The Boundedness of Intrinsic Square Functions on the Weighted Herz Spaces

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 “zero-aperture” version of ) and the -function (could be viewed as an “infinite aperture” 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 [10–17]. 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 [18–22] 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