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On Two Weight Estimates for Dyadic Operators

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Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2)

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 5))

Abstract

We provide a quantitative two weight estimate for the dyadic paraproduct π b under certain conditions on a pair of weights (u, v) and b in Carl u, v , a new class of functions that we show coincides with BMO when u = v ∈ A 2 d. We obtain quantitative two weight estimates for the dyadic square function and the martingale transforms under the assumption that the maximal function is bounded from L 2(u) into L 2(v) and v ∈ RH 1 d. Finally we obtain a quantitative two weight estimate from L 2(u) into L 2(v) for the dyadic square function under the assumption that the pair (u, v) is in joint \(\mathcal{A}_{2}^{d}\) and u −1 ∈ RH 1 d, this is sharp in the sense that when u = v the conditions reduce to u ∈ A 2 d and the estimate is the known linear mixed estimate.

In memory of our good friend and mentor Cora Sadosky

The first author was supported by the University of Alabama RGC grant.

The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT & Future Planning(2015R1C1A1A02037331).

The third author was supported by the AVG program, 459895/2013-3, funded by Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq.

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Notes

  1. 1.

    Note that in [54, Corollary 1.4] the statement is not exactly this one. The authors are using a well-known change of variables that we are not using in this paper: their w is our v, their σ is our u −1, their \([w,\sigma ]_{A_{2}^{d}}\) corresponds to our \([\sigma ^{-1},w]_{A_{2}^{d}}\) and hence equals to our \([u,v]_{A_{2}^{d}}\). Finally in their case M(⋅σ) acts on g ∈ L 2 (σ), in our case M acts on f ∈ L 2(u), and clearly g ∈ L 2 (σ) = L 2(u −1) if and only if f = gσ = gu −1 ∈ L 2 (u) with equal norms.

  2. 2.

    The conditions on the function \(\Phi \) are satisfied by the functions \(\Phi (L) = L\log ^{1+\sigma }L\) and Llog Llog log1+σ L (for sufficiently large σ > 0), but not by \(\Phi (L) = L\log L\).

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Authors and Affiliations

  1. Department of Mathematics, University of Alabama, 345 Gordon Palmer Hall, Tuscaloosa, AL, 35487, USA

    Oleksandra Beznosova

  2. Faculty of Basic Sciences, Mathematics, Keimyung University, 1095 Dalgubeol-daero, Daegu, 42601, Korea

    Daewon Chung

  3. Instituto de Matemática, Universidade Federal do Rio Grande do Sul, Av. Bento Goncalves,9500 – Prédio 43-111, Agronomia, Caixa Postal 15080, 91509-900, Porto Alegre, RS, Brazil

    Jean Carlo Moraes

  4. Department of Mathematics and Statistics, University of New Mexico, 311 Terrace NE MSC01 1115, Albuquerque, NM, 87131, USA

    María Cristina Pereyra

Authors
  1. Oleksandra Beznosova
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  2. Daewon Chung
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  4. María Cristina Pereyra
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Correspondence to Oleksandra Beznosova .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico, USA

    María Cristina Pereyra

  2. Department of Mathematics, Venezuelan Institute for Scientific Research, Caracas, Venezuela

    Stefania Marcantognini

  3. Department of Mathematical Sciences, Georgia Southern University, Statesboro, Georgia, USA

    Alexander M. Stokolos

  4. Department of Mathematical and Actuarial Sciences, Roosevelt University Chicago, Chicago, Illinois, USA

    Wilfredo Urbina

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Beznosova, O., Chung, D., Moraes, J.C., Pereyra, M.C. (2017). On Two Weight Estimates for Dyadic Operators. In: Pereyra, M., Marcantognini, S., Stokolos, A., Urbina, W. (eds) Harmonic Analysis, Partial Differential Equations, Banach Spaces, and Operator Theory (Volume 2). Association for Women in Mathematics Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-51593-9_5

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