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Necessary/sufficient conditions in weighted theory

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Pith's one-line read For A_infinity weights, the two-weight A_p condition implies the Pivotal condition.

desk verdict A mostly-sound dictionary paper with new counterexamples and a clean reduction of Ap to Pivotal under A-infinity, but the proof of the headline NTV application is too terse and one construction hinges on an unverified imported lemma. read the letter →

arxiv 2009.12091 v1 pith:VWUNIDIQ submitted 2020-09-25 math.CA

Christos Grigoriadis This is my paper · ORCID
classification math.CA MSC 42B2042B25
keywords weightedinequalitiesApconditionCpPivotalEnergyA_infinityweightsNTVconjecturetwo-weighttheory
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes an essentially complete dictionary of implications among the standard weight conditions: doubling, one-weight A_p, A_infinity, C_p, two-weight A_p, and the buffer Energy and Pivotal conditions. Its most notable result is that when the measures are A_infinity weights, the classical two-weight A_p condition forces the Pivotal condition, so T1 testing suffices to prove the NTV conjecture for p=2 in the A_infinity case. The paper also constructs a doubling C_p weight that is not A_infinity, showing the separation is strict, and proves the sharp threshold on the doubling constant below which C_p forces A_infinity. This matters because it clarifies exactly when the buffer Pivotal condition is automatic and when it must be imposed.

What carries the argument

The Pivotal condition V(omega,$\sigma$)_p is the central object: it measures, over all cube decompositions of I0, the weighted sum of omega(I_r) times Poisson-tail factors P(I_r, 1_{I_0} $\sigma$)^p; it is the buffer estimate needed for T1 arguments. The key mechanism is the reduction from A_p to dyadic testing for the maximal operator when $\sigma$ is A_infinity, which then controls Pivotal sums through the maximal function. The other load-bearing structure is the triadic construction of [6], modified with finite stopping heights and parent ratios delta_1, delta_2; its worst-interval property ensures the C_p density estimate with an exponential gain 2^k, while the overall maximal-function integral grows by the same factor, producing exactly the balance needed for C_p.

What would settle it

Directly compute the ratios in (4.7) for the modified triadic weight at a finite stage: if the worst interval fails w(E_{n_k})/w(J_{n_k}) about 1/2 or the uniform estimate w(E)/w(I) less than a constant times 2^k |E|/|I|, then the C_p conclusion collapses and the strict inclusion would need another construction.

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Extended reading notes

Core claim

The central discovery is the implication A_p(omega,$\sigma$) plus $\sigma$ in A_infinity implies the Pivotal condition V(omega,$\sigma$)_p, and consequently the dual contributes a short proof of the NTV conjecture for A_infinity weights with T1 testing assumptions. The proof derives a dyadic testing bound for the maximal operator from A_p and then uses that bound to control Pivotal sums, removing the need for an explicit energy or pivotal side condition. In the one-weight setting, the paper separates C_p from A_infinity by building a triadic weight with doubling constant about $3^{{np}}$ that satisfies C_p but fails the A_infinity density ratio; conversely, any doubling C_p weight with constant below $3^{{np}}$ is A_infinity, so the construction is sharp. The paper also maps the tailed A_p conditions: for doubling measures the classical, one-tailed, and two-tailed Ap conditions are equivalent, while for non-doubling measures the strict hierarchy holds.

Load-bearing premise

The strict-separation construction relies on the imported lemma that the finite triadic weight has a worst interval J_{n_k} with w(E_{n_k})/w(J_{n_k}) close to 1/2 and |E_{n_k}|/|J_{n_k}| close to $2^{{-k}}$, uniformly over all subintervals.

