[1] P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domain of R", J.
Funct. Anal. 201 (2003), 148-184.
[2] P. Auscher, E. Russ and P. Tchamitchian, Hardy Sobolev spaces on strongly Lipschitz domain of Rn, I.
Funct. Anal. 218 (2005), 54--109.
[3]
A.
Benyi, W. Damian, K. Moen and R. H. Torres, Compactness properties of commutators of bilinear
fractional integrals, Math. Z. 280 (2015), 569-582.
[4]
A.
Benyi and R.H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc. 141
(2013), 3609-3621.
[5] M. Cao and K. Yabuta, Characterizations ofVMOl!,w(Rn) space, Preprint.
[6] D.-C. Chang, The dual of Hardy spaces on a bounded domain in Rn, Forum Math. 6 (1994), 65-81.
[7] D.-C. Chang, S. G. Krantz and E. M. Stein, HP theory on a smooth domain in RN and elliptic boundary
value problems, I. Funct. Anal. 114 (1993), 286-347.
[8] J. Chen, Y. Chen and G. Hu, Compactness for the commutators of singular integral operators with rough
variable kernels, I. Math. Anal. Appl. 431 (2015), 597-621.
[9] Y. Chen and Y. Ding, Compactness of the commutators of parabolic singular integrals, Sci. China Math.
53 (2010), 2633-2648.
[10] Y. Chen, Y. Ding and X. Wang, Compactness of commutators of Riesz potential on Morrey spaces, Potential Anal. 30 (2009), 301-313.
[11] A. Clop and V. Cruz, Weighted estimates for Beltrami equations, Ann. Acad. Sci. Fenn. Math. 38 (2013),
91-113.
[12] R. Coifman, P. L. Lions, Y. Meyer and S. Semmes, Compensated compactness and Hardy spaces, I. Math.
Pures Appl. 72 (1993), 247-286.
[13] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math.
Soc. 83 (1977), 569-645.
[14] R.R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables,
Ann. Math. 103 (1976), 611-635.
[15] D. G. Deng, X. T. Duong, A. Sikora and L. X. Yan, Comparison of the classical BMO with the BMO
spaces associated with operators and applications, Rev. Mat. lberoam. 24 (2008), 267-296.
[16] D. G. Deng, X. T. Duong, L. Song, C. Tan and L. Yan, Functions of vanishing mean oscillation associated
with operators and applications, Michigan Math. J. 56 (2008), 529-550.
174
[17] X. T. Duong, I. Holmes, J. Li, B. D. Wick and D. Yang, Two weight commutators in the Dirichlet and
Neumann Laplacian settings, J. Funct. Anal. 276 (2019), 1007-1060.
[18] X. Duong, J. Li, S. Mao, H. Wu and D. Yang, Compactness of Riesz transform commutator associated
with Bessel operators, J. Anal. Math. 135 (2018), 639--673.
[19] X. T. Duong and L. X. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpola-
tion, and applications, Comm. Pure Appl. Math. 58 (2005), 1375-1420.
[20] X. T. Duong and L. X. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel
bounds, J. Amer. Math. Soc. 18 (2005), 943-973.
[21] N. Dunford and J. Schwartz, Linear operators. I, Interscience, New York and London, 1964.
[22] C. Pefferman, and E. M. Stein, HP spaces of several variables, Acta Math. 129 (1972), 137-193.
[23] I. Holmes, R. Rahm and S. Spencer, Commutators with fractional integral operators, Studia Math. 233
(2016), 279-291.
[24] T. lwaniec, Nonlinear commutators and Jacobians, J. Fourier Anal. Appl. 3 (2007), 775-796.
[25] T. Iwaniec and C. Sbordone, Riesz transform and elliptic PDEs with VMO coefficients, J. Anal. Math. 74
(1998), 183-212.
[26] P. W. Jones and J-L. Journe, On weak convergence in H 1(R"), Proc. Amer.Math. Soc. 120 (1994), 137138.
[27] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961),
415-426.
[28] J. Li and B. D. Wick, Characterizations of Ht(R") and BMOl!.N(R") via weak factorizations and com-
mutators, J. Funct. Anal. 272 (2017), 5384--5416.
[29] R. A. Macfas and C. Segovia, Lipschitzfunctions on spaces of homogeneous type, Adv. Math. 33 (1979),
257-270.
[30] D. Palagachev and L. Softova, Singular integral operators, Morrey spaces and fine regularity of solutions
to PDEs, Potential Anal. 20 (2004), 237-263.
[31] M. Reed and B. Simon, Methods of modern mathematical physics I: functional analysis, Academic Press,
New York 1980.
[32] D. Sarason, Functions of vanishing mean oscillation, Trans. Amer. Math. Soc. 207 (1975), 391-405.
[33] W. A. Strauss, Partial differential equation: An introduction, John Wiley Sons, Inc., New York, 2008.
[34] A. Uchiyama, On the compactness of operators of the Hankel type, Tohoku Math. J. 30 (1978), 163-171.
[35] S. L. Wang, Compactness of commutators offractional integrals (Chinese), an English summary appears
in Chinese Ann. Math. Ser. B 8 (1987), no. 4,493, Chinese Ann. Math. Ser. A 8 (1987), 475-482.
[36] H. Wu and D. Yang, Characterizations of weighted compactness of commutators via CMO(R"), Proc.
Amer. Math. Soc. 146 (2018), 4239-4254.
[37] K. Yabuta, Singular Integrals (in Japanese), Iwanarni, 2010.
[38] K. Yosida, Functional Analysis, Springer, Berlin (1995).
RESEARCH CENTER FOR MATHEMATICS AND DATA SCIENCE, KWANSEI GAKUIN UNIVERSITY, GAKUEN
669-1337, JAPAN.
E-mail address: kyabuta3@kwansei.ac. jp
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