We introduce a ``weak $CMO$'' space $WCMO$ and consider the boundedness of the singular integral operators from $CMO$ to $WCMO$. Our result is the best possible.
"Some weak-type estimates for singular integral operators on $CMO$ spaces." Hokkaido Math. J. 39 (1) 115 - 126, February 2010. https://doi.org/10.14492/hokmj/1274275022