Multilinear Riesz Potential on Morrey-Herz Spaces with Non-Doubling Measures

Abstract

The authors consider the multilinear Riesz potential operator defined by. . , f m , m, n the nonnegative integers with n ≥ 2, m ≥ 1, 0 < α < mn, and μ is a nonnegative n-dimensional Borel measure. In this paper, the boundedness for the operator I α,m on the product of homogeneous Morrey-Herz spaces in nonhomogeneous setting is found.

Inclusion Properties of The Homogeneous Herz-Morrey Spaces With Variable Exponent

“…where 𝑚 𝑘 ( 𝛾, 𝑓 ) = |{ 𝑥 ∈ 𝐴 𝑘 : |𝑓(𝑥)| > 𝛾 }|. Some authors have investigated those spaces in various terms of discussion (see [12] - [15]). Meanwhile, this article aims to discuss in terms inclusion properties and inclusion relation of the homogeneous Herz-Morrey spaces and homogeneous weak Herz-Morrey spaces with variable exponent.…”

Inclusion Properties of The Homogeneous Herz-Morrey

“…where ( , ) = |{ ∈ : | ( )| > }|. Some authors have studied those spaces in different term of discussion (see [7], [8], [9], [10]). Meanwhile, in this article, the authors would like to discuss the inclusion properties and inclusion relation of the homogeneous Herz-Morrey spaces and the homogeneous weak Herz-Morrey spaces.…”

Rough bilinear fractional integrals with variable kernels

“…Some transference results of B 1,α on the n-torus were also obtained in [2] (see also [3] for a version on a compact Lie group). On the other hand, the study of the operators K Ω,α (as well as B Ω,α ) and its related operators with rough Ω were recently attracted several authors (see [4,13]). The following theorem is the main result in [4].…”

“…Since the arguments are simple, their details are omitted. However, we want to remark that some weighted estimates for multilinear fractional integrals were recently obtained in the condition s > n/(n − α) (p is not necessary larger than n/(n − α)), see [13].…”

“…More elegant estimates must be introduced (see [4,12,13]). Our third observation is that all multilinear operators studied in the literature involving a function Ω are assumed that Ω is independent of the x-variable.…”