Two-Weight, Weak-Type Norm Inequalities for Fractional Integral Operators and Commutators on Weighted Morrey and Amalgam Spaces

Hua Wang

doi:10.1155/2020/3235942

Abstract

Let 0<γ<n and Iγ be the fractional integral operator of order γ, Iγfx=∫ℝnx−yγ−nfydy and let b,Iγ be the linear commutator generated by a symbol function b and Iγ, b,Iγfx=bx⋅Iγfx−Iγbfx. This paper is concerned with two-weight, weak-type norm estimates for such operators on the weighted Morrey and amalgam spaces. Based on weak-type norm inequalities on weighted Lebesgue spaces and certain Ap-type conditions on pairs of weights, we can establish the weak-type norm inequalities for fractional integral operator Iγ as well as the corresponding commutator in the framework of weighted Morrey and amalgam spaces. Furthermore, some estimates for the extreme case are also obtained on these weighted spaces.

Highlights

In [7], Martell considered the case when q > p and gave a verifiable condition which is sufficient for the twoweight, weak-type (p, q) inequality for fractional integral operator Ic. e condition (10) given below is simpler than the one in eorem 1
In view of eorems 2–5, it is a natural and interesting problem to find some sufficient conditions for which the two-weight, weak-type norm inequalities hold for the operators Ic and [b, Ic], in the endpoint case p 1
Let p′ be the conjugate index of p whenever p > 1; that is, 1/p + 1/p′ 1

Summary

Introduction

In [7], Martell considered the case when q > p and gave a verifiable condition which is sufficient for the twoweight, weak-type (p, q) inequality for fractional integral operator Ic. e condition (10) given below (in the Euclidean setting of [7]) is simpler than the one in eorem 1. In view of eorems 2–5, it is a natural and interesting problem to find some sufficient conditions for which the two-weight, weak-type norm inequalities hold for the operators Ic and [b, Ic], in the endpoint case p 1. In [17,18,19], the author studied the two-weight, weak-type (p, p) inequalities for fractional integral operator, as well as its commutators on weighted Morrey and amalgam spaces, under some Ap-type conditions (4) and (7) on the pair (w, ]). As a continuation of the works mentioned above, in this paper, we consider related problems about two-weight, weak-type (p, q) inequalities for Ic and [b, Ic], under some other Ap-type conditions (10) and (12) on (w, ]) and 1 < p < q

Statement of Our Main Results

Notation and Definitions

Proofs of Theorems 6 and 7

Proofs of Theorems 8 and 9

Proofs of Theorems 10 and 11