{"id":"b2990c82-7153-40ca-a7bc-9d150761ebf1","arxiv_id":"2504.19060","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Pseudo-differential, trace, extension, and Calderon-Zygmund operators are shown to be bounded on generalized matrix-weighted Besov-Triebel-Lizorkin-type spaces with matrix A-infinity weights.","lead":"This paper proves boundedness results for pseudo-differential, trace, and Calderon-Zygmund operators on a broad new family of matrix-weighted Besov-Triebel-Lizorkin function spaces with matrix A-infinity weights. The results generalize earlier operator bounds from narrower matrix A_p weights, so a reader interested in weighted function spaces gets a more unified theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim depends on unproved companion Theorem 3.6; index ranges in all six operator theorems are only as reliable as that external result.","rationale":"The reader's weakest assumption correctly identifies the almost diagonal operator theorem and the molecular/wavelet characterizations imported from [13] as the load-bearing support for the central boundedness claims. My reading of the manuscript confirms that the proofs of Theorem 3.10 use Lemma 3.16 and Lemma 3.17, the trace theorems use Lemma 3.23, and the Calderón–Zygmund theorem uses Theorem 3.6 through Lemma 3.43, Lemma 3.46, and Lemma 3.47. There is no independent proof of these tools in the present text, and [13] is cited as 'Submitted or arXiv:2505.02136.' This is a genuine verification gap rather than a demonstrated mathematical error; I found no concrete internal inconsistency in the reductions themselves. The paper is honest about the limitation, explicitly noting in Remark 3.7 that its index ranges are weaker than previous results for W∈A_p when p∈(1,∞), and it also omits several routine verifications ('we omit the details'). These omissions do not, by themselves, undermine the argument. The proposed test—reducing Theorem 3.6 to the classical scalar unweighted almost diagonal criteria—would settle the most direct way the imported machinery could be wrong, without requiring access to the full companion preprint. Because the reader's conditional verdict already reflects exactly this dependency, I do not recommend changing the verdict; the correct disposition remains CONDITIONAL on verification of the companion machinery. I therefore set verdict_should_be to UNCHANGED, meaning no adjustment to the reader's verdict is needed after the stress-test pass.","tokens_in":54362,"tokens_out":22088,"duration_ms":218022,"concrete_test":"Independently re-derive Theorem 3.6 in the scalar unweighted limit: take m=1, W≡1, and υ(Q)=|Q|^τ for some τ∈[0,∞), for both a=b and a=f. From first principles, or by comparison with the classical almost diagonal criteria in Frazier–Jawerth [24], compute the sharp D,E,F thresholds for ˙b^{s,τ}_{p,q} and ˙f^{s,τ}_{p,q}. Verify that Theorem 3.6 reduces exactly to D>J+n[τ−1/p]+, E>n/2+s+n[τ−1/p]+, and F>J−n/2−s−n(τ−1/p)_+, with J given by (3.4). If the reduced thresholds differ by any term, the companion theorem's index formula contains an error that propagates directly to the operator ranges in Theorems 3.10, 3.24, 3.32, 3.35, and 3.51. A successful match would still not prove the matrix-A∞ case but would remove the most concrete failure mode.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Every main theorem in this paper—Theorems 3.10, 3.24, 3.30, 3.32, 3.35, and 3.51—is proved by reducing operator boundedness to the almost diagonal operator theorem, molecular characterization, and wavelet characterization imported from the companion preprint [13], specifically Theorem 3.6 and Lemmas 3.16 and 3.23 of the present paper (cited as [13, Theorems 5.6, 5.17, 5.20]). The paper does not reproduce those proofs, and [13] is itself a submitted preprint. In particular, the thresholds D_˙a, E_˙a, F_˙a in Theorem 3.6 determine exactly which pseudo-differential, trace, and Calderón–Zygmund operators are bounded and on which index ranges. If any of the piecewise constants in (3.4) or any of the terms involving d_lower and d_upper in Theorem 3.6 is incorrect, the stated ranges in all six main theorems would shift. The authors themselves flag in Remark 3.7 that their Theorem 3.6 gives weaker ranges than the earlier matrix-A_p result [9, Theorem 13.1] when p∈(1,∞), due to the extra d_upper/p term; this self-admitted non-sharpness makes the exact threshold formula especially load-bearing rather than a cosmetic matter. The concern is not an internal inconsistency in this paper but an unverified external dependency without which the central claims cannot be certified from the present manuscript alone.