Fractional maximal operators with weighted Hausdorff content

TL;DR

This paper establishes weighted estimates for fractional maximal operators on weighted Hausdorff content spaces, generalizing previous results and including new Fefferman-Stein type inequalities with arbitrary weights.

Contribution

It provides a general framework for weighted estimates of fractional maximal operators on weighted Hausdorff content spaces, extending prior work and including new inequalities.

Findings

Derived weighted estimates for fractional maximal operators.

Established Fefferman-Stein type inequalities with arbitrary weights.

Unified previous results as special cases, including Tang's result.

Abstract

Let $n\ge 2$ be the spatial dimension. The purpose of this note is to obtain some weighted estimates for the fractional maximal operator ${\mathfrak M}{\alpha}$ of order $\alpha$, $0\le\alpha<n$, on the weighted Choquet-Lorentz space $L^{p,q}(H_{w}^{d})$, where the weight $w$ is arbitrary and the underlying measure is the weighted $d$-dimensional Hausdorff content $H^{d}_{w}$, $0<d\le n$. Concerning a dependence of two parameters $\alpha$ and $d$, we establish a general form of the Fefferman-Stein type inequalities for ${\mathfrak M}_{\alpha}$. Our results contain the works of Adams, \cite{Ad} and of Orobitg and Verdera \cite{OV} as the special cases. Our results also imply the Tang result \cite{Ta}, if we assume the weight $w$ is in the Muckenhoupt $A_{1}$-class.