Weak type (1,1) bound criterion for maximal truncated singular integral operators with rough kernels

This paper proves the weak type (1,1) boundedness for maximal truncated singular integral operators with rough non-convolution kernels of the form $Ω(x-y) K(x,y)$, where $Ω$ is homogeneous of degree zero with $Ω\in L\log L(\sS^{d-1})$ and $K$ is a standard Calderón-Zygmund kernel. This extends the r