On two weight estimates for iterated commutators

TL;DR

This paper extends the bump conjecture to iterated commutators, providing new necessary conditions for their two-weight boundedness and related results, even for first-order cases.

Contribution

It introduces a new bump type necessary condition for two-weight boundedness of iterated commutators and extends existing conjectures to these operators.

Findings

New necessary bump conditions for iterated commutators

Extension of bump conjecture to higher-order commutators

Results related to a converse of Bloom's theorem

Abstract

In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators $T_b^m$. Our results are new even for the first order commutator $T_b^1$. A new bump type necessary condition for the two-weighted boundedness of $T_b^m$ is obtained as well. We also provide some results related to a converse to Bloom's theorem.