{"id":"f42a16c7-c7e0-4c46-99e0-450c6b9bb923","arxiv_id":"1908.03291","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A survey with new proofs, corrections of known errors, and improved exponent ranges for maximal characterizations and Calderon-Zygmund boundedness on mixed-norm Hardy spaces.","lead":"This survey of mixed-norm function spaces proves an unproved inequality of Bagby, corrects several errors in prior results, and improves the maximal function characterizations of anisotropic mixed-norm Hardy spaces. Specialists will read it for the fixes and the new proof techniques.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.10's proof of (ii)⇒(i) invokes Bownik's scalar anisotropic Hardy-space result for mixed-norm L^{p⃗}; the missing mixed-norm analogue is the central unproven step.","rationale":"The reader's verdict of CONDITIONAL is appropriate, and my concern supports it, but for a different reason than the one emphasized in the reader's weakest_assumption. The reader focused on Lemma 4.3 failing when some p_i = ∞, citing Remark 4.4. However, Theorem 4.10 and Lemma 4.12 only claim p⃗ ∈ (0,∞)^n, so the ∞-endpoint is outside the theorem's scope. The genuinely load-bearing gap is the use of [14, Proposition 3.10], a scalar anisotropic Hardy-space result, to justify the mixed-norm implication (ii)⇒(i) in Theorem 4.10. The proof obtains the radial grand maximal M^0_I(f) ∈ L^{p⃗}, while Definition 4.4 requires M_{N_{p⃗}}(f) ∈ L^{p⃗} for a potentially much larger N_{p⃗}. Since S_{N_{p⃗}} ⊂ S_I, it would suffice to know M_I(f) ∈ L^{p⃗}; but the passage from the radial to the non-tangential grand maximal in mixed norms is precisely the type of result the theorem claims, and no mixed-norm version of Bownik's proposition is stated or proved. This is compounded by Lemma 4.12, whose proof is omitted but which itself hinges on choosing λ < p_- to apply Lemma 4.3 with exponent vector p⃗/λ ∈ (1,∞)^n. The gap is likely fillable, since the standard scalar argument may extend to mixed norms using Lemma 4.3, which is why I would not move the verdict to REJECT. The paper should either supply the mixed-norm proof or explicitly state the mixed-norm version of the cited proposition as a lemma with proof. The survey's other contributions—Bagby's inequality proof, the corrections in §4.2.1, and the revised Calderón–Zygmund results—appear substantially supported; the concern is confined to the claimed improvement in Theorem 4.10, which is the paper's headline new result.","tokens_in":66226,"tokens_out":16342,"duration_ms":164495,"concrete_test":"Provide a self-contained proof of the following mixed-norm companion of Bownik's Proposition 3.10: for N ≥ ⌊1/p_-⌋+2ν+3, M_N(f) ∈ L^{p⃗}(R^n) if and only if M^0_N(f) ∈ L^{p⃗}(R^n), using only the pointwise estimates in Lemmas 4.9, 4.13, 4.14 and the mixed-norm Hardy–Littlewood maximal inequality Lemma 4.3. If such a proof cannot be supplied without additional hypotheses (for example, a_+ = a_- or p_i > 1 for all i), then the range of N in Theorem 4.10 is unsupported. Alternatively, check whether [14, Proposition 3.10] itself contains a vector-exponent version; if it does not, the citation does not cover the mixed-norm conclusion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the proof of Theorem 4.10 (§4.2.2), the authors establish, via Lemmas 4.13 and 4.12, that ||M^{0(K,0)}_I(f)||_{L^{p⃗}} ≤ C ||M^{(K,0)}_φ(f)||_{L^{p⃗}} and hence, after K→∞, that the radial grand maximal function M^0_I(f) lies in L^{p⃗} for I = ⌊1/p_-⌋+2ν+3. They then write: “By this and [14, p. 17, Proposition 3.10] with A as in (4.7), we further conclude that, if (ii) holds true, then (i) also holds true.” This is the load-bearing step, and it is not justified. Bownik's Proposition 3.10 is a statement about scalar anisotropic Hardy spaces H^p_A: it gives the equivalence of the radial and non-tangential grand maximal conditions in L^p(R^n). The present space H^{p⃗}_a is defined in Definition 4.4 with the much larger N_{p⃗} from (4.6), and the target condition is membership in the mixed-norm space L^{p⃗}. To conclude (i) from