Estimates for fractional type Marcinkiewicz integrals with non-doubling measures

Guanghui Lu,Jiang Zhou

doi:10.1186/1029-242x-2014-285

Abstract

Under the assumption that μ is a non-doubling measure on satisfying the growth condition, the authors prove that the fractional type Marcinkiewicz integral ℳ is bounded from the Hardy space to the Lebesgue space for with kernel satisfying a certain Hormander-type condition. In addition, the authors show that for , ℳ is bounded from the Morrey space to the space and from the Lebesgue space to the space . MSC:46A20, 42B25, 42B35.

Highlights

Let μ be a nonnegative Radon measure on Rd which satisfies the following growth condition: for all x ∈ Rd and all r >, μ B(x, r) ≤ C rn, ( . )where C and n are positive constants and n ∈ (, d], B(x, r) is the open ball centered at x and having radius r
It is well known that the doubling condition on underlying measures is a key assumption in the classical theory of harmonic analysis
In recent years, many classical results concerning the theory of Calderón-Zygmund operators and function spaces have been proved still valid if the underlying measure is a nonnegative Radon measure on Rd which only satisfies ( . )

Summary

Introduction

Let μ be a nonnegative Radon measure on Rd which satisfies the following growth condition: for all x ∈ Rd and all r > , μ B(x, r) ≤ C rn,. If there exists a positive constant C such that for any x ∈ supp(μ) and r > , μ(B(x, r)) ≤ Cμ(B(x, r)), the μ is said to be doubling measure. The kernel K is said to satisfy a Hörmander-type condition if there exist cs > and Cs > such that for any x ∈ Rd and > cs|x|, sup kε k. We denote by Hs the class of kernels satisfying this condition. ). The purpose of this paper is to get some estimates for the fractional type Marcinkiewicz integral M with kernel K satisfying Dy Rd |x – y|n–α is bounded from Lp(μ) to Lr(μ) (see [ ])

Suppose that

Then define

Choose p and q such that

Here we have used the fact that

For α n and

We conclude from y