Choquet integrals, Hausdorff content and sparse operators

Abstract

Let $H^d$, $0<d<n$, be the dyadic Hausdorff content of the $n$-dimensional Euclidean space ${\mathbb{R}}^n$. It is shown that $H^d$ counts a~Cantor set of the unit cube $[0,1{)}^n$ as $\approx 1$, which implies unboundedness of the sparse operator ${\mathcal{A}}_{\mathcal{S}}$ on the Choquet space ${\mathcal{L}}^p(H^d)$, $p>0$. In this paper we verify that the sparse operator ${\mathcal{A}}_{\mathcal{S}}$ maps ${\mathcal{L}}^p(H^d)$, $1\le p<\infty$, into an associate space of Orlicz-Morrey space ${{\mathcal{M}}_{{\Phi }_0}^{p^{\prime }}(H^d)}^{\prime }$, ${\Phi }_0(t)=t\log (e+t)$. We also give another characterizations of those associate spaces using the tiling $\mathcal{T}$ of ${\mathbb{R}}^n$.