagonal thresholds; its dimension exponents need to be finite for all W ∈ A_{p,∞}. These are the two things a referee must verify. Everything else is routine or appropriately cited, though there are many \"we omit the details\" passages—mostly harmless.\n\nOne smaller complaint: the abstract claims that even in the scalar setting all results are new. That is not true for the p-invariance, which in the scalar case reduces to Bownik's theorem, as the authors themselves point out in Remark 6.5(i). The overclaim is minor and does not affect the substance.\n\nThis paper deserves a serious referee, not a desk reject. The mathematics looks coherent, the novelty is real, and the external dependence is explicit. I would send it to peer review with a note that the referee should certify the cited lemmas from [15], especially Corollary 5.8 for p < 1. If those estimates hold, the paper is a strong advance; if not, the F-space results need to be restricted or reproved.","headline":"Genuinely new and well-built paper whose full-range F-space theory rests on one unverified imported inequality; referee must check [15, Cor 5.8] before the claims stand.","tokens_in":109378,"tokens_out":2602,"would_cite":true,"duration_ms":35515,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46E35","47A56","42B25","42C40","46E40","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Matrix-weighted Besov–Triebel–Lizorkin spaces admit a full real-variable theory, and their p-invariance forces scalar-like weights.","keywords":["matrix weight","generalized Besov–Triebel–Lizorkin-type space","φ-transform","Peetre maximal function","Littlewood–Paley function","almost diagonal operator","wavelet","Sobolev-type embedding"],"falsifier":"To test the p-invariance theorem, take any matrix weight $W \\in \\mathcal{A}_{p,\\infty}$ with $W$ not equivalent to $E_W I_m$ (for example the diagonal weight $\\mathrm{diag}(|x|^\\alpha, |x|^\\beta)$ with $\\alpha < \\beta \\le 0$) and, for a fixed nonzero vector $z$, compare the values of $\\|\\{1_{Q=R} z\\}\\|_{\\dot f^{s,\\upsilon_{1/q,W}}_{q,q}(W)}$ and $\\|\\{1_{Q=R} z\\}\\|_{\\dot f^{s,\\upsilon_{1/p,W}}_{p,q}(W)}$ as $Q$ ranges over dyadic cubes; if the ratio is unbounded, the theorem's necessity is confirmed. To test the optimality claim, attempt to construct an almost increasing positive function $\\upsilon$ on dyadic cubes, not obeying the $(\\delta_1,\\delta_2;\\omega)$ growth condition, for which all $(D,E,F)$-almost diagonal operators are nevertheless bounded on $\\dot a^{s,\\upsilon}_{p,q}(W)$; Proposition 5.9 says no such $\\upsilon$ exists.","tokens_in":108363,"feed_emoji":"📐","tokens_out":9962,"duration_ms":76065,"temperature":0.7,"pith_summary":"This paper introduces a unified scale of matrix-weighted Besov–Triebel–Lizorkin-type spaces $\\dot A^{s,\\upsilon}_{p,q}(W)$ in which the 'type' structure is controlled by a growth function $\\upsilon$ on dyadic cubes and the weighting by a matrix $\\mathcal{A}_{p,\\infty}$ weight $W$. The authors aim to show that this scale carries the complete real-variable theory—$\\varphi$-transform, Peetre maximal functions, Littlewood–Paley functions, molecules, and wavelets—and that the growth-function condition is exactly the right hypothesis: it is both sufficient and necessary for the boundedness of almost diagonal operators. As applications they give necessary and sufficient conditions for the equivalence of Triebel–Lizorkin-type spaces across the integrable index $p$ and for Sobolev-type embeddings. If the claims are correct, this settles the open p-invariance question for matrix weights and shows that genuine matrix-valued weights break the classical scalar p-invariance identity unless they reduce to scalar multiples of the identity.","feed_headline":"Matrix-weighted spaces now have a complete real-variable theory","feed_subtitle":"A unified growth-function scale settles the open p-invariance question and gives sharp Sobolev embeddings.","key_machinery":"The load-bearing machinery is threefold: the $(\\delta_1,\\delta_2;\\omega)$-order growth functions $\\upsilon$ on dyadic cubes, the matrix $\\mathcal{A}_{p,\\infty}$ weights, and sequences of reducing