M^0_I(f)∈L^{p⃗}, one needs a mixed-norm analogue of Bownik's radial-to-non-tangential equivalence, e.g. that M_I(f) ∈ L^{p⃗} (then M_{N_{p⃗}} ≤ M_I since S_{N_{p⃗}} ⊂ S_I). The inequality M_I ≲ M^0_I in L^{p⃗} is exactly the kind of maximal-function equivalence that Theorem 4.10 is meant to prove; citing the scalar-p version of [14] leaves a circular or at least undocumented gap. Lemma 4.12, on which the argument depends, is also stated with “the details are omitted”; its proof requires choosing λ < p_- so that p⃗/λ ∈ (1,∞)^n before applying Lemma 4.3, and this restriction is not stated. The paper's own Remark 4.4 shows Lemma 4.3 fails at p_i = ∞, but Theorem 4.10 excludes ∞, so that endpoint is not the real problem; the real problem is the unproved mixed-norm radial-to-non-tangential step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This survey reviews recent developments in function spaces with mixed norms, covering mixed Lebesgue spaces, iterated and weak mixed-norm spaces, mixed Morrey spaces, and anisotropic mixed-norm Hardy spaces H^pvec_a(R^n). Its new contributions are: a detailed proof of Bagby's extended Hardy–Littlewood maximal inequality for mixed Lebesgue norms (Theorem 2.12); corrections to the range of exponents in the boundedness of the anisotropic Hardy–Littlewood maximal operator, including a counterexample when one exponent is infinity (Remark 4.4 and Lemma 4.3); a corrected and sealed proof of the Lusin area function characterization of H^pvec_a (Section 4.2.1); an improved maximal-function characterization of H^pvec_a with a smaller admissible index N (Theorem 4.10, Section 4.2.2); and revised versions of the boundedness of anisotropic Calderón–Zygmund operators (Theorems 4.18 and 4.19). The paper is largely expository but contains several genuinely new or corrected statements.","tokens_in":66686,"tokens_out":10682,"duration_ms":111132,"significance":"If the main new proofs are correct, the paper makes a useful contribution to the mixed-norm harmonic analysis literature. The detailed proof of Bagby's inequality fills a known gap, and the correction of Lemma 4.3 with an explicit counterexample is valuable given that the erroneous range (1,∞]^n appears in prior work. Theorem 4.10, if established, would genuinely improve the maximal-function characterization of anisotropic mixed-norm Hardy spaces by lowering the required smoothness/moment index. The paper also carefully records corrections to the proof of the Lusin-area characterization in [42], which is a service to the community. The survey portions are extensive and well organized. However, as detailed in the major comments, the proof of the main new theorem contains a load-bearing step that is currently justified only by an appeal to a scalar result, and several auxiliary lemmas or revised theorems are stated without proof. These issues are significant but appear fixable within the manuscript's own framework.","major_comments":[{"comment":"After establishing (4.59) and letting K → ∞, the proof concludes from M^0_I(f) ∈ L^pvec(R^n) that f ∈ H^pvec_a(R^n) by citing [14, p. 17, Proposition 3.10]. Bownik's Proposition 3.10 is a statement about scalar anisotropic Hardy spaces H^p_A on L^p(R^n); it does not by itself justify a conclusion in the mixed-norm space L^pvec. What is needed is a mixed-norm analogue, for example a pointwise estimate of the form M_I(f)(x) ≤ C [M_HL^a((M^0_I(f))^r)(x)]^{1/r} for some r < p_-, followed by an application of Lemma 4.3. Without such an estimate, the implication (ii)⇒(i) is not documented. This is load-bearing because Definition 4.4 defines H^pvec_a using an index N_pvec from (4.6) which can be larger than the I appearing in the argument, so one needs control of the non-tangential grand maximal function M_I, not merely the radial one. Please provide the missing mixed-norm argument or state and prove the precise proposition being invoked.","section":"§4.2.2, proof of Theorem 4.10, implication (ii)⇒(i)"},{"comment":"Lemma 4.12 is stated with the comment 'the details are omitted', yet it is used in both directions of the proof of Theorem 