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

Eddy Kwessi

doi:10.1515/dema-2021-0013

Abstract

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 {H}^{1} strictly contains the special atom space.

Highlights

The space of functions of bounded means oscillation has taken central stage in the mathematical literature after the work of Charles Fefferman [1], where he showed that it is the dual space of the real Hardy space 1, a long sought-after result
Since the dual space ( 1)∗ of 1 is BMO and B1 ⊂ 1, it follows that the dual space (B1)∗ of B1 must be a superset of BMO
It was already proved that (B1)* ≅ Λ∗′, where Λ∗′ is the space of derivative of functions in the Zygmund class Λ∗, see, for example, [3] and [4]

Summary

Introduction

The space of functions of bounded means oscillation has taken central stage in the mathematical literature after the work of Charles Fefferman [1], where he showed that it is the dual space of the real Hardy space 1, a long sought-after result. We can define the space of functions of bounded mean oscillation and its weak counterpart. The space of functions of bounded mean oscillation is defined as the space of locally integrable functions f for which the operator M# is bounded, that is, BMO( n) = {f ∈ Ll1oc( n) : M#( f ) ∈ L∞( n)}. The space of functions of weak bounded mean oscillation is defined as the space of locally integrable functions f for which the operator M is bounded, that is, BMOw( n) = {f ∈ Ll1oc( n) : Mf ∈ L∞( n)}. Following the work of Girela in [6] on the space of analytic functions of bounded means oscillations, we introduce their weak counterpart. We can define the space BMOAw of analytic function of weak bounded mean oscillation.

Main results

Discussion