operators $A = \\{A_Q\\}$ of order $p$ for $W$. A growth function controls how the normalization constant $\\upsilon(Q)$ compares across cubes of different sizes and locations, and the paper's optimality result says this is precisely the regularity needed for almost diagonal matrices to act boundedly on the corresponding sequence spaces. The reducing operators replace the matrix weight $W$ by a sequence of positive definite matrices with the property $|A_Q z| \\approx (\\int_Q |W^{1/p} z|^p dx)^{1/p}$; the key identity is the coincidence $\\dot a^{s,\\upsilon}_{p,q}(W) = \\dot a^{s,\\upsilon}_{p,q}(A)$ (and its function-space analogue), which transfers every statement about matrix-weighted spaces to a statement about the unweighted sequence spaces $\\dot a^{s,\\upsilon}_{p,q}$ applied to $\\{|A_Q t_Q|\\}$. The discrete Littlewood–Paley $g^*_\\lambda$ characterization (Proposition 3.19) then supplies the device that makes almost diagonal operators bounded on these scalar sequence spaces, and the matrix-weighted Fefferman–Stein inequality transfers that bound back to the matrix-weighted spaces.","core_discovery":"The paper's central claim is that the spaces $\\dot A^{s,\\upsilon}_{p,q}(W)$ admit a complete real-variable theory for every matrix $\\mathcal{A}_{p,\\infty}$ weight $W$ and every $(\\delta_1,\\delta_2;\\omega)$-order growth function $\\upsilon$, and that this scale is optimal in a precise sense. The $\\varphi$-transform characterization (Theorem 2.5) identifies the function spaces with sequence spaces $\\dot a^{s,\\upsilon}_{p,q}(W)$; the coincidence theorems (Theorems 3.5 and 3.7) show that both the function and sequence spaces are, up to equivalent quasi-norms, the 'averaging' spaces built from a sequence of reducing operators of order $p$ for $W$, so the matrix weight acts through scalar quantities $|A_Q t_Q|$. On these sequence spaces the paper proves boundedness of almost diagonal operators (Theorems 5.2 and 5.6), and Proposition 5.9 shows the growth-function hypothesis is necessary as well as sufficient—if almost diagonal operators are bounded and $\\upsilon$ is almost increasing, then $\\upsilon$ must satisfy the growth condition. The applications give an 'if and only if' for p-invariance of the Triebel–Lizorkin-type spaces (Theorems 6.4 and 6.6): the identity $\\dot f^{s,\\upsilon_{1/q,W}}_{q,q}(W) = \\dot f^{s,\\upsilon_{1/p,W}}_{p,q}(W)$ holds exactly when $W \\sim E_W I_m$, i.e. when the matrix weight is scalar-like. The Sobolev-type embedding theorem (Theorem 6.11) similarly reduces an embedding between two such spaces to a pointwise estimate on the weights.","pith_inferences":["The same averaging-space reduction should transfer further operator-boundedness results from the unweighted BTL-type setting to matrix weights, including trace and pseudodifferential theorems, without re-proving the scalar theory.","The necessity of $W \\sim E_W I_m$ suggests that genuine matrix-valued weights cannot support a $p$-independent Triebel–Lizorkin scale; a possible replacement is to work with operator-valued weights and reducing operators that capture the full matrix structure rather than a scalar normalization.","The optimality statement for growth functions indicates that the boundary cases $\\delta_1 = 1/p$ and $\\delta_2 = 1/p$ are the delicate ones; constructing counterexamples near these indices would provide sharp tests of the almost-diagonal bounds."],"forward_implications":["Every future operator question (trace, pseudodifferential, Calderón–Zygmund) on the new spaces can be reduced to the unweighted sequence-space level via the coincidence theorem, so the same bounds will hold for all growth functions and all matrix $\\mathcal{A}_{p,\\infty}$ weights.","The p-invariance theorem gives a complete answer: the classical scalar identity $\\dot f^s_{\\infty,q} = \\dot f^{s,1/p}_{p,q}$ survives in the matrix setting exactly when the weight is equivalent to $E_W I_m$, and fails otherwise.","Because the growth condition is necessary as well as sufficient for almost diagonal boundedness, any attempt to widen the class of normalization functions beyond $(\\delta_1,\\delta_2;\\omega)$-growth will lose the