On Sharpness of $L\log L$ Criterion for Weak Type $(1,1)$ boundedness of rough operators

In this note, we show that the $L\log L$ hypothesis is the strongest size condition on a homogeneous rough function on the sphere which ensures the weak type $(1,1)$ boundedness of the corresponding singular integral $T_{\mathrm{\Omega }}$, provided $T_{\mathrm{\Omega }}$ is bounded in $L^2$.