4.10. The proof is not completely immediate: one must choose λ < p_- so that the exponent vector pvec/λ lies in (1,∞)^n before applying Lemma 4.3, and the range of admissible λ depends on N. Since this lemma is central to the main improvement claimed in the paper, the omitted derivation should be supplied or at least sketched in sufficient detail. The closely related Lemma 4.11 is also stated without proof; a one-line argument or reference would suffice there.","section":"§4.2.2, Lemma 4.12"},{"comment":"The revised boundedness results for anisotropic β-order Calderón–Zygmund operators are asserted by saying that replacing ⌊β⌋ by ⌈β⌉−1 in the proofs of [42, Theorems 6.8 and 6.9] suffices, with details omitted. This is a substantive modification: Definition 4.13 changes the regularity and moment conditions, and the admissible range of p_- in Theorem 4.18 is altered accordingly. The authors should at least indicate where the kernel estimates are affected by the new definition and why the exponent range in p_- still closes. As written, this is an unproved claimed improvement.","section":"§4.4, Theorems 4.18 and 4.19"}],"minor_comments":[{"comment":"Lemma 4.11 is stated with 'we omit the details'; since it is a standard monotone convergence property for L^pvec, either a reference or a one-sentence proof would improve the exposition.","section":"Lemma 4.11"},{"comment":"The role of N in Theorem 4.10 should be clarified: since H^pvec_a is defined in Definition 4.4 with the index N_pvec from (4.6), the statement should explicitly say that (i) refers to membership in that fixed space, while the N appearing in the theorem is the index used for the maximal functions in (ii) and (iii).","section":"Theorem 4.10, statement"},{"comment":"There is a typo in 'curial role'; it should be 'crucial role'.","section":"Remark 2.2"},{"comment":"The counterexample in Remark 4.4 has some notational slips: 'for any x2 ∈ Rn' should be 'for any x2 ∈ R', and the integral expression '∫_{R^2}' appears where a one-dimensional integral is intended. These do not affect the mathematical content.","section":"Remark 4.4"},{"comment":"The claim that the range of N in Theorem 4.2 is a proper subset of the range in Theorem 4.10 would benefit from a short justification, since the comparison depends on a-, a+, ν, and p_-.","section":"Remark 4.6"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the proof of Theorem 4.10: the current citation of Bownik's scalar Proposition 3.10 leaves a genuine gap in the mixed-norm setting. I believe the gap is likely fixable using the paper's own machinery (Lemmas 4.3, 4.9, 4.10, and 4.12), so I recommend major revision rather than rejection. I would advise the editor to have the revised proof of Theorem 4.10 checked by an expert in anisotropic Hardy spaces, since the omitted step is the central new contribution."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The genuinely useful parts are the Bagby inequality proof, the corrections to Nogayama's and Huang-Liu-Yang-Yuan's statements, and the explicit counterexample showing the Hardy-Littlewood maximal operator can fail on mixed-norm L^p when one exponent is infinity. Those are real, checkable contributions, and the survey portions are competently organized. The citation pattern is fine: the self-citations point to published theorems being corrected, not to hidden assumptions.\n\nThe soft spot is Theorem 4.10, the advertised improvement. The proof of (ii)⇒(i) establishes ||M0_I(f)||_{L^p} ≤ C||M_φ(f)||_{L^p} and then invokes [14, Proposition 3.10] to conclude f ∈ H^p_a. Bownik's proposition is a scalar L^p statement. What is needed is a mixed-norm version: from the radial grand maximal function M0_I(f) lying in L^{p_vec}, conclude the non-tangential grand maximal function with the large N_{p_vec} from Definition 4.4 also lies in L^{p_vec}. That inequality is not supplied. Since N_{p_vec} can be much larger than I when n>1, the monotonicity M_{N_{p_vec}} ≤ M_I does not close the gap; in fact the non-tangential maximal function is larger than the radial one, so the reverse comparison is the nontrivial direction. This step is load-bearing for the claimed improvement, and the stress-test note points at exactly the right place.