molecular and wavelet characterizations.","The Sobolev-type embedding criterion turns a function-space embedding between two weighted spaces into a pointwise checkable condition on the two weights, giving a practical test for embeddings in applications."],"supporting_citations":[{"why":"Supplies the $\\varphi$-transform framework, the sequence-space norms $\\dot f^s_{\\infty,q}$ and $\\dot f^{s,1/p}_{p,q}$, and the scalar invariance identity that the paper generalizes to matrix weights.","marker":"[38]"},{"why":"Establishes the theory of matrix $\\mathcal{A}_{p,\\infty}$ weights used throughout: reducing operators with sharp norm estimates, the Fefferman–Stein inequality for matrix weights, and the dimension bounds that feed the almost-diagonal constants.","marker":"[15]"},{"why":"Develops matrix-weighted Triebel–Lizorkin spaces and the averaging-space technique that the present paper extends to growth functions and $\\mathcal{A}_{p,\\infty}$ weights.","marker":"[41]"},{"why":"Introduces the matrix $\\mathcal{A}_p$ and $\\mathcal{A}_{p,\\infty}$ classes and reducing operators, and poses the question about sequence-space coincidence answered in Theorem 3.5.","marker":"[99]"},{"why":"Gives the inhomogeneous matrix-weighted BTL-type theory whose almost-diagonal boundedness result is the template for Theorem 5.6 in the homogeneous growth-function setting.","marker":"[16]"},{"why":"Provides the sharp boundedness of almost diagonal operators on matrix-weighted BTL-type sequence spaces, which Theorem 5.2 extends and sharpens.","marker":"[13]"},{"why":"Introduces the scalar Besov–Triebel–Lizorkin-type spaces and poses the open question on p-invariance of Besov-type spaces settled by Proposition 6.8.","marker":"[106]"}],"fun_headline_variants":["Matrix-weighted spaces get optimal real-variable theory","Sharp p-invariance and Sobolev embeddings for matrix weights","Optimal scale: complete theory for matrix-weighted Besov-Triebel-Lizorkin","Necessary and sufficient growth conditions for matrix-weighted spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole structure rests on the existence of reducing operators for matrix $\\mathcal{A}_{p,\\infty}$ weights with sharp norm estimates and on the matrix-weighted Fefferman–Stein inequality for such weights; if those estimates fail in full generality, the coincidence of matrix-weighted and averaging spaces, and with it the $\\varphi$-transform, molecule, and wavelet characterizations, would no longer hold.","fun_headline_variants_meta":{"raw":{"variants":["Matrix-weighted spaces get optimal real-variable theory","Sharp p-invariance and Sobolev embeddings for matrix weights","Optimal scale: complete theory for matrix-weighted Besov-Triebel-Lizorkin","Necessary and sufficient growth conditions for matrix-weighted spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000877,"raw_usage":{"total_tokens":3899,"prompt_tokens":1158,"completion_tokens":2741,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":774,"completion_tokens_details":{"reasoning_tokens":2667}},"tokens_in":774,"tokens_out":2741,"duration_ms":20748,"temperature":1.0,"reasoning_tokens":2667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:59:55.182045+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To test the p-invariance theorem, take any matrix weight $W \\in \\mathcal{A}_{p,\\infty}$ with $W$ not equivalent to $E_W I_m$ (for example the diagonal weight $\\mathrm{diag}(|x|^\\alpha, |x|^\\beta)$ with $\\alpha < \\beta \\le 0$) and, for a fixed nonzero vector $z$, compare the values of $\\|\\{1_{Q=R} z\\}\\|_{\\dot f^{s,\\upsilon_{1/q,W}}_{q,q}(W)}$ and $\\|\\{1_{Q=R} z\\}\\|_{\\dot f^{s,\\upsilon_{1/p,W}}_{p,q}(W)}$ as $Q$ ranges over dyadic cubes; if the ratio is unbounded, the theorem's necessity is confirmed. To test the optimality claim, attempt to construct an almost increasing positive function $\\upsilon$ on dyadic cubes, not obeying the $(\\delta_1,\\delta_2;\\omega)$ growth condition, for which all $(D,E,F)$-almost diagonal operators are nevertheless bounded on $\\dot a^{s,\\upsilon}_{p,q}(W)$; Proposition 5.9 says no such $\\upsilon$ exists.","supporting_citations":[],"review_version":1}