New estimates for bounded oscillation operators

TL;DR

This paper introduces a generalized framework for bounded oscillation operators, extending their scope to vector-valued spaces, and establishes new sparse domination and decay estimates with applications in harmonic analysis.

Contribution

It presents a new, simplified approach to bounded oscillation operators, including a broader class with parameters and vector-valued considerations, and introduces novel sparse estimation techniques.

Findings

Proved sparse domination and exponential decay estimates.

Established boundedness of maximally modulated Calderón-Zygmund operators on BMO.

Recovered recent results and introduced new sparse estimation methods.

Abstract

In this paper we consider a generalized version of bounded oscillation operators, involving new parameters in the definition, as well as considering the operators on vector-valued function spaces. With this definition we will capture some more operators to the class of Bounded oscillation operators. We prove sparse domination and exponential decay estimates, with various applications in harmonic analysis operators. We provide a new simplified approach, separating certain set theoretic proposition, which become a basic tool in the proofs of the main results. For a "narrowed" class of bounded oscillation operators we also obtain new type of sparse estimation, involving mean oscillation instead of integral averages in the definition of sparse operators. Among with new corollaries we recover also series of results obtained in recent years. In particular, we prove the boundedness of maximally modulated Calder\'on-Zygmund operators on spaces BMO.