The Riesz-Herz equivalence for capacitary maximal functions

We prove a Riesz-Herz estimate for the maximal function associated to a capacity C on ℝn, MCf(x)=sup Q∋xC(Q)−1∫Q|f|, which extends the equivalence (Mf)∗(t)≃f∗∗(t) for the usual Hardy-Littlewood maximal function Mf. The proof is based on an extension of the Wiener-Stein estimates for the distribution function of the maximal function, obtained using a convenient family of dyadic cubes. As a byproduct we obtain a description of the norm of the interpolation space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(L^{1},{\mathcal{L}}^{1,C})_{1/p',p}$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal{L}}^{1,C}$\end{document} denotes the Morrey space based on a capacity.