\n\nMinor issues: Lemmas 4.11 and 4.12 have details omitted, and Theorems 4.18-4.19 are asserted via a routine modification argument. Those are secondary and probably fixable, but they do add to the sense that the paper is slightly rougher than its main theorem requires.\n\nOverall, this is a useful survey with credible corrections and one new result that needs a repaired proof. It deserves a serious referee: someone should either supply a proof of the mixed-norm radial-to-non-tangential comparison or the claim in Theorem 4.10 should be downgraded to what actually follows. I would not desk-reject it, but I would not accept it as is.","headline":"The Bagby proof and the corrections are solid, but Theorem 4.10's main step cites a scalar result where a mixed-norm analogue is needed.","tokens_in":692,"tokens_out":1933,"would_cite":true,"duration_ms":80752,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["42B35","42B30","42B25","42B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves a sharper maximal-function characterization of anisotropic mixed-norm Hardy spaces and supplies missing proofs and corrections.","keywords":["mixed norm","weak Lebesgue space","Morrey space","Hardy space","maximal function","Littlewood–Paley function","Calderón–Zygmund operator","anisotropic quasi-homogeneous norm"],"falsifier":"For the endpoint question the paper itself flags, repeat the computation of Remark 4.4: with $n=2$, $\\vec p=(p_1,\\infty)$, and $f(x_1,x_2)=x_2^{1-1/p_1}/x_1$ on the sector $x_1\\ge x_2>0$ and zero elsewhere, the mixed norm of $f$ is finite but the anisotropic Hardy\\,--\\,Littlewood maximal function has infinite $L^{(p_1,\\infty)}$ norm, refuting the endpoint version of the maximal-operator bound. For the improved characterization itself, a decisive check would be to test whether the threshold $\\lfloor1/p_-\\rfloor+2\\nu+3$ can be lowered further or whether a counterexample appears at that boundary.","tokens_in":66010,"feed_emoji":"","tokens_out":11930,"duration_ms":113577,"temperature":0.7,"pith_summary":"Mixed-norm Lebesgue spaces let different coordinate directions carry different integrability, which makes them natural for space-time estimates in partial differential equations and for problems with anisotropic geometry. This survey argues that the associated anisotropic mixed-norm Hardy spaces $H^{{\\vec p}}$_{\\vec a}(\\mathbb R^n) have a usable real-variable theory: the central new result is that membership in $H^{{\\vec p}}$_{\\vec a} is equivalent to the radial or non-tangential maximal function of $f$ lying in $L^{\\vec p}$, with a smaller smoothness cutoff on the test function than was previously available. The paper also supplies the first detailed proof of an extended Hardy\\,--\\,Littlewood maximal inequality that had been stated without proof, and it repairs errors in earlier treatments, including an endpoint counterexample showing one maximal-operator estimate fails when an exponent is infinite. If these results are right, the standard tools of harmonic analysis\\,---\\,maximal functions, atoms, area functions, and singular-integral boundedness\\,---\\,transfer to function spaces with mixed norms, which is what makes them usable in PDE applications.","feed_headline":"One maximal-function test now certifies mixed-norm Hardy spaces","feed_subtitle":"New proof lowers the required smoothness cutoff and patches earlier proof gaps.","key_machinery":"The central objects are the mixed Lebesgue quasi-norm $\\|f\\|_{L^{\\vec p}(\\mathbb R^n)}=\\big(\\int_{\\mathbb R}\\cdots\\big(\\int_{\\mathbb R}|f(x_1,\\dots,x_n)|^{p_1}\\,dx_1\\big)^{p_2/p_1}\\cdots dx_n\\big)^{1/p_n}$ and the anisotropic quasi-homogeneous balls that become rectangles under coordinatewise dilations $x\\mapsto(t^{a_1}x_1,\\dots,t^{a_n}x_n)$. The argument is carried by a comparison of truncated weighted maximal functions: at each dyadic scale $K$ the auxiliary maximal functions $M^{0(K,L)}_\\phi$ and $M^{(K,L)}_\\phi$ are