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves boundedness results for pseudodifferential, trace, and Calderón–Zygmund operators on the generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces Ȧ^{s,υ}_{p,q}(W) with matrix A_{p,∞} weights, which were introduced in the companion paper [13]. The main results are Theorem 3.10 (pseudodifferential operators), Theorems 3.24, 3.30, 3.32, and 3.35 (trace and extension operators), and Theorem 3.51 (Calderón–Zygmund operators). The proofs are structured as reductions: using the ϕ-transform, molecular, and wavelet characterizations from [13], each operator is shown to map suitable molecules or atoms to synthesis molecules, with the admissible index ranges controlled by the almost-diagonal-operator thresholds D_ȧ, E_ȧ, F_ȧ from Theorem 3.6. The trace theorem is stated as an if-and-only-if result in terms of a two-weight matrix inequality (3.21).","tokens_in":54626,"tokens_out":14645,"duration_ms":141342,"significance":"If the companion results in [13] are sound, the paper delivers a broad and coherent framework for operator boundedness on matrix-weighted Besov–Triebel–Lizorkin-type spaces with A_{p,∞} weights. It subsumes or extends earlier A_p results of Bu–Hytönen–Yang–Yuan when p∈(0,1], and in the scalar unweighted case it recovers or improves several Besov-type and Triebel–Lizorkin-type results of Sawano–Yang–Yuan. The proofs are detailed and the reduction strategy is transparent, and the authors are explicitly honest about the comparison with prior work, including the non-sharpness issue for p∈(1,∞) recorded in Remark 3.7. The main limitation is the heavy reliance on the companion preprint [13], whose proofs are not included in the manuscript.","major_comments":[{"comment":"All six main theorems are proved by reducing operator boundedness to the almost diagonal operator theorem and the molecular/wavelet characterizations restated from the companion preprint [13] as Theorem 3.6, Lemma 3.16, and Lemma 3.23. The index thresholds D_ȧ, E_ȧ, F_ȧ in Theorem 3.6 determine the exact hypotheses of every main theorem, and the constants involving d_lower and d_upper are load-bearing for the stated parameter ranges. Since [13] is a submitted preprint and its proofs are not reproduced here, the present manuscript cannot be fully certified on its own: any correction or revision of those index formulas would shift the ranges in all six theorems. Please either include complete proofs of the supporting results (at least of the specific consequences used), or condition the paper on the acceptance of [13] and state this dependence explicitly in the introduction.","section":"Theorem 3.6 and Lemmas 3.16, 3.23 (used in Theorems 3.10, 3.24, 3.30, 3.32, 3.35, 3.51)"},{"comment":"The necessity direction of the trace theorems tests boundedness on a single function built from one wavelet coefficient, namely the sequence (3.22). This is a valid and efficient testing strategy, but the argument implicitly relies on the wavelet characterization in Lemma 3.23 to identify the norm of that test function, and Lemma 3.23 is itself one of the imported results from [13]. This is not a separate error, but it reinforces the first major comment: the if-and-only-if statements are only as reliable as the companion results, and the dependence should be made visible in the statement of the theorems rather than only in the proofs.","section":"Theorem 3.24 and Theorem 3.32 (necessity direction)"}],"minor_comments":[{"comment":"There are typos in the title and first lines: “Optima l Scale” and “defintion” should read “Optimal Scale” and “definition”; please proofread the manuscript.","section":"Title and page 