%0 Journal Article
%T The Boundedness of Intrinsic Square Functions on the Weighted Herz Spaces
%A Hua Wang
%J Journal of Function Spaces
%D 2014
%R 10.1155/2014/274521
%X We will obtain the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley -function, and -function on the weighted Herz spaces with general weights. 1. Introduction and Main Results Let and . The classical square function (Lusin area integral) is a familiar object. If is the Poisson integral of , where denotes the Poisson kernel in , then we define the classical square function (Lusin area integral) by (see [1, 2]) where denotes the usual cone of aperture one: Similarly, we can define a cone of aperture for any : and corresponding square function The Littlewood-Paley -function (could be viewed as a ˇ°zero-apertureˇ± version of ) and the -function (could be viewed as an ˇ°infinite apertureˇ± version of ) are defined, respectively, by (see, e.g., [3, 4]) The modern (real-variable) variant of can be defined in the following way (here we drop the subscript if ). Let be real, radial and have support contained in , and let . The continuous square function is defined by (see, e.g., [5, 6]) In 2007, Wilson [7] introduced a new square function called intrinsic square function which is universal in a sense (see also [8]). This function is independent of any particular kernel , and it dominates pointwise all the above-defined square functions. On the other hand, it is not essentially larger than any particular . For , let be the family of functions defined on such that has support containing in , , and for all , For and , we set Then we define the intrinsic square function of (of order ) by the following formula: We can also define varying-aperture versions of by the formula The intrinsic Littlewood-Paley -function and the intrinsic -function will be given, respectively, by In [8], Wilson showed the following weighted boundedness of the intrinsic square functions. Theorem A. Let , , and . Then there exists a constant independent of such that Moreover, in [9], Lerner obtained sharp norm inequalities for the intrinsic square functions in terms of the characteristic constant of for all . For further discussions about the boundedness of intrinsic square functions on various function spaces, we refer the readers to [10¨C17]. Before stating our main results, let us first recall some definitions about the weighted Herz and weak Herz spaces. For more information about these spaces, one can see [18¨C22] and the references therein. Let and let for any . Denote for , if , and , where is the characteristic function of the set . For any given weight function on and , we denote by the space of all functions satisfying
%U http://www.hindawi.com/journals/jfs/2014/274521/