Science China Mathematics

, Volume 57, Issue 5, pp 903–962

# Orlicz-Hardy spaces and their duals

Articles

## Abstract

We establish the theory of Orlicz-Hardy spaces generated by a wide class of functions. The class will be wider than the class of all the N-functions. In particular, we consider the non-smooth atomic decomposition. The relation between Orlicz-Hardy spaces and their duals is also studied. As an application, duality of Hardy spaces with variable exponents is revisited. This work is different from earlier works about Orlicz-Hardy spaces HΦ(ℝn) in that the class of admissible functions ϕ is largely widened. We can deal with, for example,
$$\Phi (r) \equiv \left\{ \begin{gathered} r^{p1} \left( {\log \left( {e + {1 \mathord{\left/ {\vphantom {1 r}} \right. \kern-\nulldelimiterspace} r}} \right)} \right)^{q1} , 0 < r \leqslant 1, \hfill \\ r^{p2} \left( {\log \left( {e + r} \right)} \right)^{q2} , r > 1, \hfill \\ \end{gathered} \right.$$
with p1, p2 ∈ (0,∞) and q1, q2 ∈ (−∞,∞), where we shall establish the boundedness of the Riesz transforms on HΦ(ℝn). In particular, ϕ is neither convex nor concave when 0 < p1 < 1 < p2 < ∞, 0 < p2 < 1 < p1 < ∞ or p1 = p2 = 1 and q1, q2 > 0. If Φ(r) ≡ r(log(e+r))q, then HΦ(ℝn) = H(log H)q(ℝn). We shall also establish the boundedness of the fractional integral operators Iα of order α ∈ (0,∞). For example, Iα is shown to be bounded from H(log H)1 − α/n (ℝn) to Ln/(nα)(log L)(ℝn) for 0 < α < n.

### Keywords

Hardy space Orlicz space atomic decomposition Campanato space bounded mean oscillation

