Browse > Article
http://dx.doi.org/10.4134/BKMS.b200202

WEIGHTED NORM ESTIMATES FOR THE DYADIC PARAPRODUCT WITH VMO FUNCTION  

Chung, Daewon (Major in Mathematics Faculty of Basic Sciences Keimyung University)
Publication Information
Bulletin of the Korean Mathematical Society / v.58, no.1, 2021 , pp. 205-215 More about this Journal
Abstract
In [1], Beznosova proved that the bound on the norm of the dyadic paraproduct with b ∈ BMO in the weighted Lebesgue space L2(w) depends linearly on the Ad2 characteristic of the weight w and extrapolated the result to the Lp(w) case. In this paper, we provide the weighted norm estimates of the dyadic paraproduct πb with b ∈ VMO and reduce the dependence of the Ad2 characteristic to 1/2 by using the property that for b ∈ VMO its mean oscillations are vanishing in certain cases. Using this result we also reduce the quadratic bound for the commutators of the Calderón-Zygmund operator [b, T] to 3/2.
Keywords
Weighted norm estimate; dyadic paraproduct; $A_2$-weights; Carleson sequenc;
Citations & Related Records
연도 인용수 순위
  • Reference
1 O. V. Beznosova, Linear bound for the dyadic paraproduct on weighted Lebesgue space L2(w), J. Funct. Anal. 255 (2008), no. 4, 994-1007. https://doi.org/10.1016/j.jfa.2008.04.025   DOI
2 S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253-272. https://doi.org/10.2307/2154555   DOI
3 D. Chung, Sharp estimates for the commutators of the Hilbert, Riesz transforms and the Beurling-Ahlfors operator on weighted Lebesgue spaces, Indiana Univ. Math. J. 60 (2011), no. 5, 1543-1588. https://doi.org/10.1512/iumj.2011.60.4453   DOI
4 D. Chung, M. C. Pereyra, and C. Perez, Sharp bounds for general commutators on weighted Lebesgue spaces, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1163-1177. https://doi.org/10.1090/S0002-9947-2011-05534-0   DOI
5 R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. https://doi.org/10.4064/sm-51-3-241-250   DOI
6 R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569-645. https://doi.org/10.1090/S0002-9904-1977-14325-5   DOI
7 O. Dragicevic, L. Grafakos, M. C. Pereyra, and S. Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005), no. 1, 73-91. https://doi.org/10.5565/PUBLMAT_49105_03   DOI
8 R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. https://doi.org/10.2307/1996205   DOI
9 T. P. Hytonen, The sharp weighted bound for general Calderon-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473-1506. https://doi.org/10.4007/annals.2012.175.3.9   DOI
10 J. Li, J. Pipher, and L. A. Ward, Dyadic structure theorems for multiparameter function spaces, Rev. Mat. Iberoam. 31 (2015), no. 3, 767-797. https://doi.org/10.4171/RMI/853   DOI
11 J. C. P. Moraes, Weighted estimates for dyadic operators with complexity, ProQuest LLC, Ann Arbor, MI, 2011.
12 B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. https://doi.org/10.2307/1995882   DOI
13 F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909-928. https://doi.org/10.1090/S0894-0347-99-00310-0   DOI
14 D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405. https://doi.org/10.2307/1997184   DOI
15 F. Nazarov, A. Volberg, Bellman function, polynomial estimates of weighted dyadic shifts, and A2 conjecture, Preprint, 2011.
16 M. C. Pereyra, Dyadic harmornic analysis and weighted inequalities: the sparse revolution, New Trends in Applied Harmonic Analysis (Vol 2), A. Aldroubi et al. (eds.) (Springer 2019), 159-239.
17 S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical Ap characteristic, Amer. J. Math. 129 (2007), no. 5, 1355-1375. https://doi.org/10.1353/ajm.2007.0036   DOI