Project Euclid - mathematics and statistics online

In this article, we prove the boundedness of the parametric Marcinkiewicz integral and its higher-order commutators generated by BMO spaces on the variable Morrey-Herz space. All the results are new even when α · is a constant.

The aim of this paper is to establish the intrinsic square function characterizations in terms of the intrinsic Littlewood–Paleyg-function, the intrinsic Lusin area function, and the intrinsicgλ∗-function of the variable Hardy–Lorentz spaceHp⋅,qℝn, forp⋅being a measurable function onℝnsatisfying0<p−≔ess infx∈ℝnpx≤ess supx∈ℝnpx≕p+<∞and the globally log-Hölder continuity condition andq∈0,∞, via its atomic and Littlewood–Paley function characterizations.

Let $(X,d,\mu )$ be a space of homogenous type and L be a non-negative self-adjoint operator on $L^2\left(X\right)$ with heat kernels satisfying Gaussian upper bounds. In this paper, we introduce the variable Besov–Morrey space associated with the operator L and prove that this space can be characterized via the Peetre maximal functions. Then, we establish its atomic decomposition.