Two-weight \(L^p\)-\(L^q\) bounds for positive dyadic operators: unified approach to \(p \leq q\) and \(p>q\). (English) Zbl 1357.42017
Let \(\mathcal{D}\) be a collection of dyadic cubes. Among other results, the authors characterized the boundedness of the operator \(T(\cdot \sigma): L^p(\sigma)\rightarrow L^q (\omega)\) defined by
\[
T( f\sigma)=\sum\limits_{Q\in \mathcal{D} }\lambda_Q\displaystyle\int_Q fd\sigma\cdot 1_Q.
\]
They also studied the bilineair case \(T(\cdot \sigma_1, \cdot \sigma_2): L^{p_1}(\sigma_1)\times L^{p_2}(\sigma_2)\rightarrow L^q (\omega)\) defined by
\[
T(f_1 \sigma_1, f_2 \sigma_2)= \sum\limits_{Q\in \mathcal{D} }\lambda_Q\displaystyle\int_Q f_1d\sigma_1\cdot \displaystyle\int_Q f_2d\sigma_2\cdot1_Q.
\]
Reviewer: Yaogan Mensah (Lomé)
Keywords:
two weight norm inequality; positive dyadic operators; discrete Wolff potential; testing conditions; bilinear operators; fractional integralsReferences:
| [1] | Cascante, C., Ortega, J.M., Verbitsky, I.E.: Nonlinear potentials and two weight trace inequalities for general dyadic and radial kernels. Indiana Univ. Math. J 53(3), 845-882 (2004) · Zbl 1077.31007 · doi:10.1512/iumj.2004.53.2443 |
| [2] | Cascante, C., Ortega, J.M., Verbitsky, I.E.: On Lp−Lq trace inequalities. J. Lond. Math. Soc. 74(2), 497-511 (2006) · Zbl 1109.46036 · doi:10.1112/S0024610706023064 |
| [3] | Hänninen, T.S., Hytönen, T.P.: Operator-valued dyadic shifts and the T(1) theorem. Monatshefte für Mathematik (2016). doi:10.1007/s00605-016-0891-3 · Zbl 1345.42013 |
| [4] | Hytönen, T.P.: The A2 theorem: remarks and complements. In: Harmonic Analysis and Partial Differential Equations, Contemporary Mathematics, vol. 612, pp. 91-106. American Mathematical Society, Providence, RI. arXiv:1212.3840 [math.CA] (2014) · Zbl 1354.42022 |
| [5] | Lacey, M.T., Sawyer, E.T., Uriarte-Tuero, I.: Two weight inequalities for discrete positive operators. Preprint (2009). arXiv:0911.3437 [math.CA] · Zbl 1279.42016 |
| [6] | Li, K., Sun, W.: Characterization of a two weight inequality for multilinear fractional maximal operators To appear. Houst. J. Math. (2013). arXiv:1305.4267 [math.CA] · Zbl 0538.47020 |
| [7] | Li, K., Sun, W.: Two weight norm inequalities for the bilinear fractional integrals. Manuscripta Mathematica (2015). doi:10.1007/s00229-015-0800-4 · Zbl 1341.42029 |
| [8] | Nazarov, F., Treil, S., Volberg, A.: The Bellman functions and two-weight inequalities for Haar multipliers. J. Amer. Math. Soc. 12(4), 909-928 (1999). doi:10.1090/S0894-0347-99-00310-0 · Zbl 0951.42007 · doi:10.1090/S0894-0347-99-00310-0 |
| [9] | Sawyer, E.T.: A characterization of a two-weight norm inequality for maximal operators. Stud. Math. 75(1), 1-11 (1982) · Zbl 0508.42023 |
| [10] | Sawyer, E.T.: Weighted Lebesgue and Lorentz norm inequalities for the Hardy operator. Trans. Amer. Math. Soc. 281(1), 329-337 (1984). doi:10.2307/1999537 · Zbl 0538.47020 · doi:10.2307/1999537 |
| [11] | Sawyer, E.T.: A characterization of two weight norm inequalities for fractional and Poisson integrals. Trans. Amer. Math. Soc. 308(2), 533-545 (1988) · Zbl 0665.42023 · doi:10.1090/S0002-9947-1988-0930072-6 |
| [12] | Sawyer, E.T., Wheeden, R.L., Zhao, S.: Weighted norm inequalities for operators of potential type and fractional maximal functions. Potential Anal. 5(6), 523-580 (1996) · Zbl 0873.42012 · doi:10.1007/BF00275794 |
| [13] | Tanaka, H.: A characterization of two-weight trace inequalities for positive dyadic operators in the upper triangle case. Potential Anal. 41(2), 487-499 (2014). doi:10.1007/s11118-013-9379-0 · Zbl 1298.42020 · doi:10.1007/s11118-013-9379-0 |
| [14] | Tanaka, H.: The n linear embedding theorem. Potential Analysis (2015). doi:10.1007/s11118-015-9531-0. To appear · Zbl 0873.42012 |
| [15] | Tanaka, H.: The trilinear embedding theorem. Stud. Math. 2857, 239-248 (2015). doi:10.4064/sm227-3-3 · Zbl 1333.42045 · doi:10.4064/sm227-3-3 |
| [16] | Verbitsky, I.E.: Weighted norm inequalities for maximal operators and Pisier’s theorem on factorization through Lp∞. Integr. Equ. Oper. Theory 15(1), 124-153 (1992). doi:10.1007/BF01193770 · Zbl 0782.47027 · doi:10.1007/BF01193770 |
| [17] | Vuorinen, E.: Lp(μ)Lq(ν) characterization for well localized operators. Preprint (2014). arXiv:1412.2127 [math.CA] · Zbl 1077.31007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