Editorial extensions

If this is right

  • For A_infinity weights, the two-weight A_p condition implies the Pivotal condition and its dual, so T1 testing alone suffices for boundedness of alpha-fractional singular integrals.
  • The strict separation of C_p from A_infinity is sharp: any doubling C_p weight with doubling constant below 3^{np} is automatically A_infinity.
  • For doubling measures, the classical two-weight A_p condition is equivalent to the tailed A_p conditions, so the usual hierarchy collapses for doubling weights.
  • The Pivotal and Energy conditions do not imply the tailed A_p conditions for 1<p<=2, so the buffer conditions are genuinely weaker in that direction.
  • The fractional analogues of these implications hold without modification, extending the dictionary to alpha-fractional singular integrals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The sharp threshold 3^{np} suggests that quantitative A_infinity-type density estimates should hold for doubling C_p weights with constant strictly below the threshold, controlled explicitly by the doubling and Cp constants.
  • The transfer of the proofs to fractional conditions implies the same lattice of implications holds for fractional A_alpha^p and Pivotal conditions, which could simplify future two-weight theorems for fractional operators.
  • The equivalence between the two-tailed Ap and the conjunction of both one-tailed conditions likely extends to offset or punctured A2 variants for doubling measures, though the non-doubling examples show limits.
  • The constructed C_p counterexample may be adaptable to higher dimensions by taking products with the Lebesgue measure, potentially giving higher-dimensional examples of doubling Cp weights outside A_infinity.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper studies the lattice of one- and two-weight conditions in weighted theory. Its main positive claims are: Theorem 1 constructs a doubling weight satisfying the C_p condition but not A_infty; Theorem 2 shows that a doubling C_p weight with sufficiently small doubling constant is A_infty; Theorem 4 shows that for reverse doubling measures the classical two-weight A_p condition implies the tailed A_p conditions; Corollary 2 states that for sigma in A_infty, the two-weight A_p condition implies the p-Pivotal condition; Theorem 5 asserts that this gives a short proof of the NTV conjecture for p=2 for A_infty weights by invoking existing T1 theorems; and Theorem 6 gives a counterexample showing that the Pivotal condition does not imply the one-tailed A_p condition. The paper also contains several remarks and a claimed dictionary of implications summarized in diagrams in Sections 2 and 3.

Significance. If the claims are correct, the most significant contribution is the short route from A_infty membership plus two-weight A_p to boundedness, through the Pivotal condition, stated as Theorem 5. The construction in Theorem 1 is nontrivial and addresses a natural question about the C_p class. The proofs of Theorems 2, 8 and 9, and the self-contained counterexample in Theorem 6, are clear and credible. However, as written the headline boundedness theorem is conditional on an unexamined external T1 theorem, and the construction in Theorem 1 relies on an imported lemma whose applicability is not demonstrated through the required limits. These issues are central and require substantial revision.

major comments (5)
  1. [Section 5.3, Theorem 5] The proof of Theorem 5 is a single sentence: 'apply the main theorem from [31] (or the one in [16])'. The hypotheses of those theorems are never stated, so the reader cannot verify that the conditions listed in Theorem 5, together with Corollary 2 and Remark 1.1, actually imply those hypotheses. In particular, [31] is described as a theorem 'with an energy side condition', but the paper never states the exact form of that energy condition or proves that the Pivotal condition obtained from Corollary 2 implies it. Remark 5.6 asserts the fractional analogue of Corollary 2 without proof. Because Theorem 5 is the central boundedness claim of the paper, this gap is load-bearing.
  2. [Section 4.3, Theorem 1, around (4.7)] The C_p verification for the constructed weight depends on the estimates (4.7), which are asserted by reference to [6] for the finite-stage construction. The proof then passes to limits in i_k and k, but the limiting weight is never defined explicitly, and the constants in (4.12) through (4.15) are not shown to persist through both limiting processes. If the imported Garnett-Killip-Schul lemma does not apply to the modified construction, the estimates establishing the C_p condition collapse. This needs a complete limiting argument.
  3. [Section 5.1, Theorem 3(3)] The proof of the equivalence A_t^1_p ∩ A_t^{1,*}_p ⇔ A_t^2_p is not a complete proof as written. It asserts without proof that there exist k1,k2 with P(I,sigma) approximately twice a truncated series, and then displays a chain of 'approximately equal to' and 'greater than or similar to' steps with uncontrolled constants. Since this equivalence is a central entry in the claimed dictionary of implications, the argument needs to be made rigorous.
  4. [Section 5.3, Theorem 8 and Corollary 2] The proof of Theorem 8 uses the A_infty decay property to obtain sigma(A_t^m) ≤ (1/2) sigma(I_t^m), but the subsequent summation step 'Fix m in N, k > -m ... by taking m to infinity' is not clearly justified: the levels are not defined for negative indices, and the passage to the full double sum is written in a way that does not display the dependence on the maximal cubes at each level. Additionally, the estimate (5.4) in Corollary 2 controls P(I,sigma) by inf_{x in I} M_d sigma(x) using intervals (2^m+1)I, which are not dyadic relatives of I; a dyadic-grid argument is needed here, since Theorem 8 is stated for the dyadic maximal operator.
  5. [Section 5.3, Remark 5.6] Remark 5.6 states that the proof of Corollary 2 extends to the fractional A_p and Pivotal conditions 'as stated in [31]' without proof. Since Theorem 5 explicitly uses fractional conditions, this remark is load-bearing for the fractional version of the boundedness claim and cannot be left as an unproved assertion.
minor comments (5)
  1. [Section 5.1, proof of Theorem 3(1)] The notation v^k_{-100k} is undefined, since v_n^k was defined only for nonnegative n; presumably v^{100k}_{-100k} is intended.
  2. [Section 5.1, definitions (1.10)-(1.12)] The symbol P_alpha is used both for the reproducing Poisson integral in (1.11)-(1.12) and for the 'standard Poisson integral' introduced immediately afterwards; this should be disambiguated.
  3. [Theorem 5] The class of alpha-fractional singular integrals is not defined in the paper; the reader is left to guess the exact operator class from [31].
  4. [Section 4.3, parameter choices] After choosing delta_2 = 3^{-p}/2, the paper does not explicitly verify that this is compatible with the earlier condition 1/3 > delta_2 > 0; this is true but should be stated.
  5. [Section 2, lattices] The diagram summaries in Section 2 include entries such as A_alpha_infty ∩ D and C_p, but several of these inclusions are asserted without references or proofs in the body of the paper; adding pointers to the relevant theorems or remarks would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the headline implication is proved from a self-contained maximal-function argument, and the boundedness step is delegated to independent prior T1 theorems.

