On entangled and multi-parameter commutators

Abstract

We complement the recent theory of general singular integrals $T$ invariant under the Zygmund dilations $(x_1,x_2,x_3)\mapsto (s{x}_1,t{x}_2,st{x}_3)$ 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 $\mathrm{BMO}$ 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.

27 pages