A view from above on JNp(Rn)

Abstract

For a symmetric convex body $K\subset R^n$ and $1\le p<\infty$, we define the space $S^p\left(K\right)$ to be the tent generalization of ${\text{JN}}_p\left(R^n\right)$, i.e., the space of all continuous functions f on the upper-half space $R_{+}^{n+1}$ such that${\Vert f\Vert }_{S^p\left(K\right)}:={(\underset{C}{\sup }\sum _{B\in C}|f_B{|}^p)}^{\frac{1}{p}}<\infty ,$ 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 $S_0^1\left(K\right)$, the closure of the space of continuous functions with compact support in $S^1\left(K\right)$, consists of all Radon measures on $R_{+}^{n+1}$ with uniformly bounded total variation on cones with base K and vertex in $R^n$. 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 dyadic ${\text{JN}}_p$ spaces.