2"},{"comment":"The notation A := {1}_{Q∈D(Rn)} is ambiguous; it should be A := {1_Q}_{Q∈D(Rn)} or explicitly described as the sequence of identity matrices on the scalar-valued setting.","section":"Remark 3.26"},{"comment":"In the proof of Theorem 3.51, the abbreviation “WPB” appears where “WBP” (the weak boundedness property defined earlier) is meant.","section":"Theorem 3.51 proof"},{"comment":"The deduction that H ≥ 0 and G ≥ 0 imply T(1)=0 and T*(1)=0 via Definition 3.40(iv) is only implicit; it should be stated explicitly because it is exactly what makes [19, Theorem 1] applicable.","section":"Theorem 3.51(ii)"},{"comment":"The phrase “optimal scale” in the title and abstract refers to the scale of spaces and growth functions, but Remark 3.7 shows that the operator-index ranges are not optimal for p∈(1,∞). Please add a clarifying sentence in the introduction or in Remark 3.7 so that readers do not interpret the operator bounds as sharp in the parameter ranges.","section":"Title and Remark 3.7"}],"recommendation":"major_revision","confidential_remarks":"The principal risk is external: the companion preprint [13] contains the almost diagonal operator theorem and the molecular/wavelet characterizations on which all six main theorems depend. I recommend that the editor ask the authors either to include the proofs of these supporting results (or a detailed verification of the specific consequences used) or to provide the latest accepted version of [13] for the refereeing process. Under those premises, the operator-theoretic arguments in the manuscript appear correct and are presented in a clear, structured way."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does what it says: it proves boundedness of pseudo-differential, trace, extension, and Calderón–Zygmund operators on the generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces from the companion paper [13]. The genuinely new content is Theorems 3.10, 3.24, 3.30, 3.32, 3.35 and 3.51, which cover matrix A_{p,∞} weights and general growth functions. The authors are right that even the case A^s_{p,q}(W) with W∈A_{p,∞} is new. The proofs are structured: each theorem reduces to checking that the operator maps atoms or wavelets to synthesis molecules, with Lemma 3.17 and Corollary 3.50 doing that work and the trace/extension arguments given in detail. The paper is also honest: Remark 3.7 admits that the index ranges are weaker than the earlier matrix-A_p result in [9] for p∈(1,∞), because of the extra d_upper/p term. That is a real limitation, but it is not hidden. The citation pattern looks appropriate: the heavy self-citations are to the companion paper where the machinery actually lives, and the comparisons to [7] and [9] are fair.\n\nThe soft spot is exactly the one the stress test identifies. Every main theorem depends on Theorem 3.6, the almost-diagonal operator boundedness result, and on the molecular and wavelet characterizations imported from [13]. Those results are not proved here, and [13] is itself a submitted preprint. This is a dependency, not circular reasoning: the present paper uses [13] as a premise and is explicit about which statements are being imported. But it means the manuscript alone does not certify the thresholds D, E, F in Theorem 3.6, and a wrong constant there would shift the parameter ranges in all six operator theorems. The authors do flag the comparison with [9] in Remark 3.7, which helps, but they do not reproduce or prove the companion results. For a final version, the companion should be available with proofs (posted with full details, ideally accepted), and the referee should have access to it. At minimum, the submission should state clearly that the paper is not self-contained.\n\nOverall this is a competent, honest, technical continuation, not a breakthrough. It deserves a serious referee. The likely outcome after review is conditional acceptance, with the main condition being resolution of the companion-paper dependency. I would bring it to a reading group that works on matrix weights, and I would cite it if I were writing in that area. It does not change how I think about the subject.