SURVEY ARTICLE

Some results and problems on commutators

  • Shanzhen LU
Expand
  • School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received date: 12 Oct 2010

Accepted date: 17 Aug 2011

Published date: 01 Oct 2011

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

In this paper, the author introduces some late results and puts forward a few problems on commutators of many important operators in harmonic analysis, included the Bochner-Riesz operator below the critical index, the strongly singular integral operator, the pseudo-differential operator, a class of convolution operators with oscillatory kernel, the Marcinkiewicz integral operator, and the fractional integral operator with rough kernel.

Cite this article

Shanzhen LU . Some results and problems on commutators[J]. Frontiers of Mathematics in China, 2011 , 6(5) : 821 -833 . DOI: 10.1007/s11464-011-0157-2

1
Alvarez J, Bagby R J, Kurtz D S, Pérez C. Weighted estimates for commutators of linear operators. Studia Math, 1993, 104: 195-209

2
Alvarez J, Hounie J. Estimates for the kernel and continuity properties of pseudodifferential operators. Ark Mat, 1990, 28: 1-22

DOI

3
Auscher P, Taylor M. Paradifferential operators and commutator estimates. Comm PDE, 1995, 20: 1743-1775

DOI

4
Carleson L, Sjölin P. Oscillatory integrals and a multiplier problem for the disc. Studia Math, 1972, 44: 287-299

5
Chanillo S. A note on commutators. Indiana Univ Math J, 1982, 31: 7-16

DOI

6
Chanillo S. Remarks on commutators of pseudo-differential operators. Contemp Math, 1997, 205: 33-37

7
Chiarenza F, Frasca M, Longo P. W2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans Amer Math Soc, 1993, 336: 841-853

DOI

8
Coifman R R, Rochberg R, Weiss G. Factorization theorems for Hardy spaces in several variables. Ann Math, 1976, 103: 611-635

DOI

9
Ding Y, Lu S Z. Higher order commutators for a class of rough operators. Ark Mat, 1999, 37: 33-44

DOI

10
Ding Y, Lu S Z, Yabuta K. On commutators of Marcinkiewicz integrals with rough kernel. J Math Anal Appl, 2002, 275: 60-68

DOI

11
Drobot V, Naparstek A, Sampson G. (Lp, Lq) mapping properties of convolution transforms. Studia Math, 1976, 55: 41-70

12
Fazio G Di, Ragusa M A. Interior estimates in Morrey spaces for strongly solutions to nondivergence form equations with discontinuous coefficients. J Funct Anal, 1993, 112: 241-256

DOI

13
Fefferman C. Inequality for strongly singular convolution operators. Acta Math, 1970, 124: 9-36

DOI

14
Fu Z W, Liu Z G, Lu S Z, Wang H B. Characterization for commutators of N-dimensional Hardy operators. Sci China, Ser A, 2007, 10: 1418-1426

DOI

15
Hu G E, Lu S Z. The commutator of the Bochner-Riesz operator. Tohoku Math J, 1996, 48: 259-266

DOI

16
Hu G E, Lu S Z. A weighted L2 estimate for the commutator of the Bochner-Riesz operator. Proc Amer Math Soc, 1997, 125: 2867-2873

DOI

17
Janson S. Mean oscillation and commutators of singular integrals operators. Ark Mat, 1978, 16: 263-270

DOI

18
Li J F. <DissertationTip/>. Beijing: Beijing Normal Univ, 2005

19
Li J F, Lu S Z. Lp estimates of multilinear operators on strongly singular integral operators. Nagoya Math J, 2006, 181: 1-22

DOI

20
Li J F, Lu S Z. Applications of the scale changing method to boundedness of certain commutators. Int J Appl Math Sci, 2004, 1: 1-12

DOI

21
Lin Y. Commutators of pseudo-differential operators. Sci China, Ser A, 2008, 51: 453-460

DOI

22
Lin Y, Lu S Z. Pseudo-differential operators on Sobolev and Lipschitz spaces. Acta Math Sinica, Eng Ser, 2010, 26: 131-142

DOI

23
Lu S Z, Mo H X. The boundedness of commutators for the Marcinkiewicz integrals. Acta Math Sinica, 2006, 49: 481-490 (in Chinese)

24
Lu S Z, Mo H X. Boundedness of higher-order Marcinkiewicz-type integrals. Int J Math Math Sci, 2006, Article ID: 2006, 31705: 1-21

25
Lu S Z, Xia X. A note on commutators of Bochner-Riesz operator. Front Math China, 2007, 2(3): 439-446

DOI

26
Lu S Z, Xia X. Boundedness for commutators of Bochner-Riesz operators below a critical index. Sci China, Ser A, 2007, 50: 801-813

DOI

27
Lu S Z, Yang D C. The central BMO spaces and Littlewood-Paley operators. Approx Theory and Appl, 1995, 11: 72-94

28
Lu S Z, Zhang P. Lipschitz estimates for generalized commutators of fractional integrals with rough kernel. Math Nachr, 2003, 252: 70-85

DOI

29
Paluszyński M. Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss. Indiana Univ Math J, 1995, 44: 1-17

30
Segovia C, Torrea J. Higher order commutators for vector-valued Calderón-Zygmund operators. Trans Amer Math Soc, 1993, 336: 537-556

DOI

31
Shi X L, Sun Q Y. Weighted norm inequalities for Bochner-Riesz operators and singular integral operators. Proc Amer Math Soc, 1992, 116: 665-673

DOI

32
Sjolin P. Convolution with oscillating kernels. Indiana Univ Math J, 1981, 30: 47-56

DOI

33
Stein E M. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993

34
Torchinsky A, Wang S L. A note on the Marcinkiewicz integral. Colloq Math, 1990, 60/61: 235-243

35
Xia X, Lu S Z. Boundedness of commutators of an oscillatory integral operator. Studia Math, 2008, 186: 15-27

DOI

36
Xia X, Lu S Z. Commutators of Bochner-Riesz operators for a class of radial functions. Acta Math Sci, 2011, 31(B): 477-482

Options
Outlines

/