Logarithmic mean oscillation on the polydisc, endpoint results for multi-parameter paraproducts, and commutators on BMO

TL;DR

This paper investigates the boundedness of multiparameter paraproducts and commutators on dyadic BMO spaces, introducing logarithmic mean oscillation on the polydisc to establish endpoint results in harmonic analysis.

Contribution

It introduces a new notion of logarithmic mean oscillation on the polydisc and applies it to endpoint boundedness results for multiparameter paraproducts and commutators.

Findings

Boundedness of multiparameter paraproducts on dyadic BMO spaces.

Endpoint boundedness of iterated commutators on BMO.

Introduction of logarithmic mean oscillation concept for polydiscs.

Abstract

We study boundedness properties of a class of multiparameter paraproducts on the dual space of the dyadic Hardy space H_d^1(T^N), the dyadic product BMO space BMO_d(T^N). For this, we introduce a notion of logarithmic mean oscillation on the polydisc. We also obtain a result on the boundedness of iterated commutators on BMO([0,1]^2).