full rationale

I walked the claimed derivation chain. The headline implication (Corollary 2: A_infty plus two-weight A_p implies the p-Pivotal condition) is proved through Theorem 8, whose proof is self-contained: it uses the two-weight A_p condition on maximal dyadic cubes and the A_infty density estimate (4.2); neither of these inputs is equivalent to the pivotal conclusion. The pivotal condition is then bounded by the dyadic Sawyer testing condition using the pointwise estimate P(I,sigma) <= inf M_d sigma in (5.4) and summing over the decomposition, with no fitted parameter relabeled as a prediction. The boundedness step in Theorem 5 invokes the prior T1 theorems [31] and [16]; these are external prior works by other authors, not self-citations of the present paper, and no step reduces by construction to an input of this paper. Even if the paper does not spell out all hypotheses needed to apply [31], such as the energy side condition, that is an unverified-application/correctness risk rather than a circular reduction. Remark 5.6 asserts a fractional analogue of Corollary 2 without proof, but an omitted proof is not circularity. Theorem 1's construction imports the Garnett-Killip-Schul estimate (4.7) from [6]; this is an independent external lemma, and the paper then provides its own estimates (4.8)-(4.15) to verify the C_p condition. There are citations to the author's advisors, but no load-bearing step reduces to a bare self-citation chain, and no uniqueness theorem is imported from the authors' own prior work. Therefore the paper's central claims have independent content and no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical or mathematical entities. Its free parameters are construction constants in the counterexample proofs, chosen to satisfy inequalities. The main results depend on standard A_infty machinery and on substantial external theorems whose hypotheses are not fully restated.