\n\nRecommendation: send to peer review; require that the companion paper be available or its relevant proofs included.","headline":"A solid, honest technical continuation; the operator boundedness results are real and new, but all six theorems lean on the companion paper's almost-diagonal theorem, so peer review should happen with that dependency visible.","tokens_in":55212,"tokens_out":3244,"would_cite":true,"duration_ms":32772,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B20","46E35","35S05","47A56","46E40","42B35"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes boundedness of pseudo-differential, trace, and Calderón–Zygmund operators on generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces, including a trace theorem that is an equivalence.","keywords":["matrix weight","matrix A∞ weight","generalized Besov–Triebel–Lizorkin-type space","pseudo-differential operator","trace operator","Calderón–Zygmund operator","almost diagonal operator","molecular and wavelet characterization"],"falsifier":"In the scalar unweighted case ($m=1$, $W\\equiv 1$, $\\upsilon(Q)=|Q|^\\tau$) the thresholds in Theorem 3.6 must reduce to the known sharp almost-diagonal indices for classical Besov–Triebel–Lizorkin-type spaces; a mismatch for any choice of $p,q,\\tau$ would falsify the criterion and pull down all six theorems. Alternatively, for $n=1$ and $p=2$, choose explicit matrix weights $W,V$ for which the comparison (3.21) fails yet $s$ lies above the stated threshold; if the trace operator is still bounded from $\\dot B^{s,\\upsilon(2)}_{2,q}(W)$ to $\\dot B^{s-\\gamma/2,\\upsilon(1)}_{2,q}(V)$, the equivalence in Theorem 3.24 is false.","tokens_in":54122,"feed_emoji":"🧮","tokens_out":14728,"duration_ms":129187,"temperature":0.7,"pith_summary":"This paper aims to show that the main operator classes of harmonic analysis—pseudo-differential operators with homogeneous symbols, trace and extension operators across a hyperplane, and Calderón–Zygmund singular integrals—are bounded on the generalized matrix-weighted Besov–Triebel–Lizorkin-type spaces $\\dot A^{s,\\upsilon}_{p,q}(W)$ introduced in the companion paper [13]. The weights are matrix-valued $A_{p,\\infty}$ weights, the current broadest matrix-weight class, and the growth function $\\upsilon$ lets the results cover Morrey-type, classical, and weighted scales at once. Every theorem is proved by the same route: replace the operator by an almost diagonal matrix on the associated sequence space, apply the boundedness criterion from [13, Theorem 3.6], and reassemble the output through synthesis molecules. The trace result is an iff statement, so it identifies exactly which matrix-weight comparison is needed for a bounded boundary trace. If the proofs are correct, one uniform framework yields all these operator bounds, recovering and in several cases improving earlier scalar and matrix-weighted results.","feed_headline":"Six theorems prove bounded operators on matrix-weighted spaces","feed_subtitle":"Six theorems extend Besov–Triebel–Lizorkin bounds to matrix A∞ weights, with a sharp trace result.","key_machinery":"The load-bearing mechanism is the boundedness of almost diagonal operators on the matrix-weighted sequence spaces $\\dot a^{s,\\upsilon}_{p,q}(W)$. An infinite matrix $U=\\{u_{Q,R}\\}$ is $(D,E,F)$-almost diagonal when $|u_{Q,R}|$ is controlled by $[1+|x_Q-x_R|/(\\ell(Q)\\vee\\ell(R))]^{-D}$ times powers of the scale ratio $\\ell(Q)/\\ell(R)$, and Theorem 3.6 gives explicit thresholds on $D,E,F$ in terms of the smoothness $s$, the growth function $\\upsilon$, and the $A_{p,\\infty}$-lower and upper dimensions of $W$. This single criterion powers the molecular, $\\phi$-transform, and wavelet characterizations from [13]; each operator is converted into its coefficient matrix between analysis and synthesis molecules, the matrix is shown to be almost diagonal, and boundedness follows. Pseudo-differential operators enter through the lemma that $|Q|^{\\eta/n}T_\\theta(\\psi_Q)$ is a molecule, trace and extension operators enter through compactly supported wavelets and comparisons of reducing operators, and Calderón–Zygmund operators enter through the atom-to-molecule