Compactness of bilinear singular integral with mild kernel regularity

TL;DR

This paper extends the understanding of compactness criteria for bilinear singular integral operators with mild kernel regularity, matching classical conditions and introducing a new weak compactness property.

Contribution

It generalizes previous results to a broader class of operators and introduces a novel weak compactness condition.

Findings

Characterization of compactness matches classical bilinear T1 theorem conditions.

Introduces a new weak compactness property for bilinear operators.

Extends previous work to operators with mild kernel regularity.

Abstract

This paper extends the characterization of compactness established in \cite{cao2024} to bilinear singular integral operators with mild kernel regularity. The exponent we obtain coincides with the best known sufficient condition for the classical bilinear $T1$ theorem. A novel weak compactness property condition is also introduced.