free parameters (3)
  • delta_1 = chosen in (3^{-p}, 1/3)
    Determines mass decay and growth on the right thirds of the nested triadic intervals in Theorem 1. The C_p estimate (4.10) requires delta_1 > 3^{-p} for convergence of the outer tail series, while doubling requires delta_1 < 1/3.
  • delta_2 = chosen <= 3^{-p}, e.g. 3^{-p}/2
    Determines mass distribution inside the interval I^l_{n_k}. The inner-series divergence in (4.9) requires the ratio 3^{-p}/delta_2 > 1, i.e. delta_2 < 3^{-p}.
  • n_k and i_k = large integers chosen in the proof
    The sequences n_k and i_k are chosen so that each relevant interval gains a factor 2^k in the maximal integral, which is needed for the C_p estimates (4.12) and (4.14).
assumptions (4)
  • standard math A_infty weights are doubling and satisfy the quantitative A_infty property w(E)/w(I) <= C(|E|/|I|)^epsilon
    Used throughout Theorem 8 and Corollary 2 as a standard characterization, cited to Duoandikoetxea.
  • domain assumption Garnett-Killip-Schul lemma: the finite triadic construction admits a worst interval with controlled measure and size ratios that persists in the limit
    The C_p verification in Theorem 1 relies on this external lemma at (4.7). The paper does not reprove it for the modified construction.
  • domain assumption The main theorem of Sawyer-Shen-Uriarte-Tuero [31] (and Lacey-Wick [16]) holds under the hypotheses of Theorem 5 and yields boundedness once the pivotal or energy condition is known
    Theorem 5 is proved entirely by reduction to this external theorem. Its hypotheses are not stated in the paper, so the application cannot be checked from the text alone.
  • standard math For doubling measures, the Energy condition E(I,omega)^2 has a positive lower bound, making the Pivotal condition equivalent to the Energy condition
    Invoked in Remark 5.9 to connect Pivotal and Energy for doubling measures. The statement is plausible but not proved in this paper.

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Pith. "Pith review of Necessary/sufficient conditions in weighted theory." pith.science (2026). https://pith.science/paper/VWUNIDIQ

@misc{pith2026200912091,
  author       = {Pith},
  title        = {Pith review of: Necessary/sufficient conditions in weighted theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VWUNIDIQ}},
  note         = {Machine review of arXiv:2009.12091}
}
read the original abstract

We provide an essentially complete dictionary of all implications among the basic and fundamental conditions in weighted theory such as the doubling, one weight A_p(w), A_\infty and C_p conditions as well as the two weight A_p and the "buffer" Energy and Pivotal conditions. The most notable implication is that in the case of A_\infty weights the two weight A_p condition implies the p-Pivotal condition hence giving an elegant and short proof of the known NTV-conjecture with p=2 for A_\infty weights in terms of existing T1 theory. We also provide a quite technical construction inspired by [6] proving that we can have doubling weights satisfying the C_p condition which are not in A_\infty.

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Works this paper leans on

39 extracted references · 39 canonical work pages

  1. [31]

    Sawyer, Shen Chun-Yen, Uriarte-Tuero Ignacio: A two weight theorem for α-fractional singular integrals with an energy side condition

    E. Sawyer, Shen Chun-Yen, Uriarte-Tuero Ignacio: A two weight theorem for α-fractional singular integrals with an energy side condition. Rev. Mat. Iberoam. 32 (2016), 79-174. doi: 10.4171/RMI/882

    work page doi:10.4171/rmi/882 2016

  2. [16]

    Two Weight Inequalities for Riesz Transforms: Uniformly Full Dimension Weights

    M. Lacey, B. Wick (2013), Two Weight Inequalities for Riesz Transforms: Uniformly Full Dimension Weights, arXiv e-prints , arXiv:1312.6163

    work page Pith review arXiv 2013

  3. [6]

    Gustafson, A generalized Poincar´ e inequality for a class of constant coefficient differential operators, Proc

    J. Garnett, R. Killip, R. Schul (2009). A doubling measure on Rd can charge a rectifi- able curve. Proceedings of the American Mathematical Society. 138. 10.1090/S0002-9939- 10-10234-2

    work page doi:10.1090/s0002-9939- 2009

  4. [1]

    D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften, volume 314, Springer 1999

    work page 1999

  5. [2]

    Coifman, C

    R. Coifman, C. Fefferman Studia Mathematica 51 (1974), 241-250 DOI: 10.4064/sm-51-3- 241-250

    work page doi:10.4064/sm-51-3- 1974

  6. [3]

    Duoandikoetxea: Fourier Analysis

    J. Duoandikoetxea: Fourier Analysis. - Grad. Stud. Math. 29, Amer. Math. Soc., Providence, 2000

    work page 2000

  7. [4]

    Fefferman, B

    C. Fefferman, B. Muckenhoupt (1974). Two Nonequivalent Conditions for Weight Functions. Proceedings of the American Mathematical Society, 45(1), 99-104. doi:10.2307/2040615

    work page doi:10.2307/2040615 1974

  8. [5]

