On entangled and multi-parameter commutators

TL;DR

This paper characterizes the boundedness and compactness of commutators of Zygmund dilation-invariant singular integrals on Lebesgue spaces, revealing that compactness implies the symbol must be constant, a novel insight even for bi-parameter cases.

Contribution

It provides necessary and sufficient conditions for commutator boundedness and compactness for a broad class of Zygmund type singular integrals, clarifying previous ambiguities.

Findings

Compactness of commutators implies the symbol is constant.

Complete characterizations for $p \,\leq\, q$ in Zygmund singular integrals.

Surprising result that compactness always forces the symbol to be constant.

Abstract

We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1, x_2, x_3) \mapsto (s x_1, tx_2, st x_3)$ by proving necessary and sufficient conditions for the boundedness and compactness of commutators $[b,T]$ from $L^p \to L^q$. Previously, only the $p=q$ upper bound in terms of a Zygmund type little $\operatorname{BMO}$ space was known for general operators, and it appears that there has been some confusion about the corresponding lower bound in recent literature. We give complete characterizations whenever $p \le q$ for a general class of non-degenerate Zygmund type singular integrals. Some of the results are somewhat surprising in view of existing papers - for instance, compactness always forces $b$ to be constant. Even in the simpler situation of bi-parameter singular integrals it appears that this has not been observed previously.