proposition of the paper.","core_discovery":"The central claim is that the full set of operator-boundedness phenomena survives on $\\dot A^{s,\\upsilon}_{p,q}(W)$ with matrix $A_{p,\\infty}$ weights. Theorem 3.10 constructs an extension $\\widetilde T_\\theta$ of a pseudo-differential operator with symbol $\\theta \\in \\dot S^\\eta_{1,1}$ and proves it is bounded from $\\dot A^{s+\\eta,\\upsilon}_{p,q}(W)$ to $\\dot A^{s,\\upsilon}_{p,q}(W)$ whenever $F_{\\dot a^{s,\\upsilon}_{p,q}(W)} < n/2$, with a vanishing-moment condition covering the critical case. Theorem 3.24 proves that the trace operator extends to a bounded map from $\\dot B^{s,\\upsilon(n+1)}_{p,q}(W)$ on $\\mathbb R^{n+1}$ to $\\dot B^{s-\\gamma/p,\\upsilon(n)}_{p,q}(V)$ on $\\mathbb R^n$ if and only if, for every dyadic cube $Q'$ and vector $z$, $\\int_{Q'} |V^{1/p}(x)z|^p\\,dx \\le C 2^{j_{Q'}\\gamma}\\int_{P(Q',0)} |W^{1/p}(x)z|^p\\,dx$; Theorems 3.30, 3.32, and 3.35 give the companion extension operators and the Triebel–Lizorkin version. Theorem 3.51 proves that Calderón–Zygmund operators in the class $CZO_\\sigma(E,F,G,H)$ are bounded whenever the indices satisfy (3.62), by showing they map sufficiently regular atoms to synthesis molecules. Specializing $\\upsilon(Q)=|Q|^\\tau$ recovers matrix-weighted Besov–Triebel–Lizorkin-type spaces, so the results are new already for matrix-weighted Besov–Triebel–Lizorkin spaces with matrix $A_{p,\\infty}$ weights.","pith_inferences":["The iff trace theorem suggests treating (3.21) as the definition of a traceability class for pairs of matrix weights $(W,V)$ across a hyperplane; classifying which pairs satisfy it for given $p,\\gamma$ would be a natural next step the paper does not isolate.","The extra term $d^{upper}_{p,\\infty}(W)/p$ in the index thresholds appears only because the weights are $A_{p,\\infty}$ rather than $A_p$; since the paper observes it vanishes for $p\\le 1$, testing diagonal-matrix examples for $p>1$ should show whether the term is necessary or an artefact of the reducing-operator estimate.","The same molecule-and-wavelet scheme is directly portable: any setting with an almost-diagonal boundedness theorem and an atom-to-molecule lemma (mixed norms, product spaces, inhomogeneous spaces) should inherit matching pseudo-differential, trace, and Calderón–Zygmund boundedness results.","Since all proofs are reductions to almost diagonal matrices, the constants in the operator bounds are controlled by the same quantities as the sequence-space criterion; extracting explicit dependence on the $A_{p,\\infty}$ characteristic would make the results quantitative."],"forward_implications":["Homogeneous pseudo-differential operators of order $\\eta$ shift the smoothness scale: they boundedly map $\\dot A^{s+\\eta,\\upsilon}_{p,q}(W)$ into $\\dot A^{s,\\upsilon}_{p,q}(W)$, exactly as in classical Besov–Triebel–Lizorkin theory.","The trace and extension theorems make $\\dot B^{s,\\upsilon(n+1)}_{p,q}(W)$ and $\\dot B^{s-\\gamma/p,\\upsilon(n)}_{p,q}(V)$ exact boundary counterparts, with $Tr \\circ Ext$ the identity, precisely under the matrix-weight comparison (3.21) or its reverse.","Calderón–Zygmund operators are bounded on every space of this family whose indices satisfy (3.62), so the result covers matrix-weighted Besov–Triebel–Lizorkin spaces $\\dot A^s_{p,q}(W)$ with $W\\in A_{p,\\infty}$—cases that are new even without the Morrey-type parameter.","Because $\\upsilon(Q)=|Q|^\\tau$ and $\\upsilon(Q)=\\int_Q w$ are included, the same proofs deliver the results simultaneously for matrix-weighted Besov–Triebel–Lizorkin, Besov–Triebel–Lizorkin-type, and weighted spaces.","In the scalar unweighted limit the theorems match or improve prior Besov–Triebel–Lizorkin-type results, in particular removing the upper bound on $\\tau$ that earlier pseudo-differential and trace theorems required."],"supporting_citations":[{"why":"Companion paper that defines the spaces and supplies Theorem 3.6 and the molecular, transform, and wavelet characterizations used in every proof.","marker":"[13]"},{"why":"Earlier matrix-$A_p$ treatment of molecules, wavelets, trace, pseudo-differential, and Calderón–Zygmund