### MSC(2010)

42B30 46E30 42B35

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### References

1. 1.
Anh B T, J. Li, Orlicz-Hardy spaces associated to operators satisfying bounded H -functional calculus and Davies-Gaffney estimates. J Math Anal Appl, 2011, 373: 485–501
2. 2.
Antonov N Y. Convergence of Fourier series. East J Approx, 1996, 2: 187–196
3. 3.
Bergh J, Jörgen L. Interpolation Spaces: An Introduction. Grundlehren der Mathematischen Wissenschaften. Berlin-New York: Springer-Verlag, 1976
4. 4.
Bui H Q. Some aspects of weighted and non-weighted Hardy spaces. Kokyuroku Res Inst Math Sci, 1980, 383: 38–56Google Scholar
5. 5.
Bui H Q. Weighted Besov and Triebel spaces: Interpolation by the real method. Hiroshima Math J, 1982, 12: 581–605
6. 6.
Bui T A, Cao J, Ky L D, et al. Musielak-Orlicz-Hardy spaces associated with operators satisfying reinforced off diagonal estimates. Anal Geom Metric Spaces, 2012, 1: 69–129
7. 7.
Cao J, Chang D C, Yang D, et al. Weighted local Orlicz-Hardy spaces on domains and their applications in inhomogeneous Dirichlet and Neumann problems. Trans Amer Math Soc, 2013, 365: 4729–4809
8. 8.
Cianchi A. Strong and weak type inequalities for some classical operators in Orlicz spaces. J London Math Soc (2), 1999, 60: 187–202
9. 9.
Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569–645
10. 10.
Cruz-Uribe D V, Wang L A. Variable Hardy spaces. ArXiv:1211.6505, 2014Google Scholar
11. 11.
Curbera P, García-Cuerva J, Martell J, et al. Extrapolation with weights, rearrangement-invariant function spaces, modular inequalities and applications to singular integrals. Adv Math, 2006, 203: 256–318
12. 12.
DeVore R A, Sharpley R C. Maximal functions measuring smoothness. Mem Amer Math Soc, 1984, 47: viii+115
13. 13.
Duoandikoetxea J. Fourier Analysis. Providence, RI: Amer Math Soc, 2001
14. 14.
Edmunds D E, Gurka P, Opic B. Double exponential integrability of convolution operators in generalized Lorentz-Zygmund spaces. Indiana Univ Math J, 1995, 44: 19–43
15. 15.
Fefferman C, Stein E. H p spaces of several variables. Acta Math, 1972, 129: 137–193
16. 16.
Gallardo D. Orlicz spaces for which the Hardy-Littlewood maximal operator is bounded. Publ Mat, 1988, 32: 261–266
17. 17.
García-Cuerva J, Rubio de Francia J L. Weighted Norm Inequalities and Related Topics. Amsterdam-New York-Oxford: North-Holland, 1985
18. 18.
Goldberg D. A local version of real Hardy spaces. Duke Math J, 1979, 46: 27–42
19. 19.
Grafakos L. Classical Fourier Analysis. New York: Springer, 2008
20. 20.
Grafakos L, Tao T, Terwilleger E. L p bounds for a maximal dyadic sum operator. Math Z, 2004, 246: 321–337
21. 21.
Grevholm B. On the structure of the spaces $$\mathcal{L}_k ^{p,\lambda }$$. Math Scand, 1971, 26: 241–254
22. 22.
Hou S, Yang D, Yang S. Lusin area function and molecular characterizations of Musielak-Orlicz Hardy spaces and their applications. Commun Contemp Math, 2013, doi: 10.1142/S0219199713500296Google Scholar
23. 23.
Hou S, Yang D, Yang S. Musielak-Orlicz BMO-type spaces associated with generalized approximations to the identity. ArXiv:1303.6366, 2013Google Scholar
24. 24.
Iwaniec T, Onninen J. H 1-estimates of Jacobians by subdeterminants. Math Ann, 2002, 324: 341–358
25. 25.
Iwaniec T, Sbordone C. Weak minima of variational integrals. J Reine Angew Math, 1994, 454: 143–161
26. 26.
Iwaniec T, Verde A. A study of Jacobians in Hardy-Orlicz spaces. Proc Roy Soc Edinburgh Sect A, 1999, 129: 539–570
27. 27.
Kokilashvili V, Krbec M. Weighted Inequalities in Lorentz and Orlicz Spaces. River Edge, NJ: World Scientific Publishing, 1991
28. 28.
Jiang R, Yang D. New Orlicz-Hardy spaces associated with divergence form elliptic operators. J Funct Anal, 2010, 258: 1167–1224
29. 29.
Komori Y, Shirai S. Weighted Morrey spaces and a singular integral operator. Math Nachr, 2009, 282: 219–231
30. 30.
Krantz S G. Fractional integration on Hardy spaces. Studia Math, 1982, 73: 87–94
31. 31.
Ky L D. New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators. Integral Equations Operator Theory, 2014, 78: 115–150
32. 32.
Lerner A K. A simple proof of the A 2 conjecture. Int Math Res Not, 2013, 2013: 3159–3170
33. 33.
Liang Y, Huang J, Yang D. New real-variable characterizations of Musielak-Orlicz Hardy spaces. J Math Anal Appl, 2012, 395: 413–428
34. 34.
Liang Y, Yang D. Musielak-Orlicz Campanato spaces and applications. J Math Anal Appl, 2013, 406: 307–322
35. 35.
Liang Y, Yang D, Yuan W, et al. A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Diss Math, 2013, 489: 114ppGoogle Scholar
36. 36.
Luxenberg W. Banach Function Spaces. Delft: Technische Hogeschool te Delft, 1955Google Scholar
37. 37.
Miyamoto T, Nakai E, Sadasue G. Martingale Orlicz-Hardy spaces. Math Nachr, 2012, 285: 670–686
