RESEARCH ARTICLE

Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators

  • Dachun YANG , 1,2 ,
  • Dongyong YANG 1,2
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  • 1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China
  • 2. School of Mathematical Sciences, Xiamen University, Xiamen 361005, China

Received date: 23 Jul 2014

Accepted date: 22 Sep 2014

Published date: 24 Jun 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

Let φ be a growth function, and let A:=-(-ia)(-ia)+V be a magnetic Schrödinger operator on L2(n),n2, where α:=(α1,α2,,αn)Lloc2(n,n) and 0VLloc1(n). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space HA,φ(n), defined by the Lusin area function associated with {e-t2A}t>0, in terms of the Lusin area function associated with {e-tA}t>0, the radial maximal functions and the nontangential maximal functions associated with {e-t2A}t>0 and {e-tA}t>0, respectively. The boundedness of the Riesz transforms LkA-1/2,k{1,2,,n}, from HA,φ(n) to Lφ(n) is also presented, where Lk is the closure of xk-iαk in L2(n). These results are new even when φ(x,t):=ω(x)tp for all xnand t ∈(0,+) with p ∈(0, 1] and ωA(n) (the class of Muckenhoupt weights on n).

Cite this article

Dachun YANG , Dongyong YANG . Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schrödinger operators[J]. Frontiers of Mathematics in China, 2015 , 10(5) : 1203 -1232 . DOI: 10.1007/s11464-015-0432-8

1
Auscher P, Duong X T, McIntosh A. Boundedness of Banach space valued singular integral operators and Hardy spaces. Unpublished Manuscript, 2005

2
Bonami A, Grellier S, Ky L D. Paraproducts and products of functions in BMO(Rn) and H1(Rn) through wavelets. J Math Pures Appl (9), 2012, 97: 230-241

3
Bonami A, Iwaniec T, Jones P, Zinsmeister M. On the product of functions in BMO and H1. Ann Inst Fourier (Grenoble), 2007, 57: 1405-1439

DOI

4
Bui T A, Cao J, Ky L D, Yang D C, Yang S B. Weighted Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Taiwanese J Math, 2013, 17: 1127-1166

DOI

5
Bui T A, Cao J, Ky L D, Yang D C, Yang S B. Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off-diagonal estimates. Anal Geom Metr Spaces, 2013, 1: 69-129

DOI

6
Bui T A, Li J. Orlicz-Hardy spaces associated to operators satisfying bounded H functional calculus and Davies-Gaffney estimates. J Math Anal Appl, 2011, 373: 485-501

DOI

7
Cao J, Chang D C, Yang D C, Yang S B. Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems. Trans Amer Math Soc, 2013, 365: 4729-4809

DOI

8
Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569-645

DOI

9
Cruz-Uribe D, Neugebauer C J. The structure of the reverse Hölder classes. Trans Amer Math Soc, 1995, 347: 2941-2960

10
Diening L. Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull Sci Math, 2005, 129: 657-700

DOI

11
Diening L, Hästö P, Roudenko S. Function spaces of variable smoothness and integrability. J Funct Anal, 2009, 256: 1731-1768

DOI

12
Duong X T, Li J. Hardy spaces associated to operators satisfying Davies-Gaffney estimates and bounded holomorphic functional calculus. J Funct Anal, 2013, 264: 1409-1437

DOI

13
Duong X T, Ouhabaz E M, Yan L X. Endpoint estimates for Riesz transforms of magnetic Schrödinger operators. Ark Mat, 2006, 44: 261-275

DOI

14
Duong X T, Xiao J, Yan L X. Old and new Morrey spaces with heat kernel bounds. J Fourier Anal Appl, 2007, 13: 87-111

DOI

15
Duong X T, Yan L X. Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J Amer Math Soc, 2005, 18: 943-973

DOI

16
Duong X T, Yan L X. Commutators of Riesz transforms of magnetic Schrödinger operators. Manuscripta Math, 2008, 127: 219-234