results that the present theorems extend and, for $p\\le1$, coincide with.","marker":"[7]"},{"why":"Source of the matrix $A_{p,\\infty}$ class, the lower and upper dimension parameters, and the sharp reducing-operator estimate entering all index thresholds.","marker":"[11]"},{"why":"Previous sharp almost-diagonal boundedness theorem for matrix-$A_p$ spaces whose index ranges are compared in Remark 3.7.","marker":"[9]"},{"why":"Classical discrete transform, atoms, molecules, and the almost-diagonal framework adapted here to matrix-weighted type spaces.","marker":"[24]"},{"why":"Supplies the homogeneous-symbol pseudo-differential calculus and the decay estimates behind the molecule lemma for $T_\\theta$.","marker":"[36]"},{"why":"Compactly supported orthonormal wavelets with vanishing moments, used for the wavelet characterization and the trace and extension construction.","marker":"[17]"},{"why":"L2 boundedness criterion for singular integrals used to ensure the Calderón–Zygmund extension agrees with the originally given operator.","marker":"[19]"}],"fun_headline_variants":["Sharp trace and full operator bounds on matrix-weighted spaces","Six theorems extend operator bounds to matrix A∞ spaces","Pseudo-differential, trace, C–Z bounded on matrix-weighted spaces","Matrix A∞ weights: all operator bounds survive"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the companion paper's boundedness criterion for almost diagonal matrices on the sequence spaces—together with the molecular and wavelet characterizations built on it—is correct; every main theorem is only that criterion applied after checking that the operator's coefficient matrix is almost diagonal.","fun_headline_variants_meta":{"raw":{"variants":["Sharp trace and full operator bounds on matrix-weighted spaces","Six theorems extend operator bounds to matrix A∞ spaces","Pseudo-differential, trace, C–Z bounded on matrix-weighted spaces","Matrix A∞ weights: all operator bounds survive"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000842,"raw_usage":{"total_tokens":3751,"prompt_tokens":1113,"completion_tokens":2638,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":729,"completion_tokens_details":{"reasoning_tokens":2571}},"tokens_in":729,"tokens_out":2638,"duration_ms":20124,"temperature":1.0,"reasoning_tokens":2571,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:02:37.571443+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the scalar unweighted case ($m=1$, $W\\equiv 1$, $\\upsilon(Q)=|Q|^\\tau$) the thresholds in Theorem 3.6 must reduce to the known sharp almost-diagonal indices for classical Besov–Triebel–Lizorkin-type spaces; a mismatch for any choice of $p,q,\\tau$ would falsify the criterion and pull down all six theorems. Alternatively, for $n=1$ and $p=2$, choose explicit matrix weights $W,V$ for which the comparison (3.21) fails yet $s$ lies above the stated threshold; if the trace operator is still bounded from $\\dot B^{s,\\upsilon(2)}_{2,q}(W)$ to $\\dot B^{s-\\gamma/2,\\upsilon(1)}_{2,q}(V)$, the equivalence in Theorem 3.24 is false.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier matrix-$A_p$ treatment of molecules, wavelets, trace, pseudo-differential, and Calderón–Zygmund results that the present theorems extend and, for $p\\le1$, coincide with."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous sharp almost-diagonal boundedness theorem for matrix-$A_p$ spaces whose index ranges are compared in Remark 3.7."},{"cited_title":"Grafakos and R","cited_arxiv_id":null,"evidence_quote":"Supplies the homogeneous-symbol pseudo-differential calculus and the decay estimates behind the molecule lemma for $T_\\theta$."},{"cited_title":"Daubechies, Orthonormal bases of compactly support ed wavelets, Comm","cited_arxiv_id":null,"evidence_quote":"Compactly supported orthonormal wavelets with vanishing moments, used for the wavelet characterization and the trace and extension construction."},{"cited_title":"David and J.-L","cited_arxiv_id":null,"evidence_quote":"L2 boundedness criterion for singular integrals used to ensure the Calderón–Zygmund extension agrees with the originally given operator."}],"review_version":1}