Some estimates of intrinsic square functions on the weighted Herz-type Hardy spaces

Hua Wang

doi:10.1186/s13660-015-0583-x

Abstract

In this paper, by using the atomic decomposition theory of weighted Herz-type Hardy spaces, we obtain some strong type and weak type estimates for intrinsic square functions including the Lusin area function, Littlewood-Paley $\mathcal{G}$ -function and $\mathcal{G}^{*}_{\lambda}$ -function on these spaces.

Highlights

Introduction and main resultsLet Rn++ = Rn × (, ∞) and φt(x) = t–nφ(x/t)
A weight function w is said to belong to the reverse Hölder class RHr if there exist two constants r > and C > such that the following reverse Hölder inequality holds:
A theory of Hardy spaces associated with Herz spaces has been developed in [, ]. These new Herz-type Hardy spaces may be regarded as a local version at the origin of the classical Hardy spaces Hp(Rn) and are good substitutes for Hp(Rn) when we study the boundedness of non-translation invariant operators

Summary

Introduction and main results

A weight function w is said to belong to the reverse Hölder class RHr if there exist two constants r > and C > such that the following reverse Hölder inequality holds: w(x)r dx. In , Lu and Yang [ , ] introduced the following weighted Herz-type Hardy spaces HKqα,p(w , w ) (HKqα,p(w , w )) and established their central atomic decompositions. We will use Lu and Yang’s central atomic decomposition theory for weighted Herz-type Hardy spaces in [ , ] (see [ ]). We characterize weighted Herz-type Hardy spaces in terms of central atoms in the following way. Applying Hölder’s inequality, Theorem A and the size condition of central atom a with supp a ⊆ B , we have.

Changing the order of integration yields

Changing the order of summation gives us that