Articles

MIXED LIPSCHITZ SPACES AND THEIR APPLICATIONS

  • Shaoyong HE ,
  • Jiecheng CHEN
Expand
  • 1. Department of Mathematics, Huzhou University, Huzhou 313000, China;
    2. Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

Received date: 2020-01-22

  Revised date: 2020-09-09

  Online published: 2022-04-22

Supported by

Supported by Zhejiang Provincial Natural Science Foundation of China (LQ22A010018) and National Natural Science Foundation of China (12071437).

Abstract

The purpose of this paper is to introduce bi-parameter mixed Lipschitz spaces and characterize them via the Littlewood-Paley theory. As an application, we derive a boundedness criterion for singular integral operators in a mixed Journé class on mixed Lipschitz spaces. Key elements of the paper are the development of the Littlewood-Paley theory for a special mixed Besov spaces, and a density argument for the mixed Lipschitz spaces in the weak sense.

Cite this article

Shaoyong HE , Jiecheng CHEN . MIXED LIPSCHITZ SPACES AND THEIR APPLICATIONS[J]. Acta mathematica scientia, Series B, 2022 , 42(2) : 690 -714 . DOI: 10.1007/s10473-022-0217-6

References

[1] Chang S-Y A, Fefferman R. A continuous version of duality of H1 with BMO on the bidisc. Ann Math, 1980, 112(1):179-201
[2] Chang S-Y A, Fefferman R. The Calderón-Zygmund decomposition on product domains. Amer J Math, 1982, 104(3):455-468
[3] Chang S-Y A, Fefferman R. Some recent developments in fourier analysis and Hp theory on product domains. Bull Amer Math Soc, 1985, 12(1):1-43
[4] David G, Journé J L. A boundedness criterion for generalized Calderón-Zygmund operators. Ann Math, 1984, 120(2):371-397
[5] Ding W, Lu G. Boundedness of inhomogeneous Journé's type operators on multi-parameter local Hardy spaces. Nonlinear Analysis, 2020, 197:1-31
[6] Ding W, Lu G, Zhu Y. Discrete Littlewood-Paley-Stein characterization of multi-parameter local Hardy spaces. Forum Math, 2019, 31(6):1467-1488
[7] Ding W, Lu G, Zhu Y. Multi-parameter local Hardy spaces. Nonlinear Analysis, 2019, 184:352-380
[8] Ding W, Zhu Y. Mixed Hardy spaces and their applications. Acta Mathematica Scientia, 2020, 40(4):945-969
[9] Fefferman R. Singular integrals on product Hp spaces. Rev Mat Iberoam, 1985, 1(2):25-31
[10] Fefferman R. Calderón-Zygmund theory for product domains:Hp spaces. Proc Nat Acad Sci, 1986, 83(4):840-843
[11] Fefferman R. Harmonic analysis on product spaces. Ann Math, 1987, 126(1):109-130
[12] Fefferman R, Pipher J. Multiparameter operators and sharp weighted inequalities. Amer J Math, 1997, 119(2):337-369
[13] Fefferman R, Stein E M. Singular integrals on product spaces. Adv Math, 1982, 45(2):117-143
[14] Frazier M, Jawerth B. A discrete transform and decomposition of distribution. J Funct Anal, 1990, 93(1):34-170
[15] Frazier M, Jawerth B, Weiss G. Littlewood-Paley theory and the study of function spaces. CBMS Regional conference series in Mathematics, Vol 79. Providence, RI:American Mathematical Society, 1991
[16] Goldberg D. A local version of real Hardy spaces. Duke Math J, 1979, 46(1):27-42
[17] Gundy R, Stein E M. Hp theory for the poly-disk. Proc Nat Acad Sci, 1979, 76(3):1026-1029
[18] Han Y, Han Y. Boundedness of composition operators associated with mixed homogeneities on Lipschitz spaces. Math Res Lett, 2016, 23(5):1387-1403
