# The space $JN_p$: nontriviality and duality

## TL;DR

This paper investigates the properties of the function space $JN_p$, establishing its nontriviality, its relation to $L^p$, and its duality with a new Hardy-type space $HK_{p'}$, revealing its complex structure.

## Contribution

The paper constructs a function in $JN_p$ not in $L^p$, shows $JN_p$ and $L^p$ coincide for monotone functions, and characterizes $JN_p$ as the dual of a new Hardy-like space.

## Key findings

- $JN_p$ contains functions not in $L^p$
- $JN_p$ and $L^p$ coincide for monotone functions
- $JN_p$ is dual to a new Hardy-type space $HK_{p'}$

## Abstract

We study a function space $JN_p$ based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that $L^p\subset JN_{p}\subsetneq L^{p,\infty}$, but otherwise the structure of $JN_p$ is largely a mystery. Our first main result is the construction of a function that belongs to $JN_p$ but not $L^p$, showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes $JN_{p}$ and $L^p$ do coincide. Our second main result describes $JN_p$ as the dual of a new Hardy kind of space $HK_{p'}$.