$Tb$ theorems for Triebel-Lizorkin spaces over special spaces of homogeneous type and their applications

The author first establishes the $Tb$ theorems for Triebel-Lizorkin spaces by the discrete Calderón type reproducing formula and the Plancherel-P\^{o}lya characterization for the Triebel-Lizorkin spaces. As an application of the $Tb$ theorems, new characterizations of Triebel-Lizorkin spaces with minimum regularity and cancellation conditions are given over special spaces of homogeneous type.