Fractional maximal operators with weighted Hausdorff content

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 $H_w^d$, $0<d\le n$. 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.