Boundedness of some sublinear operators on Herz spaces

IntroductionIt is well known that Beurling [2] and Herz 11 introduced some new spaces that characterize certain properties of functions.These new spaces are called the Herz spaces.Many studies involving these spaces can be found in the literature.One of the main reasons is that Hardy space theory associated with Herz spaces is very rich.Actually, these new Hardy spaces are a sort of local version of the ordinary Hardy spaces; the former, sometimes, are good substitutes of the latter when considering, for example, the boundedness of non-translation invariant singular integral operators.This paper is motivated by previous work of Lu, Hernfindez and the second author (see 14] and 10]), and also by more applications, such as the boundedness ofbilinear operators and the regularity of solutions of the Laplacian and the wave equations on Herz-type spaces.See 12] and 16].Our main interest is to study the boundedness of some sublinear operators on these spaces under certain weak size conditions (see (2.1) and (2.2) below).These conditions are similar to those introduced by Soria and Weiss in 18], and are satisfied by most of the operators in harmonic analysis (see 18]).Let us first introduce some notations.Let Bk {x E ]n.Ixl _< 2k} and Ak B \ Bk-1 for k E Z. Let ) XA for k Z, where )e is the characteristic function of the set E. where Definitionl.1.LetotN, 0<p<cx3and0<q < (a) The homogeneous Herz space/'P (]1n) is defined by /,p(n) {f Loc(ln \ {0})" Ilfllzcg'') < 1, 2kaP P Ilfllt,(,) IlfxkllLq(n) < with the usual modifications made when p cx and/or q .(b) The non-homogeneous Herz space K' (n) is defined by K'P(Nn) {f Loc(Rn) IlfllKg'() < },