Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures

Yinsheng Jiang,Tie Li,Yaoming Niu

doi:10.4236/jamp.2020.81005

Abstract

Let μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces .Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

Highlights

Let μ be a positive Radon measures on d satisfying only the growth condition, that is, there exists a constant C > 0 and n ∈ (0, d ] such that μ (Q) ≤ Cl (Q)n (1)for any cube Q ⊂ d with sides parallel to the coordinate axes
This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces
We show that the iterated commutators generated by multilinear singular integrals operators (Tm )Πb are bounded from

Summary

Introduction

Let μ be a positive Radon measures on d satisfying only the growth condition, that is, there exists a constant C > 0 and n ∈ (0, d ] such that μ (Q) ≤ Cl (Q)n (1). In 2005, Sawano and Tanaka [10] gave a natural definition of Morrey spaces for Radon measures which might be non-doubling but satisfied the growth condition, and they investigated the boundedness in these spaces of some classical operators in harmonic analysis. Tolsa [5] developed the theory of Calderón-Zygmund operators and their commutators with RBMO functions in the setting of non-doubling measures. It points out that Perez and Pradolini [21] introduced a said iterated commutators generated by the multilinear singular integral operators with Calderón-Zygmund type and vector function b ∈ RBMOm and obtained the boundedness from Lp1 × × Lpm to Lp with 1 p= 1 p1 + +1 pm for 1 < p1, , pm < ∞ (they considered the weighted case). L1 m,∞ (μ ) , for all σ ⊂ {1, 2, , m} , the commutators ( ) Tm ∏bσ are bounded from

Proof of Main Results

Conclusions