Sharp weighted bounds for the Hardy–Littlewood maximal operators on Musielak–Orlicz spaces

详细信息    查看全文

• 作者：Haibo Lin
• 关键词：Weighted bound ; Hardy–Littlewood maximal operator ; Musielak ; Orlicz space
• 刊名：Archiv der Mathematik
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：106
• 期：3
• 页码：275-284
• 全文大小：493 KB
• 参考文献：1.Birnbaum Z., Orlicz W.: Über die verallgemeinerung des begriffes der zueinander konjugierten potenzen. Studia Math. 3, 1–67 (1931)
  2.Buckley S.M.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Amer. Math. Soc. 340, 253–272 (1993)CrossRef MathSciNet MATH
  3.J. Cao, D. Chang, D. Yang, and S. Yang, Riesz Transform Characterizations of Musielak-Orlicz-Hardy Spaces, Trans. Amer. Math. Soc. arXiv:​1401.​7373v2 (accepted).
  4.Diening L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces, Bull. Sci. Math. 129, 657–700 (2005)MathSciNet MATH
  5.J. Duoandikoetxea, Fourier Analysis, American Mathematical Society, Providence, 2001.
  6.J. García-Cuerva, Weighted H p spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 1–63.
  7.J. García-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.
  8.L. Grafakos, Modern Fourier Analysis, second ed., in: Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009.
  9.S. Hou, D. Yang, and S. Yang, Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications, Commun. Contemp. Math. 15 (2013), no. 6, 1350029, 37 pp.
  10.T. P. Hytönen, The sharp weighted bound for general Calderón-Zygmund operators, Ann. of Math. (2) 175 (2012), 1473–1506.
  11.T. P. Hytönen and C. Pérez, Sharp weighted bounds involving \({A_{\infty}}\) , Anal. PDE 6 (2013), 777–818.
  12.T. P. Hytönen, C. Pérez, and E. Rela, Sharp reverse Hölder property for \({A_{\infty}}\) weights on spaces of homogeneous type, J. Funct. Anal. 263 (2012), 3883–3899.
  13.S. Janson, Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation, Duke Math. J. 47 (1980), 959–982.
  14.L. D. Ky, New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators, Integral Equations Operator Theory 78 (2014), 115–150.
  15.Lerner A.K.: Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals. Adv. Math. 226, 3912–3926 (2011)CrossRef MathSciNet MATH
  16.A. K. Lerner, A simple proof of the A 2 conjecture, Int. Math. Res. Not. 14 (2013), 3159–3170.
  17.Y. Liang, J. Huang, and D. Yang, New real-variable characterizations of Hardy spaces of Musielak-Orlicz type, J. Math. Anal. Appl. 395 (2012), 413–428.
  18.Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226.CrossRef MathSciNet MATH
  19.J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag, Berlin, 1983, iii+222 pp.
  20.Orlicz W.: Über eine gewisse Klasse von Räumen vom Typus B. Bull. Intern. Acad. Polonaise. Ser. A 8, 207–220 (1932)
  21.Strömberg J.-O.: Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ. Math. J. 28, 511–544 (1979)CrossRef MathSciNet
  22.J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math. 1381, Springer-Verlag, Berlin, 1989, vi+193 pp.
  23.J. M. Wilson, Weighted inequalities for the dyadic square function without dyadic \({A_{\infty}}\) , Duke Math. J. 55 (1987), 19–50.
  24.Wilson J.M.: Weighted norm inequalities for the continuous square function. Trans. Amer. Math. Soc. 314, 661–692 (1989)CrossRef MathSciNet MATH
  25.J. M. Wilson, Weighted Littlewood-Paley theory and exponential-square integrability, Lecture Notes in Mathematics 1924, Springer, Berlin, 2008.
  
• 作者单位：Haibo Lin (1)
  
  1. College of Science, China Agricultural University, Beijing, 100083, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Birkh盲user Basel
• ISSN：1420-8938

文摘

Let \({\varphi}\) be a Musielak–Orlicz function satisfying that, for any \({(x,\,t)\in{\mathbb R}^n \times [0, \infty)}\), \({\varphi(\cdot,\,t)}\) belongs to the Muckenhoupt weight class \({A_\infty({\mathbb R}^n)}\) with the critical weight exponent \({q(\varphi) \in [1,\,\infty)}\) and \({\varphi(x,\,\cdot)}\) is an Orlicz function with uniformly lower type \({p^{-}_{\varphi}}\) and uniformly upper type \({p^+_\varphi}\) satisfying \({q(\varphi) < p^{-}_{\varphi}\le p^{+}_{\varphi} < \infty}\). In this paper, the author obtains a sharp weighted bound involving \({A_\infty}\) constant for the Hardy–Littlewood maximal operator on the Musielak–Orlicz space \({L^{\varphi}}\). This result recovers the known sharp weighted estimate established by Hytönen et al. in [J. Funct. Anal. 263:3883–3899, 2012].