Logarithmic mean oscillation on the Polydisc, Multi-parameter paraproducts and iterated commutators

TL;DR

This paper introduces a new concept of bounded logarithmic mean oscillation on the N-torus, characterizes it via multi-parameter paraproducts, and explores conditions for boundedness of iterated commutators with Hilbert transforms.

Contribution

It defines a novel bounded logarithmic mean oscillation space and links it to multi-parameter paraproducts and commutator boundedness in harmonic analysis.

Findings

Equivalent definition of bounded logarithmic mean oscillation in terms of paraproduct boundedness

Characterization of the space via dyadic little BMO and product BMO

Sufficient condition for boundedness of iterated commutators with Hilbert transforms

Abstract

We introduce another notion of bounded logarithmic mean oscillation in the N-torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little BMO of Cotlar-Sadosky to the product BMO of Chang-Fefferman. We also obtain a sufficient condition for the boundedness of iterated commutators with Hilbert transforms betweeen the strong notions of these two spaces.