{"id":"e455696d-c007-4271-85fe-20aea9f991ba","arxiv_id":"2501.09912","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Under boundedness of the local Hardy-Littlewood maximal operator, a function lies in a ball Banach function space exactly when its wavelet square function does, with equivalent norms.","lead":"This paper shows that a large family of function spaces can be described by how big a function's wavelet coefficients are, provided a standard maximal operator is bounded on the space and its dual. The method transfers weighted inequalities to many spaces at once, covering Lorentz, Herz, Morrey, Orlicz, and variable-exponent spaces, and it relaxes assumptions in a recent extension operator result.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.2's proof is a bare citation to [22, Theorem 4.6]; it never verifies the hypotheses for f∈X and silently identifies the local Bessel potential (1−t0²Δ)^{s/2} with the global Bessel potential in Definition 2.1.","rationale":"The reader correctly identifies the unverified pairings (2.5)–(2.6) as the critical assumption. My stress test refines that concern: even if [22, Theorem 4.6] is valid as a weighted statement, the proof of Theorem 2.2 does not justify that the pairings are in F_loc for arbitrary f∈X, and it silently replaces the global Bessel potential used in Definition 2.1 with the local Bessel potential appearing in the cited theorem. The missing norm equivalence is a nontrivial local/global issue, not a cosmetic rewording; the paper's own Corollary 2.4 illustrates the same confusion by asserting that Mloc-boundedness on X and X′ is equivalent to boundedness of the global Riesz transforms, which is false for weighted Lebesgue spaces with weights in A_{p,loc} but not A_p. These issues are addressable by a reformulation of the Sobolev space definition or a complete proof of the bridge, so the appropriate verdict remains CONDITIONAL rather than UNCHANGED as if the proof were sound. I do not see evidence of intentional misuse, and the extrapolation framework itself is standard; the concern is a gap in the presented argument, not a failure of the overall method.","tokens_in":18679,"tokens_out":26647,"duration_ms":279788,"concrete_test":"Inspect Ref. [22, Theorem 4.6] and verify three points: (i) whether its weighted inequality is stated for every f∈X or only for functions in a weighted Sobolev space; (ii) whether the Bessel potential is (1−t0²Δ)^{s/2} or (1−Δ)^{s/2}; (iii) whether the weights run over all A_{p,loc} with constants depending only on [w]_{A_{p,loc}}. If the theorem uses (1−t0²Δ)^{s/2}, attempt to prove ||(1−Δ)^{s/2}f||_X ≤ C ||(1−t0²Δ)^{s/2}f||_X under Mloc-boundedness; a concrete check is to test this equivalence on X=L^p(w) with w(x)=e^{|x|} using a sequence of smooth unit-height bumps supported at N. Separately, verify Corollary 2.4's equivalence: for the same X, Mloc is bounded while the Hilbert transform is unbounded, disproving the stated 'or equivalently' clause.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim, Theorem 2.2, is proved in one sentence: (2.5) and (2.6) are asserted to follow from [22, Theorem 4.6], and the proof ends. Three gaps are left open. First, [22, Theorem 4.6] is a weighted statement for local Muckenhoupt weights, but no argument shows that an arbitrary f∈X (which need not belong to any single weighted L^p(w)) satisfies the hypotheses, nor that the untruncated pair ( (1−t0²Δ)^{s/2}f, Vf+W_sf ) lies in F_loc rather than merely in some approximating family. Second, the theorem's statement uses W^s_X defined via the global Bessel potential (1−Δ)^{s/2} (Definition 2.1), while the pairings use (1−t0²Δ)^{s/2}; the equivalence of these two norms under only Mloc-boundedness is neither proved nor obvious, since the multiplier separating them is a global zero-order operator. The same local/global slippage appears in Corollary 2.4, which asserts that Mloc-boundedness on X and X′ is equivalent to boundedness of the global Riesz transforms Rj; this is false: for X=L^p(w) with w∈A_{p,loc}∖A_p, e.g. w(x)=e^{|x|}, Mloc is bounded but Rj is not. Thus the proof as written does not establish the stated wavelet characterization; it could be repaired by reformulating W^s_X with (1−t0²Δ)^{s/2} and proving the missing bridge, or by citing a version of [22] that directly uses (1−Δ)^{s/2}.