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 the unboundedness of the sparse operator \({{\mathcal {A}}}_{{{\mathcal {S}}}}\) on the Choquet space \({\mathcal L}^p(H^d)\), \(p>0\). In this paper, the sparse operator \({\mathcal A}_{{{\mathcal {S}}}}\) is proved to map \({{\mathcal {L}}}^p(H^d)\), \(1\le p<\infty \), into an associate space of the Orlicz-Morrey space \({{{\mathcal {M}}}^{p'}_{\Phi _0}(H^d)}'\), \(\Phi _0(t)=t\log (e+t)\). Further, another characterization of those associate spaces is given by means of the tiling \({{\mathcal {T}}}\) of \({{\mathbb {R}}}^n\).