The Riesz-Herz equivalence for capacitary maximal functions

We prove a Riesz-Herz estimate for the maximal function associated to a capacity C on ℝ n , M C f(x)=sup Q∋x C(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 \((L^{1},{\mathcal{L}}^{1,C})_{1/p',p}\), where \({\mathcal{L}}^{1,C}\) denotes the Morrey space based on a capacity.