    Garnett, John. (2007). Bounded analytic functions. Revised 1st ed. 10.1007/0-387-49763-3

    work page doi:10.1007/0-387-49763-3 2007

Show all 39 references
  1. [7]

    R. Hunt, B. Muckenhoupt, R. Wheeden, Transactions of the American Mathematical Society Vol. 176 (Feb., 1973), pp. 227-251. 24 GRIGORIADIS

    1973

  2. [8]

    Hyt¨ onen, The two-weight inequality for the Hilbert transform with general measures

    T. Hyt¨ onen, The two-weight inequality for the Hilbert transform with general measures. Proceedings of the London Mathematical Society (2013). 10.1112/plms.12136

    2013 doi

  3. [9]

    Iwaniec and G

    T. Iwaniec and G. Martin, Quasiconformal mappings and capacity, Indiana Univ. Math. J. 40 (1991), no. 1, 101122

    1991

  4. [10]

    Kakaroumpas, A note on a two-weight estimate for the dyadic square function, Proc

    S. Kakaroumpas, A note on a two-weight estimate for the dyadic square function, Proc. Amer. Math. Soc. 147 (2019), 3377-3388

    2019

  5. [11]

    Lacey, Two weight inequality for the Hilbert transform: A real variable characterization, II, Duke Math

    M. Lacey, Two weight inequality for the Hilbert transform: A real variable characterization, II, Duke Math. J. Volume 163, Number 15 (2014), 2821

    2014

  6. [12]

    Lacey, E

    M. Lacey, E. Sawyer, I. Uriarte-Tuero, A two weight inequality for the Hilbert transform assuming an Energy Hypothesis, Journal of Functional Analysis, Volume 263, Issue 2, 2012, Pages 305-363

    2012

  7. [13]

    204 (2010), 273-292

    Lacey, Michael T., Sawyer, Eric T., Uriarte-Tuero, Ignacio, Astala’s conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane, Acta Math. 204 (2010), 273-292

    2010

  8. [14]

    Lacey, E

    M. Lacey, E. Sawyer.,C. Shen, I. Uriarte-Tuero, Two weight inequality for the Hilbert trans- form: A real variable characterization I, Duke Math. J, Volume 163, Number 15 (2014), 2795

    2014

  9. [15]

    Lacey, E

    M. Lacey, E. Sawyer,C. Shen,I. Uriarte-Tuero, B. Wick (2013) Two Weight Inequalities for the Cauchy Transform from R to C+, arXiv e-prints arXiv:1310.4820

    2013 arXiv

  10. [17]

    (1981) Norm Inequalities Relating the Hilbert Transform to the Hardy- Littlewood Maximal Function

    Muckenhoupt B. (1981) Norm Inequalities Relating the Hilbert Transform to the Hardy- Littlewood Maximal Function. In: Butzer P.L., Sz.-Nagy B., G¨ orlich E. (eds) Functional Analysis and Approximation. ISNM 60: International Series of Numerical Mathematics

    1981

  11. [18]

    Nazarov, A counterexample to Sarason’s conjecture, preprint (1997)

    F. Nazarov, A counterexample to Sarason’s conjecture, preprint (1997)

    1997

  12. [19]

    Nazarov and A

    F. Nazarov and A. Volberg, The Bellman function, the two weight Hilbert transform, and the embeddings of the model space K, J. dAnalyse Math. 87 (2002), 385-414

    2002

  13. [20]

    Nazarov, S

    F. Nazarov, S. Treil, A. Volberg, The Tb -theorem on non-homogeneous spaces. Acta Math. 190 (2003), no. 2, 151–239

    2003

  14. [21]

    Nazarov, S

    F. Nazarov, S. Treil, Volberg, A., Two weight inequalities for individual Haar multipliers and other well localized operators, Math. Res. Lett. 15 (3) (2008) 583-597

    2008

  15. [22]

    Nikolski and S

    N. Nikolski and S. Treil, Linear resolvent growth of rank one perturbation of a unitary operator does not imply its similarity to a normal operator, J. Anal. Math. 87 (2002), 415–431. MR1945291

    2002

  16. [23]

    Peherstorfer, A

    F. Peherstorfer, A. Volberg and P. Yuditskii, CMV matrices with asymptotically constant coefficients, Szeg¨ o-Blaschke class, Scattering Theory, J. Funct. Anal. 256 (2009), no. 7, 21572210

