Compactness of the commutators of intrinsic square functions on weighted Lebesgue spaces

The aim of this paper is to study the compactness for the commutators of intrinsic square functions, including the intrinsic gλ∗ -function and the intrinsic Littlewood–Paley g-function.Using a weighted version of the Frechét– Kolmogorov–Riesz theorem, the compactness for their commutators generated with the CMO functions is obtained on the weighted Lebesgue spaces.

PDF

___

[1] Beatrous F, Li SY. On the boundedness and compactness of operators of Hankel type. J Funct Anal 1993; 111: 350-379.
[2] Bényi A, Damián W, Moen K, Torres R. Compact bilinear commutators: the weighted case. Michigan Math J 2015; 64: 39-51.
[3] Bényi A, Damián W, Moen K, Torres R. Compactness properties of commutators of bilinear fractional integrals. Math Z 2015; 280: 569-582.
[4] Carro MJ, Grafakos L, Soria J. Weighted weak-type (1,1) estimates via Rabio de Francia extrapolation. J Funct Anal 2015; 269: 1203-1233.
[5] Chen JC, Chen YP, Hu GE. Compactness for the commutators of singular integral operators with rough variable kernels. J Math Anal Appl 2015; 431: 597-621.
[6] Chen JC, Hu GE. Compact commutators of rough singular integral operators. Canad Math Bull 2015; 58: 19-29.
[7] Chen YP, Ding Y. Compactness characterization of commutators for Littlewood-Paley operators. Kodai Math J 2009; 32: 256-323.
[8] Chen YP, Wang H. Compactness for the commutator of the parameterized area integral in the Morrey space. Math Ineq Appl 2015; 4: 1261-1273.
[9] Clop A, Cruz V. Weighted estimates for Beltrami equations. Ann Acad Sci Fenn Math 2013; 38: 91-113.
[10] Grafakos L. Classical Fourier Analysis. 2nd ed. New York, NY, USA: Springer Science Business Media LLC, 2008.
[11] Huang JZ, Liu Y. Some characterizations of weighted Hardy spaces. J Math Anal Appl 2010; 363: 121-127.
[12] Lerner AK. Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals. Adv Math 2011; 226: 3912-3926.
[13] Liang YY, Yang DC. Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces. T Am Math Soc 2015; 367: 3225-3256.
[14] Mao SZ, Sawano Y, Wu HX. On the compactness of commutators for rough Marcinkiewicz integral operators. Taiwanese J Math 2015; 19: 1777-1793.
[15] Uchiyama A. On the compactness of operators of Hankel type. Tóhoku Math J 1978; 30: 163-171.
[16] Villarroya P. A characterization of compactness for singular integrals. J Math Pure Appl 201; 104: 485-532.
[17] Wang H. Intrinsic square functions on the weighted Morrey spaces. J Math Anal Appl 2012; 396: 302-314.
[18] Wang LJ, Tao SP. Parameterized Littlewood-Paley operators and their commutators on Herz spaces with variable exponents. Turk J Math 2016; 40: 122-145.
[19] Wilson M. The intrinsic square function. Rev Mat Iberoam 2007; 23: 771-791.
[20] Wilson M. Weighted Littlewood-Paley Theory and Exponential-Square Integrability. Berlin, Germany: Springer- Verlag, 2007.