East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

A NOTE ON TWO WEIGHT INEQUALITIES FOR THE DYADIC PARAPRODUCT

  • Chung, Daewon (Faculty of Basic Sciences, Mathematics, Keimyung University)
  • Received : 2020.04.02
  • Accepted : 2020.05.08
  • Published : 2020.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we provide detailed proof of the Sawyer type characterization of the two weight estimate for the dyadic paraproduct. Although the dyadic paraproduct is known to be a well localized operators and the testing conditions obtained from checking boundedness of the given localized operator on a collection of test functions are provided by many authors. The main purpose of this paper is to present the necessary and sufficient conditions on the weights to ensure boundedness of the dyadic paraproduct directly.

Keywords

References

  1. P. Benge, A two-weight inequality for essentially well localized operators with general measures. J. Math. Anal. Appl, 479 (2019) no. 2, 1506-1518. https://doi.org/10.1016/j.jmaa.2019.07.009
  2. O. Beznosova, Linear bound for the dyadic paraproduct on weighted Lebesgue space $L^2(w)$. J. Func. Anal. 255 (2008), 994-1007. https://doi.org/10.1016/j.jfa.2008.04.025
  3. O. Beznosova, D. Chung, J.C. Moraes, and M.C. Pereyra, On two weight estimates for dyadic operators, Harmonic analysis, partial differential equations, complex analysis, Banach space, and operator theory. Vol 2, Assoc. Women Math. Ser. 5. Springer, Cham, (2017), 135-169.
  4. D. Chung, C. Pereyra, and C. Perez, Sharp bounds for general comutators on weighted Lebesgue spaces. Trans. Amer. Math. Soc. 364 (3) (2012), 1163-1177. https://doi.org/10.1090/S0002-9947-2011-05534-0
  5. T. 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
  6. F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and the two-weight inequalities for Haar Multiplies. J. of the AMS. 12 (1992), 909-928.
  7. E. T. Sawyer, A characterization of a two weight norm inequality for maximal operators. Studia Math 75 (1982), 1-11. https://doi.org/10.4064/sm-75-1-1-11
  8. S. Treil, A Remark on Two Weight Estimates for Positive Dyadic Operators, In: Grochenig K., Lyubarskii Y., Seip K. (eds) Operator-Related Function Theory and Time-Frequency Analysis. Abel Symposia, 9, Springer, Cham, 185-195.