A view from above on ${\text{JN}}_p({\mathbb{R}}^n)$

Abstract

For a symmetric convex body $K\subset {\mathbb{R}}^n$ and $1\le p<\infty$, we define the space $S^p(K)$ to be the tent generalization of ${\text{JN}}_p({\mathbb{R}}^n)$, i.e., the space of all continuous functions $f$ on the upper-half space ${\mathbb{R}}_{+}^{n+1}$ such that \[ \|f\|_{S^p(K)} := \big( \sup_{\mathcal{C}} \sum_{x+tK \in \mathcal{C}} |f(x,t)|^p \big)^{\frac{1}{p}} < \infty, \] where, in the above, the supremum is taken over all finite disjoint collections of homothetic copies of $K$. It is then shown that the dual of $S_0^1(K)$, the closure of the space of continuous functions with compact support in $S^1(K)$, consists of all Radon measures on ${\mathbb{R}}_{+}^{n+1}$ with uniformly bounded total variation on cones with base $K$ and vertex in ${\mathbb{R}}^n$. In addition, a similar scale of spaces is defined in the dyadic setting, and for $1\le p<\infty$, a complete characterization of their duals is given. We apply our results to study ${\text{JN}}_p$ spaces.