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves a local version of Rubio de Francia extrapolation for ball Banach function spaces (Theorem 1.3, with a self-contained appendix proof) and uses it, together with the global version (Theorem 1.1), to derive applications: a wavelet characterization of X-based Sobolev spaces (Theorem 2.2 and Corollary 2.4), a vector-valued maximal inequality (Example 2.6), and an extension-operator result on bounded Lipschitz domains (Example 2.7, claimed as a refinement of Zhu--Yang--Yuan). The advertised wavelet theorem is derived by citing the weighted wavelet result [22, Theorem 4.6] and then applying the extrapolation theorems. The paper also surveys which classical spaces (weighted Lebesgue, Lorentz, Herz, variable-exponent, Orlicz, Morrey, Besov--Bourgain--Morrey) fall under the assumptions.","tokens_in":19071,"tokens_out":11344,"duration_ms":111355,"significance":"If Theorem 2.2 and Example 2.7 can be made fully rigorous, the paper would provide a uniform and quite general wavelet characterization of Sobolev spaces built on ball Banach function spaces, requiring only boundedness of the local Hardy--Littlewood maximal operator on X and its Kothe dual, with no convexification and no absolute-continuity assumption. The appendix proof of Theorem 1.3 is a genuine, self-contained contribution, and the survey of applications to many concrete spaces is potentially useful. However, the central advertised wavelet theorem currently rests on a one-line citation whose hypotheses are not verified, and on a local/global Bessel-potential identification that is not established; the false Riesz-transform equivalence in Corollary 2.4 is an independent error. The significance is therefore conditional on substantial technical repair.","major_comments":[{"comment":"The proof of Theorem 2.2 is a single sentence: after fixing 0<t0≪1, it asserts that (2.5) and (2.6) follow from [22, Theorem 4.6], and ends. This is not a proof as written. First, the cited theorem is a weighted statement for functions lying in certain weighted Sobolev spaces L^{p,s}(w) with w in the local Muckenhoupt class, but the present theorem must hold for arbitrary f∈X, and no argument is given that an arbitrary f∈X (or f∈W^s_X) belongs to any class to which [22, Theorem 4.6] applies, nor that the quantities ((1−t0²Δ)^{s/2}f, Vf+W_s f) are a pair of measurable functions (i.e., an element of L^0(R^n)^2) as required by the definition of F_loc. Second, the theorem statement uses W^s_X defined via the global Bessel potential (1−Δ)^{s/2} in Definition 2.1, while the pairings (2.5)--(2.6) use the local operator (1−t0²Δ)^{s/2}; the equivalence of the two Sobolev norms under only M_loc-boundedness is neither proved nor obvious, since the Fourier multipliers differ by a global zero-order factor. The claimed equivalence of norms therefore does not follow from the displayed argument. The theorem may be repairable by reformulating W^s_X with (1−t0²Δ)^{s/2} and proving a bridge between the local and global potential, or by citing a weighted wavelet theorem that directly uses (1−Δ)^{s/2}, but as it stands the central claim is not established.","section":"§2.1, Theorem 2.2"},{"comment":"The statement 'Assume that M_loc is bounded on X and on X′, or equivalently, each R_j is bounded on X' is false. For X=L^p(w) with w(x)=e^{|x|}, one has w∈A_{p,loc}∖A_p, so M_loc is bounded on X and X′, but the global Riesz transforms R_j are unbounded on L^p(w); this is exactly the classical distinction between local and global Muckenhoupt classes. The equivalence stated in the corollary is Rutsky's theorem for the global maximal operator M, not for M_loc. This incorrect equivalence should be removed or replaced by the correct global statement, and the corollary should state the wavelet equivalence under the M_loc assumption alone if that is what is intended.","section":"§2.1, Corollary 2.4"},{"comment":"The assertion 'Let f∈W^k_X(D). Then f∈L^{p,k}(w,D) for some w∈A_1' is not proved and does not follow immediately from the stated assumptions. The natural route is the embedding X↪L^η(w) with w∈A_1 mentioned in Remark 1.4, taking p=η and applying the embedding to f and all its derivatives up to order k, but this argument is absent. Without it, the operator Λf is not defined and the claimed norm equivalence ‖Λf‖_X∼∑_{|α|≤k}‖Z∂^α f‖_X is unjustified. This is load-bearing for the advertised refinement of [47, Theorem 5.4], since the entire purpose of the example is to remove the absolute-continuity hypothesis while retaining the extension property.","section":"§2.3, Example 2.7"}],"minor_comments":[{"comment":"The parameter t0 appears only in the proof ('Let 0<t0≪1'), but the statement of Theorem 2.2 does not mention t0 or its relation to the wavelet scale J; the statement should either incorporate t0 explicitly or explain why the equivalence is independent of its choice.","section":"§2.1, Theorem 2.2"},{"comment":"The sentence 'we can establish֒→Lη(w) for some