    2009

  17. [24]

    Peherstorfer, A

    F. Peherstorfer, A. Volberg and P. Yuditskii, Two-weight Hilbert transform and Lipschitz property of Jacobi matrices associated to Hyperbolic polynomials, J. Funct. Anal. 246 (2007), 130

    2007

  18. [25]

    Reguera, J

    M. Reguera, J. Scurry, On joint estimates for maximal functions and singular integrals on weighted spaces, Proc. Amer. Math. Soc. 141 (2013), 1705-1717

    2013

  19. [26]

    Ruzhansky, On uniform properties of doubling measures, Proc

    M. Ruzhansky, On uniform properties of doubling measures, Proc. Amer. Math. Soc. 129 (2001), 3413–3416

    2001

  20. [27]

    Sawyer, C

    E. Sawyer, C. Shen, I. Uriarte-Tuero, Failure of necessity of the energy condition arXiv e-prints 2016 Jul arXiv:1607.06071

    2016 arXiv

  21. [28]

    Sawyer, C

    E. Sawyer, C. Shen, I. Uriarte-Tuero, A Note on Failure of Energy Reversal for Classical Fractional Singular Integrals, International Mathematics Research Notices, Volume 2015, Issue 19, 2015, Pages 9888–9920, https://doi.org/10.1093/imrn/rnu210

    2015 doi

  22. [29]

    Sawyer, C

    E. Sawyer, C. Shen, I. Uriarte-Tuero, The two weight T1 theorem for fractional Riesz trans- forms when one measure is supported on a curve, 2015, arXiv e-prints , arXiv:1505.07822

    2015 arXiv

  23. [30]

    Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero

    E. Sawyer, Chun-Yen Shen, Ignacio Uriarte-Tuero. A two weight fractional singular integral theorem with side conditions, energy and k-energy dispersed. In Harmonic analysis, partial NECESSARY/SUFFICIENT CONDITIONS IN WEIGHTED THEORY 25 differential equations, Banach spaces, and...

    2017

  24. [32]

    Sawyer, Shen Chun-Yen, Uriarte-Tuero Ignacio (2016)

    E. Sawyer, Shen Chun-Yen, Uriarte-Tuero Ignacio (2016). A Good-Lambda Lemma, two weight T1 theorems without weak boundedness, and a two weight accretive global Tb theo- rem

    2016

  25. [33]

    Sawyer, Energy conditions and twisted localizations of operators, 2018, arXiv:1801.03706

    E. Sawyer, Energy conditions and twisted localizations of operators, 2018, arXiv:1801.03706

    2018 arXiv

  26. [34]

    Sawyer 2019

    E. Sawyer 2019. A T1 theorem for general Calder´ on-Zygmund operators with comparable doubling weights, and optimal cancellation conditions. arXiv e-prints arXiv:1906.05602

    2019 arXiv

  27. [35]

    Sawyer ”Norm inequalities relating singular integrals and the maximal function.” Studia Mathematica 75.3 (1983): 253-263

    E. Sawyer ”Norm inequalities relating singular integrals and the maximal function.” Studia Mathematica 75.3 (1983): 253-263

    1983

  28. [36]

    Sawyer ”A characterization of a two-weight norm inequality for maximal operators.” Studia Mathematica 75.1 (1982): 1-11

    E. Sawyer ”A characterization of a two-weight norm inequality for maximal operators.” Studia Mathematica 75.1 (1982): 1-11

    1982

  29. [37]

    Princeton N.J.: Princeton University Press 1970

    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton N.J.: Princeton University Press 1970

    1970

  30. [38]

    Volberg, Calder´ on-Zygmund Capacities and Operators on Nonhomogeneous Spaces 2003, CBMS Regional Conference Series in Mathematics, vol

    A. Volberg, Calder´ on-Zygmund Capacities and Operators on Nonhomogeneous Spaces 2003, CBMS Regional Conference Series in Mathematics, vol. 100

    2003

  31. [39]

    Volberg and P

    A. Volberg and P. Yuditskii, On the inverse scattering problem for Jacobi matrices with the spectrum on an interval, a nite system of intervals or a Cantor set of positive length, Comm. Math. Phys. 226 (2002), 567605. Department of Mathematics, Michigan State University, East ...

    2002

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