On the equivalence between weak BMO and the space of derivatives of the Zygmund class

TL;DR

This paper establishes the duality between the weak BMO space and the space of derivatives of Zygmund class functions, clarifying their relationship and implications for the structure of Hardy and atom spaces.

Contribution

It proves the equivalence between weak BMO and derivatives of Zygmund functions, and shows that Hardy space H^1 strictly contains the special atom space.

Findings

Weak BMO is the dual of the special atom space.

The dual of the special atom space is the space of derivatives of Zygmund functions.

H^1 strictly contains the special atom space.

Abstract

In this paper, we will discuss the space of functions of weak bounded mean oscillation. In particular, we will show that this space is the dual space of the special atom space, whose dual space was already known to be the space of derivative of functions (in the sense of distribution) belonging to the Zygmund class of functions. We show in particular that this proves that the Hardy space $H^1$ strictly contains the space special atom space.