dominated pointwise by powers of the anisotropic Hardy\\,--\\,Littlewood maximal operator, and the estimate is transferred to the untruncated radial and non-tangential maximal functions by letting $K$ tend to infinity and invoking monotone convergence in mixed-norm spaces. The extended maximal inequality of Section 2.2 is proved by induction on the number of variables, with the Riesz\\,--\\,Thorin interpolation theorem for mixed Lebesgue spaces supplying the passage from endpoint cases to the full exponent range.","core_discovery":"The central claim is that the anisotropic mixed-norm Hardy space $H^{\\vec p}_{\\vec a}(\\mathbb R^n)$, defined through the non-tangential grand maximal function $M_N$, is characterized by the simpler radial and non-tangential maximal functions: for any Schwartz function $\\phi$ with $\\int\\phi\\neq 0$, a distribution $f$ belongs to $H^{\\vec p}_{\\vec a}$ if and only if $M^0_\\phi(f)\\in L^{\\vec p}(\\mathbb R^n)$, equivalently $M_\\phi(f)\\in L^{\\vec p}(\\mathbb R^n)$, and the quasi-norms are equivalent. The new proof lowers the required parameter $N$ to any integer at least $\\lfloor 1/p_-\\rfloor+2\\nu+3$, where $\\nu$ is the homogeneous dimension and $p_-$ the smallest component of $\\vec p$, improving the earlier threshold. The paper also proves in full the extended Hardy\\,--\\,Littlewood maximal inequality stated without proof in the literature, uses it to justify the boundedness of the anisotropic Hardy\\,--\\,Littlewood maximal operator and Fefferman\\,--\\,Stein vector-valued inequalities on mixed Lebesgue spaces, and corrects the exponent ranges in atomic and Littlewood\\,--\\,Paley characterizations. In particular, it records a counterexample showing the maximal operator is unbounded on $L^{(p_1,\\infty)}$, and it supplies a repaired sufficiency proof for the Lusin area function characterization.","pith_inferences":["Beyond the paper, the truncation-and-interpolation strategy behind the improved maximal characterization should carry over to mixed-norm Hardy spaces defined through expansive matrix dilations, because the pointwise estimates it invokes already exist in that setting.","The endpoint failure for $p_i=\\infty$ suggests that a genuinely useful mixed-norm Hardy theory at infinite exponents would require a maximal function with weights or an Orlicz-type norm; the paper leaves this open.","A testable consequence of the corrected atomic exponent range is that finite-atomic and molecular descriptions of $H^{\\vec p}_{\\vec a}$ should hold uniformly for all $p\\in(0,\\min\\{1,p_-\\})$, simplifying the sublinear-operator boundedness criterion.","The repaired Lusin area function proof likely applies to product or matrix-dilation analogues of these spaces, where the same type of dyadic-cube summation argument is used."],"forward_implications":["Membership in an anisotropic mixed-norm Hardy space can now be tested through a single Schwartz function with nonzero integral at smoothness cutoff $N\\ge\\lfloor1/p_-\\rfloor+2\\nu+3$, a weaker requirement than in the earlier characterization.","The detailed proof of the extended Hardy\\,--\\,Littlewood maximal inequality closes a gap in the standard derivation of maximal-operator boundedness and Fefferman\\,--\\,Stein vector-valued inequalities on mixed Lebesgue spaces.","The endpoint counterexample fixes the admissible exponent ranges in the atomic definitions, so the atomic, finite-atomic, and Littlewood\\,--\\,Paley descriptions of $H^{\\vec p}_{\\vec a}$ are consistent with the maximal-function definition.","The revised Calder\\'on\\,--\\,Zygmund results cover smoothness orders $\\beta\\in(0,\\infty)$ instead of only non-integer orders, so more singular integral operators are known to map $H^{\\vec p}_{\\vec a}$ into itself or into $L^{\\vec p}$.","The dual of $H^{\\vec p}_{\\vec a}$ is identified with a mixed-norm Campanato space for $\\vec p\\in(0,1]^n$, and with $L^{\\vec p'}$ for $\\vec p\\in(1,\\infty)^n$, extending classical $H^1$\\,--\\,BMO duality to mixed norms."],"supporting_citations":[{"why":"introduced anisotropic mixed-norm Hardy spaces