w∈A1' is incomplete due to a garbled embedding symbol; it should read 'we can establish an embedding of X into L^η(w) for some w∈A_1.'","section":"Remark 1.4"},{"comment":"The displayed inequality contains a typo: the middle expression '[R_{g+f}^{1-p}R'_h]_{A1,loc}' is not the right object; the intended factorization is [R_{g+f}^{1-p}R'_h]_{A_{p,loc}} ≤ [R_{g+f}]^{p-1}_{A_{1,loc}} [R'_h]_{A_{1,loc}}.","section":"§4, equation (4.4)"},{"comment":"Reference [34] contains the garbled name 'M. Masty/suppress lo'; this should be corrected to 'M. Mastyło'.","section":"References"},{"comment":"The sentence 'This condition also applies to the weight W' is ambiguous; it should explicitly state that for W(x)=max(1,|x|)^α the membership condition is also −n<α<n(q−1).","section":"§3.1"}],"recommendation":"major_revision","confidential_remarks":"The main gap is not a matter of provenance but of verification: Theorem 2.2's proof does not check the hypotheses of the cited [22, Theorem 4.6] for f∈X and silently identifies the local and global Bessel potentials. The false equivalence in Corollary 2.4 is independent of [22] and must be corrected. If the authors reformulate W^s_X using (1−t0²Δ)^{s/2} and supply the missing bridge, the wavelet claim is likely salvageable; I would not recommend rejection, but the current version does not prove its main theorem."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper has one solid delivered proof—Theorem 1.3, the local extrapolation theorem for ball Banach function spaces—and a useful catalogue of applications, but the two advertised results, the wavelet characterization (Theorem 2.2) and the extension-operator refinement (Example 2.7), are not actually proved as stated. I would send it to a referee, not because it is close to right as written, but because the defective pieces are identifiable and plausibly repairable.\n\nTheorem 1.3 is a genuine contribution: a local version of Rubio de Francia extrapolation for ball Banach function spaces, proved without convexification, with the A_{p,loc} weight built explicitly. The examples in Section 3 (Lorentz, Herz, Morrey, Orlicz, variable exponent) also give a useful inventory of spaces where the method applies.\n\nThe soft spots are central. Theorem 2.2 is the headline, and its proof is a bare citation to [22, Theorem 4.6] plus the assertion that (2.5) and (2.6) hold. Nothing verifies that an arbitrary f in X satisfies the hypotheses of that weighted theorem. More seriously, there is a local/global mismatch: Definition 2.1 defines W^s_X via the global Bessel potential (1−Δ)^{s/2}, while [22, Theorem 4.6] works with the local operator (1−t0^2 Δ)^{s/2}. The proof never establishes that these two operators induce comparable norms under only Mloc-boundedness, and that comparison is not automatic. The same slippage appears in Corollary 2.4, where Mloc-boundedness on X and X′ is declared equivalent to boundedness of the global Riesz transforms Rj. That equivalence is false: for X=L^p(w) with w in A_{p,loc} minus A_p, such as w(x)=e^{|x|}, Mloc is bounded but Rj is not. This is not a minor missing detail; it is a false statement. Example 2.7 is also asserted without proof: the p in L^{p,k}(D,w) is never specified, and the embedding into a weighted Sobolev space under only M-boundedness is not demonstrated.\n\nThe citation pattern is not predatory. It leans heavily on the authors' own [22], but that is an independently published theorem, so heavy reliance is a proof gap rather than misconduct.\n\nBottom line: the extrapolation core is worth refereeing, and the authors may be able to repair Theorem 2.2 by reformulating W^s_X with the local Bessel operator or by proving the local/global bridge, and by fixing Corollary 2.4. As submitted, two advertised results are not established. If it were my desk, I would send to review with a clear request to rewrite, not quietly accept.","headline":"The local extrapolation theorem is real and the example list is useful, but the two advertised applications (Theorem 2.2 and Example 2.7) are not proved as stated, and Corollary 2.4 contains a false local/global equivalence; worth a referee, but only as a request for serious revision.","tokens_in":19612,"tokens_out":3892,"would_cite":false,"duration_ms":39292,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B35","41A17","26B33"],"pacs":[],"model":"deepseek-v4-flash","headline":"On any ball Banach function space where the local Hardy-Littlewood maximal operator is bounded along with its dual, smooth wavelets give a norm equivalence between f and its square function, characterizing the X-based Sobolev space W^s_X.","keywords":["extrapolation","wavelet","Riesz