  In this note, we study the $A_p$-$A_\infty$ estimate for Calder\'on-Zygmund
operators in terms of the weak $A_\infty$ characteristics in spaces of
homogeneous type. The weak $A_\infty$ class was introduced recently by
Anderson, Hyt\"onen and Tapiola. Our estimate is new even in the Euclidean
space.
