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

where Γβ(x) = {(y, t) ∈ R + : |x− y| < βt}. Denote Gα,1(f) = Gα(f) . The intrinsic square functions were first introduced by Wilson in order to answer a conjecture proposed by Fefferman and Stein on the boundedness of the Lusin area function S on the weighted L Lebesgue space [19, 20]. The intrinsic square function has several interesting features. First, it is independent of any particular kernel, such as the Poisson kernel. It dominates pointwise the classical square function (Lusin area integral) and its real-variable generalizations. Second, although the function Gα,β(f) is defined by the kernels with uniform compact support, there is a pointwise relation between Gα,β(f) with different β : ∗Correspondence: wuxm@zjnu.cn 2010 AMS Mathematics Subject Classification: 42B20, 42B25, 47B07