[19] Han Y, Han Y, Li J, et al. Marcinkiewicz multipliers and Lipschitz Spaces on Heisenberg groups. Canad J Math, 2019, 71(3):607-627
[20] Han Y, Lee M, Lin C, et al. Calderón-Zygmund operators on product Hardy spaces. J Funct Anal, 2010, 258(8):2834-2861
[21] Han Y, Li J, Lin C, et al. Singular integrals associated with Zygmund dilations. J Geom Anal, 2019, 29(3):2410-2455
[22] Han Y, Lin C, Lu G, et al. Hardy spaces associated with different homogeneities and boundedness of composition operators. Rev Mat Iberoam, 2013, 29(4):1127-1157
[23] Han Y, Lin C, Wu X, Boundedness of singular integrals with flag kernels on weighted flag Hardy spaces. Pacific J Math, 2019, 302(2):545-598
[24] Han Y, Lu G, Ruan Z. Boundedness criterion of Journé's class of singular integrals on multiparameter Hardy spaces. J Funct Anal, 2013, 264(5):1238-1268
[25] Han Y, Lu G, Sawyer E. Flag Hardy space and Marcinkiewicz multipliers on the Heisnberg group. Anal PDE, 2014, 7(7):1465-1534
[26] Han Y, Yang D. Hp boundedness of Calderón-Zygmund operators on product spaces. Math Z, 2005, 249(4):869-881
[27] He S, Chen J. Three-parameter Hardy spaces associated with a sum of two flag singular integrals. Banach J Math Anal, 2020, 14(4):1363-1386
[28] He S, Chen J. Boundedness of multi-parameter pseudo-differential operators on multi-parameter Lipschitz spaces. J Pseudo-Differ Oper Appl, 2020, 11(4):1665-1683
[29] He S, Chen J. Inhomogeneous Lipschitz spaces associated with flag singular integrals and their applications. Math Inequal Appl, 2021, 24(4):965-987
[30] Journé J L. Calderón-Zygmund operators on product spaces. Rev Mat Iberoam, 1985, 1(3):55-91
[31] Müller D, Ricci F, Stein E M. Marcinkiewicz multipliers and multi-parameter strucure on Heisenberg(-type) groups I. Invent Math, 1995, 119(2):119-233
[32] Müller D, Ricci F, Stein E M. Marcinkiewicz multipliers and multi-parameter strucure on Heisenberg(-type) groups II. Math Z, 1996, 221(2):267-291
[33] Nagel A, Ricci F, Stein E M. Singular integrals with flag kernel and analysis on quadratic CR manifolds. J Func Anal, 2001, 181(1):29-118
[34] Nagel A, Ricci F, Stein E M, et al. Singular integrals with flag kernel on homogeneous group:I. Rev Mat Iberoam, 2012, 28(3):631-722
[35] Nagel A, Ricci F, Stein E M, et al. Algebrals of singular integrals operator with kernels controlled by multiple norms. Mem Amer Math Soc, 2018, 256(1230):vii+141
[36] Phong D H, Stein E M. Some further classes of pseudodifferential and singular integral operators arising in boundary valve problems I:composition of operators. Amer J Math, 1982, 104(1):141-172
[37] Pipher J. Journé's covering lemma and its extension to higher dimensions. Duke Math J, 1986, 53(3):683-690
[38] Ricci F, Stein E M. Multiparameter singular integrals and maximal functions. Ann Inst Fourier, 1992, 42(3):637-670
[39] Stein E M. Harmonic Analysis:Real variable methods, orthogonality, and Oscillatory integrals. Princeton, NJ:Princeton Univ Press, 1993
[40] Tan J, Han Y. Inhomogeneous multi-parameter Lipschitz spaces associated with different homogeneities and their applications. Filomat, 2018, 32(9):3397-3408
[41] Zheng T, Chen J, Dai J, et al. Calderón-Zygmund operators on homogeneous product Lipschitz spaces. 2021, 31(2):2033-2057
Options
Outlines

/