and stated the maximal-function characterization that Theorem 4.10 improves","marker":"[21, Theorem 3.4]"},{"why":"gave the boundedness of the anisotropic Hardy\\,--\\,Littlewood maximal operator on mixed Lebesgue spaces, whose exponent range is corrected here and on which the new proof relies","marker":"[42, Lemma 3.5]"},{"why":"stated without proof the extended Hardy\\,--\\,Littlewood maximal inequality that Section 2.2 proves in detail","marker":"[4]"},{"why":"introduced mixed Lebesgue spaces and provides the H\\'older, duality, and Riesz\\,--\\,Thorin interpolation results used throughout","marker":"[8]"},{"why":"provides the weighted Fefferman\\,--\\,Stein maximal inequality used as the induction step in the proof of the extended maximal inequality","marker":"[27]"},{"why":"supplies the anisotropic Hardy-space estimates and pointwise maximal-function comparisons used in the new proof of Theorem 4.10","marker":"[14]"},{"why":"established the dual-space identification for $H^{\\vec p}_{\\vec a}$ with mixed-norm Campanato spaces, stated here as Theorem 4.11","marker":"[43]"}],"fun_headline_variants":["Mixed-norm Hardy spaces: simpler maximal function test","Sharper condition for anisotropic mixed-norm Hardy spaces","Maximal operator bound proven for mixed Lebesgue spaces","Survey seals gaps in mixed-norm space theory","Counterexample shows maximal map unbounded on L^(p,∞)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The new proof depends on the averaging operator over anisotropic balls being bounded on mixed-norm Lebesgue spaces only when every direction has exponent strictly between 1 and infinity; the paper's own counterexample shows this fails when one exponent is infinity.","fun_headline_variants_meta":{"raw":{"variants":["Mixed-norm Hardy spaces: simpler maximal function test","Sharper condition for anisotropic mixed-norm Hardy spaces","Maximal operator bound proven for mixed Lebesgue spaces","Survey seals gaps in mixed-norm space theory","Counterexample shows maximal map unbounded on L^(p,∞)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000426,"raw_usage":{"total_tokens":2194,"prompt_tokens":972,"completion_tokens":1222,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":1143}},"tokens_in":588,"tokens_out":1222,"duration_ms":11290,"temperature":1.0,"reasoning_tokens":1143,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:17:41.179414+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For the endpoint question the paper itself flags, repeat the computation of Remark 4.4: with $n=2$, $\\vec p=(p_1,\\infty)$, and $f(x_1,x_2)=x_2^{1-1/p_1}/x_1$ on the sector $x_1\\ge x_2>0$ and zero elsewhere, the mixed norm of $f$ is finite but the anisotropic Hardy\\,--\\,Littlewood maximal function has infinite $L^{(p_1,\\infty)}$ norm, refuting the endpoint version of the maximal-operator bound. For the improved characterization itself, a decisive check would be to test whether the threshold $\\lfloor1/p_-\\rfloor+2\\nu+3$ can be lowered further or whether a counterexample appears at that boundary.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"stated without proof the extended Hardy\\,--\\,Littlewood maximal inequality that Section 2.2 proves in detail"},{"cited_title":"Benedek and R","cited_arxiv_id":null,"evidence_quote":"introduced mixed Lebesgue spaces and provides the H\\'older, duality, and Riesz\\,--\\,Thorin interpolation results used throughout"},{"cited_title":"Fefferman and E","cited_arxiv_id":null,"evidence_quote":"provides the weighted Fefferman\\,--\\,Stein maximal inequality used as the induction step in the proof of the extended maximal inequality"},{"cited_title":"Bownik, Anisotropic Hardy Spaces and W avelets, Mem","cited_arxiv_id":null,"evidence_quote":"supplies the anisotropic Hardy-space estimates and pointwise maximal-function comparisons used in the new proof of Theorem 4.10"},{"cited_title":"Huang, J","cited_arxiv_id":null,"evidence_quote":"established the dual-space identification for $H^{\\vec p}_{\\vec a}$ with mixed-norm Campanato spaces, stated here as Theorem 4.11"}],"review_version":1}