Bound partial paraproducts for entangled Zygmund singular integrals

Establish boundedness of partial paraproducts and thereby develop a full, non-cancellative dyadic representation or $T1$ theory for general entangled Zygmund singular integrals.

Background

Partial paraproducts are hybrid shift-paraproduct operators that arise in full T1T1 representation theorems and are described as particularly difficult in entangled settings. The paper explains that its flag methods successfully handle these operators and yield a complete flag T1T1 theorem, while the earlier Zygmund theory was restricted to cancellative singular integrals because the corresponding partial paraproducts had not been controlled. Thus, extending the methods to obtain a full Zygmund T1T1 theory remains unresolved in the stated discussion.

References

We did not know how to handle them in the entangled Zygmund setting, and this was the main reason our Zygmund theory was restricted to the cancellative case.

— Flag multiresolutions: T1, paraproducts and commutators  (2609.38092 - Foster et al., 29 Sep 2026) in Section 1, subsection “The dyadic flag $T1$ representation theorem”