Some results and problems on commutators

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.

Keywords

Commutator / BMO function / Lipschitz function

References

[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

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

[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

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

[7] Chiarenza F., Frasca M., Longo P. W ^2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans Amer Math Soc, 1993, 336, 841-853

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

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

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

[11] Drobot V., Naparstek A., Sampson G. (L ^p, L ^q) mapping properties of convolution transforms. Studia Math, 1976, 55, 41-70

[12] Di Fazio G., 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

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

[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

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

[16] Hu G. E., Lu S. Z. A weighted L ² estimate for the commutator of the Bochner-Riesz operator. Proc Amer Math Soc, 1997, 125, 2867-2873

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

[18] Li J F. Ph D Dissertation. Beijing: Beijing Normal Univ, 2005

[19] Li J. F., Lu S. Z. L ^p estimates of multilinear operators on strongly singular integral operators. Nagoya Math J, 2006, 181, 1-22

[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

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

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

[23] Lu S. Z., Mo H. X. The boundedness of commutators for the Marcinkiewicz integrals. Acta Math Sinica, 2006, 49, 481-490

[24] Lu S, Mo H Boundedness of higher-order Marcinkiewicz-type integrals. Int J Math Math Sci, 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

[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

[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

[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

[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

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

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

[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

[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