38. 38.
Mizuta Y, Nakai E, Sawano Y, et al. Littlewood-Paley theory for variable exponent Lebesgue spaces and Gagliardo-Nirenberg inequality for Riesz potentials. J Math Soc Japan, 2013, 65: 633–670
39. 39.
Nakai E. On the restriction of functions of bounded mean oscillation to the lower-dimensional space. Arch Math, 1984, 43: 519–529
40. 40.
Nakai E. Pointwise multipliers for functions of weighted bounded mean oscillation. Studia Math, 1993, 105: 106–119
41. 41.
Nakai E. Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math Nachr, 1994, 166: 95–103
42. 42.
Nakai E. Generalized fractional integrals on Orlicz-Morrey spaces. In: Banach and Function Spaces. Yokohama: Yokohama Publishers, 2004, 323–333Google Scholar
43. 43.
Nakai E. Construction of an atomic decomposition for functions with compact support. J Math Anal Appl, 2006, 313: 730–737
44. 44.
Nakai E. The Campanato, Morrey and Hölder spaces on spaces of homogeneous type. Studia Math, 2006, 176: 1–19
45. 45.
Nakai E. A generalization of Hardy spaces H p by using atoms. Acta Math Sin Engl Ser, 2008, 24: 1243–1268
46. 46.
Nakai E. Calderon-Zygmund operators on Orlicz-Morrey spaces and modular inequalities. In: Banach and Function Spaces II. Yokohama: Yokohama Publishers, 2008, 393–410Google Scholar
47. 47.
Nakai E. Singular and fractional integral operators on Campanato spaces with variable growth conditions. Rev Mat Complut, 2010, 23: 355–381
48. 48.
Nakai E, Sawano Y. Hardy spaces with variable exponents and generalized Campanato spaces. J Funct Anal, 2012, 262: 3665–3748
49. 49.
Nakano H. Modulared Semi-Ordered Linear Spaces. Tokyo: Maruzen, 1950
50. 50.
Nakano H. Topology of Linear Topological Spaces. Tokyo: Maruzen, 1951Google Scholar
51. 51.
Samko N. Weighted Hardy and potential operators in Morrey spaces. J Funct Spaces Appl, 2012, 2012: 1–21
52. 52.
O’Neil R. Fractional integration in Orlicz spaces: I. Trans Amer Math Soc, 1965, 115: 300–328
53. 53.
Peetre J. On interpolation functions: II. Acta Sci Math, 1968, 29: 91–92
54. 54.
Rauhut H, Ullrich T. Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type. J Funct Anal, 2011, 260: 3299–3362
55. 55.
Rychkov V S. Littlewood-Paley theory and function spaces with A ploc weights. Math Nachr, 2001, 224: 145–180
56. 56.
Sawano Y. Sharp estimates of the modified Hardy-Littlewood maximal operator on the nonhomogeneous space via covering lemmas. Hokkaido Math J, 2005, 34: 435–458
57. 57.
Sawano Y. A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Acta Math Sin Engl Ser, 2009, 25: 1223–1242
58. 58.
Serra C F. Molecular characterization of Hardy-Orlicz spaces. Rev Un Mat Argentina, 1996, 40: 203–217
59. 59.
Stein E M. Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton University Press, 1993
60. 60.
Stein E M, Weiss G. On the theory of harmonic functions of several variables, I: The theory of H p-spaces. Acta Math, 1960, 103: 25–62
61. 61.
Strichartz R S. A note on Trudinger’s extension of Sobolev’s inequalities. Indiana Univ Math J, 1972, 21: 841–842
62. 62.
Strömberg J O, Torchinsky A. Weighted Hardy Spaces. Berlin-New York: Springer-Verlag, 1989
63. 63.
Taibleson M H. On the theory of Lipschitz spaces of distributions on Euclidean n-space, I: Principal properties. J Math Mech, 1964, 13: 407–479
64. 64.
Taibleson M H, Weiss G. The molecular characterization of certain Hardy spaces. Asterisque, 1980, 77: 67–149
65. 65.
Torchinsky A. Interpolation of operations and Orlicz classes. Studia Math, 1976, 59: 177–207
66. 66.
Triebel H. Theory of Function Spaces. Basel: Birkhäuser, 1983
67. 67.
Trudinger N S. On imbeddings into Orlicz spaces and some applications. J Math Mech, 1967, 17: 473–483
68. 68.
Ullrich T. Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits. J Funct Spaces Appl, 2012, 2012: 1–47
69. 69.
Viviani B E. An atomic decomposition of the predual of BMO(ρ). Rev Mat Iberoamericana, 1987, 3: 401–425
70. 70.
Yang D, Yang S. Weighted local Orlicz Hardy spaces with applications to pseudo-differential operators. ArXiv:1107.3266, 2011Google Scholar
71. 71.
Yang D, Yang S. Orlicz-Hardy spaces associated with divergence operators on unbounded strongly Lipschitz domains of ℝn. Indiana Univ Math J, 2012, 61: 81–129
72. 72.
Yang D, Yang S. Local Hardy spaces of Musielak-Orlicz type and their applications. Sci China Math, 2012, 55: 1677–1720
73. 73.
Yang D, Yang S. Real-variable characterizations of Orlicz-Hardy spaces on strongly Lipschitz domains of ℝn. RevMat Iberoam, 2013, 29: 237–292
74. 74.
Yang D, Yang S. Musielak-Orlicz Hardy spaces associated with operators and their applications. J Geom Anal, 2012, doi: 10.1007/s12220-012-9344-yGoogle Scholar
75. 75.
Yang D, Yuan W, Zhuo C. Musielak-Orlicz Besov-type and Triebel-Lizorkin-type spaces. Rev Mat Complut, 2014, 27: 93–157