transform","Hardy-Littlewood maximal operator","Muckenhoupt weight","ball Banach function space","Sobolev space","extension operator"],"falsifier":"Take a ball Banach function space $X$ with $M_{\\mathrm{loc}}$ bounded on $X$ and $X'$, and a compactly supported smooth function $f$. If for some local Muckenhoupt weight $w$ the ratio $\\|Vf+W_s f\\|_{L^p(w)}/\\|(1-t_0^2\\Delta)^{s/2}f\\|_{L^p(w)}$ is unbounded as the scales vary, then the pairings (2.5)-(2.6) fail and Theorem 2.2 collapses. A concrete check is to test this ratio for wavelets of exactly borderline smoothness $K=s$ on weighted Lebesgue spaces with weights in $A_{p,\\mathrm{loc}}$ but not in $A_p$.","tokens_in":18473,"feed_emoji":"🧮","tokens_out":12938,"duration_ms":125470,"temperature":0.7,"pith_summary":"The paper proves that smooth wavelet expansions give a complete norm characterization of Sobolev-type spaces built on very general Banach function spaces, not only L^p and its weighted variants. Its main theorem says: if the local Hardy-Littlewood maximal operator is bounded on a ball Banach function space X and on its Köthe dual X', and the wavelets are smooth enough, then a function f in X belongs to the X-based Sobolev space $W^s_X$ exactly when the wavelet square function $Vf+W_s f$ lies in X, with equivalent norms. The proof is a short extrapolation step: one weighted wavelet inequality for every local Muckenhoupt weight is transferred to any X satisfying only maximal-boundedness conditions. The same technique yields vector-valued maximal inequalities and refines a recent extension-operator theorem by dropping the absolutely continuous norm assumption. One uniform mechanism now covers weighted Lebesgue, Lorentz, Herz, variable-exponent, Orlicz, Morrey, and Besov-Bourgain-Morrey spaces.","feed_headline":"Wavelet square functions characterize ball Banach Sobolev spaces","feed_subtitle":"One condition on the maximal operator yields one proof covering weighted, Herz, Orlicz, and Morrey spaces.","key_machinery":"The load-bearing mechanism is an extrapolation theorem for ball Banach function spaces (Theorem 1.3): if a pair $(f,g)$ satisfies $\\|f\\|_{L^p(w)} \\le N([w]_{A_{p,\\mathrm{loc}}})\\|g\\|_{L^p(w)}$ for every local Muckenhoupt weight $w$, then $\\|f\\|_X \\le C\\|g\\|_X$ whenever $M_{\\mathrm{loc}}$ is bounded on $X$ and on its Köthe dual $X'$. A ball Banach function space is a Banach lattice of measurable functions whose norm has finite value on balls and satisfies the Fatou property. The paper feeds two weighted pairings from [22, Theorem 4.6] into this machine, comparing the wavelet square function $Vf+W_s f$ with the Bessel potential $(1-t_0^2\\Delta)^{s/2}f$; the wavelets enter through square functions built from normalized cube indicators, with $Vf$ collecting coarse-scale coefficients and $W_s f$ the detail coefficients scaled by $2^{js}$. A short appendix proves Theorem 1.3 by dualizing with an iterated maximal operator that turns local maximal boundedness into a local $A_1$ weight.","core_discovery":"On its own terms, the paper's central claim is Theorem 2.2: fix $s>0$ and choose compactly supported wavelets $\\phi,\\psi_l$ of smoothness $K>s$. Whenever the local Hardy-Littlewood maximal operator $M_{\\mathrm{loc}}$ is bounded on a ball Banach function space $X$ and on its Köthe dual $X'$, the following equivalence holds for every $f\\in X$: $f$ lies in the $X$-based Sobolev space $W^s_X(R^n)$, defined by requiring the Bessel potential $(1-t_0^2\\Delta)^{s/2}f$ to belong to $X$, if and only if the square function $Vf+W_s f$ belongs to $X$, and the two norms are comparable. The paper also claims that the same extrapolation setup gives a vector-valued local maximal inequality and, in Example 2.7, that the known extension operator for bounded Lipschitz domains satisfies a norm equivalence using only boundedness of $M$ on $X$ and $X'$, removing the absolutely continuous norm hypothesis required by the earlier extension-operator result it refines.","pith_inferences":["The proof transfers to any operator with a known local $A_{p,\\mathrm{loc}}$ weighted bound: fractional integrals or commutators would produce analogous square-function characterizations of their natural smoothness spaces without rechecking the whole lattice class.","Because convexification is never used, the same mechanism is a plausible template for quasi-Banach lattices and for Hardy-type spaces built from such norms, once a valid extrapolation theorem is available there.","The extension-operator part suggests that trace and extension theory for Sobolev spaces on non-reflexive lattices can be built from maximal-operator control alone, avoiding