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 ${ { \mathcal M}^{ p'}_{ \Phi_0}(H^d)}'$, $\Phi_0(t)=t\log(e+t)$. We also give another characterizations of those associate spaces using the tiling ${ \mathcal T}$ of ${ \mathbb R}^n$.
Categories: math.FA