Bounded oscillation operators on BMO spaces

TL;DR

This paper investigates the boundedness of Bounded Oscillation (BO) operators on BMO spaces, establishing new local bounds and applications to harmonic analysis operators and wavelet systems.

Contribution

It proves that BO operators map L-infinity into BMO and, under certain conditions, BMO into itself, extending their boundedness properties in harmonic analysis.

Findings

BO operators map L-infinity to BMO

Under logarithmic localization, BO operators map BMO into itself

Applications include BMO estimates for Calderón-Zygmund and martingale transforms

Abstract

Bounded Oscillation (BO) operators were recently introduced in the author's paper [13], where it was proved that many operators in harmonic analysis (Calder\'on-Zygmund operators, Carleson type operators, martingale transforms, Littlewood-Paley square functions, maximal operators, etc) are $BO$ operators. $BO$ operators are defined on abstract measure spaces equipped with a basis of abstract balls. The abstract balls in their definition owe four basic properties of classical balls in $\mathbb{R}^n$, which are crucial in the study of singular operators on $\mathbb{R}^n$. Among various properties studied in these papers it was proved that $BO$ operators allow pointwise sparse domination, establishing the $A_2$-conjecture for those operators. In the present paper we study boundedness properties of $BO$ operators on $BMO$ spaces. In particular, we prove that general $BO$ operators boundedly map $L^\infty$ into $BMO$, and under a logarithmic localization condition those map $BMO$ into itself. We obtain these properties as corollaries of new local type bounds, involving oscillations of functions over the balls. We apply the results in the $BMO$ estimations of Calder\'on-Zygmund operators, martingale transforms, Carleson type operators, as well as in the unconditional basis properties of general wavelet type systems in atomic Hardy spaces $H^1$.