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

TL;DR

This paper introduces a new family of function spaces generalizing $ ext{JN}_p( ext{R}^n)$ using convex bodies, characterizes their duals, and applies these results to classical $ ext{JN}_p$ spaces.

Contribution

It defines the space $S^p(K)$ for symmetric convex bodies, characterizes the dual space of $S^1_0(K)$, and extends the analysis to dyadic spaces, advancing the understanding of $ ext{JN}_p$ spaces.

Findings

Dual of $S^1_0(K)$ consists of Radon measures with bounded variation on cones.

Complete dual characterization for dyadic $S^p(K)$ spaces.

Application of duality results to classical $ ext{JN}_p$ spaces.

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 $S^1_0(K)$, 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.