DOI

17
Fefferman C, Stein E M. Hp<?Pub Caret?> spaces of several variables. Acta Math, 1972, 129: 137-193

DOI

18
García-Cuerva J. Weighted Hp spaces. Dissertationes Math (Rozprawy Mat), 1979, 162, (63 pp)

19
Grafakos L. Modern Fourier Analysis. 2nd ed. Grad Texts in Math, Vol 250. New York: Springer, 2009

DOI

20
Hofmann S, Lu G Z, Mitrea D, Mitrea M, Yan L X. Hardy Spaces Associated to Nonnegative Self-adjoint Operators Satisfying Davies-Gaffney Estimates. Mem Amer Math Soc, Vol 214, No 1007. Providence: Amer Math Soc, 2011

21
Hofmann S, Mayboroda S. Hardy and BMO spaces associated to divergence form elliptic operators. Math Ann, 2009, 344: 37-116

DOI

22
Hofmann S, Mayboroda S, McIntosh A. Second order elliptic operators with complex bounded measurable coefficients in Lp, Sobolev and Hardy spaces. Ann Sci Éc Norm Supér (4), 2011, 44: 723-800

23
Hou S X, Yang D C, Yang S B. Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications. Commun Contemp Math, 2013, 15: 1350029, (37 pp)

24
Janson S. Generalizations of Lipschitz spaces and an application to Hardy spaces and bounded mean oscillation. Duke Math J, 1980, 47: 959-982

DOI

25
Jiang R J, Yang D C. Orlicz-Hardy spaces associated with operators satisfying Davies-Gaffney estimates. Commun Contemp Math, 2011, 13: 331-373

DOI

26
Jiang R J, Yang D C, Yang D Y. Maximal function characterizations of Hardy spaces associated with magnetic Schrödinger operators. Forum Math, 2012, 24: 471-494

DOI

27
Johnson R, Neugebauer C J. Homeomorphisms preserving Ap. Rev Mat Iberoam, 1987, 3: 249-273

DOI

28
Ky L D. Bilinear decompositions and commutators of singular integral operators. Trans Amer Math Soc, 2013, 365: 2931-2958

DOI

29
Ky L D. New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations Operator Theory, 2014, 78: 115-150

DOI

30
Lerner A K. Some remarks on the Hardy-Littlewood maximal function on variable Lp spaces. Math Z, 2005, 251: 509-521

DOI

31
Liang Y Y, Huang J Z, Yang D C. New real-variable characterizations of Musielak-Orlicz Hardy spaces. J Math Anal Appl, 2012, 395: 413-428

DOI

32
Musielak J. Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics, Vol 1034. Berlin: Springer-Verlag, 1983

33
Ouhabaz E M. Analysis of Heat Equations on Domains. Princeton: Princeton Univ Press, 2005

34
Rao M, Ren Z. Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, 146. New York: Marcel Dekker, Inc, 1991

35
Simon B. Maximal and minimal Schrödinger forms. J Operator Theory, 1979, 1: 37-47

36
Stein E M. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993

37
Stein E M, Weiss G. On the theory of harmonic functions of several variables. I. The theory of Hp-spaces. Acta Math, 1960, 103: 25-62

DOI

38
Strömberg J O. Bounded mean oscillation with Orlicz norms and duality of Hardy spaces. Indiana Univ Math J, 1979, 28: 511-544

DOI

39
Strömberg J O, Torchinsky A. Weighted Hardy Spaces. Lecture Notes in Mathematics, Vol 1381, Berlin: Springer-Verlag, 1989

40
Yang D C, Yang D Y. Real-variable characterizations of Hardy spaces associated with Bessel operators. Anal Appl (Singap), 2011, 9: 345-368

DOI

41
Yang D C, Yang S B. Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of Rn.Indiana Univ Math J, 2012, 61: 81-129

DOI

42
Yang D C, Yang S B. Musielak-Orlicz-Hardy spaces associated with operators and their applications. J Geom Anal, 2014, 24: 495-570

DOI

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