A new $A_p$-$A_{\mathrm{\infty }}$ estimate for Calder\'on-Zygmund operators in spaces of homogeneous type

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