density of test functions that fails in spaces like Morrey spaces.","A sharpness test of the smoothness condition $K>s$: with wavelets at the borderline smoothness $K=s$, the weighted comparison should degrade and the equivalence should fail for some $X$; this is directly checkable by weighted $L^p$ computation."],"forward_implications":["For every space listed in Section 3 — weighted Lebesgue, Lorentz, Herz, variable-exponent, Orlicz, Morrey, and Besov-Bourgain-Morrey — Theorem 2.2 gives $\\|f\\|_{W^s_X} \\simeq \\|Vf+W_s f\\|_X$.","The wavelet expansion of $f$ converges to $f$ in $X$ for any separable $X$ satisfying the maximal-boundedness assumption, and for nonseparable spaces such as weak Lebesgue or Morrey spaces the convergence can be captured in an $L^\\eta(w)$ space with an $A_1$ weight.","The extension operator is bounded from $W^k_X(D)$ to $X$ for bounded Lipschitz domains, with norm comparable to the sum of norms of $Z\\partial^\\alpha f$, and the absolutely continuous norm assumption in the earlier result is not needed.","The vector-valued local maximal inequality holds for $X$ whenever $M_{\\mathrm{loc}}$ is bounded on $X$ and $X'$."],"supporting_citations":[{"why":"Provides the weighted comparison of the wavelet square function with the Bessel potential for every local Muckenhoupt weight, giving the pairings (2.5) and (2.6) used in Theorem 2.2.","marker":"[22, Theorem 4.6]"},{"why":"States the ball Banach function space extrapolation theorem for local Muckenhoupt weights that the paper applies as Theorem 1.3.","marker":"[4, Theorem 3.1]"},{"why":"Supplies the general extrapolation theorem underlying Theorem 1.1, specialized to X=Y, r1=r2=1, and s1=s2=infinity.","marker":"[36, Theorem A]"},{"why":"Establishes the equivalence between boundedness of the maximal operator on X and X' and boundedness of Riesz transforms, used to connect the hypotheses with Corollary 1.2.","marker":"[38]"},{"why":"Constructs the weighted extension operator whose norm comparability is transferred to the X-setting in Example 2.7.","marker":"[6, Theorem 1.1]"},{"why":"The recent extension-operator theorem that this paper refines by removing the absolutely continuous norm assumption.","marker":"[47, Theorem 5.4]"}],"fun_headline_variants":["Wavelet square functions characterize Sobolev spaces under one maximal bound","One maximal condition yields wavelet characterization of many function spaces","Wavelet extrapolation without convexification characterizes Sobolev spaces","Refined extension operator via one maximal bound on ball Banach spaces","No convexification needed: wavelet characterization of function spaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof depends on a previously established weighted comparison: for every local Muckenhoupt weight, the wavelet square function and the Bessel-potential version of f satisfy the same weighted norm inequalities as f, with constants controlled by the weight's local characteristic. If that comparison fails at the required generality, the main theorem does not follow from the proof given.","fun_headline_variants_meta":{"raw":{"variants":["Wavelet square functions characterize Sobolev spaces under one maximal bound","One maximal condition yields wavelet characterization of many function spaces","Wavelet extrapolation without convexification characterizes Sobolev spaces","Refined extension operator via one maximal bound on ball Banach spaces","No convexification needed: wavelet characterization of function spaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000885,"raw_usage":{"total_tokens":3773,"prompt_tokens":846,"completion_tokens":2927,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":2843}},"tokens_in":462,"tokens_out":2927,"duration_ms":24497,"temperature":1.0,"reasoning_tokens":2843,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T19:32:38.950988+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a ball Banach function space $X$ with $M_{\\mathrm{loc}}$ bounded on $X$ and $X'$, and a compactly supported smooth function $f$. If for some local Muckenhoupt weight $w$ the ratio $\\|Vf+W_s f\\|_{L^p(w)}/\\|(1-t_0^2\\Delta)^{s/2}f\\|_{L^p(w)}$ is unbounded as the scales vary, then the pairings (2.5)-(2.6) fail and Theorem 2.2 collapses. A concrete check is to test this ratio for wavelets of exactly borderline smoothness $K=s$ on weighted Lebesgue spaces with weights in $A_{p,\\mathrm{loc}}$ but not in $A_p$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the equivalence between boundedness of the maximal operator on X and X' and boundedness of Riesz transforms, used to connect the hypotheses with Corollary 1.2."}],"review_version":1}