Weak type $(1,1)$ estimates for noncommutative rough singular integrals

Abstract

We establish the weak type \((1,1)\) endpoint estimate for operator-valued rough singular integrals with kernels satisfying the classical \(L\log L\) integrability and cancellation conditions. This extends Seeger's endpoint theorem [{J. Amer. Math. Soc.} \textbf{9} (1996)] to the noncommutative setting. A key ingredient is an off-diagonal analysis that controls interactions between different kernel scales and avoids an additional logarithmic loss.

arXiv comment30 pages, nonhomogeneous case added