By developing the Littlewood-Paley characterization for product homogeneous Lipschitz spaces Lip(α₁,α₂)(Rⁿ × R^m) and Lip_b(α₁,α₂)(Rⁿ × R^m), and establishing a density argument for Lip_b(α₁,α₂)(Rⁿ × R^m) in the weak sense, we give a Tb theorem for the generalized singular integral operator on Lip_b(α₁,α₂)(Rⁿ × R^m), where b(x,y) = b₁(x)b₂(y), b₁, b₂ are para-accretive functions on Rⁿ and R^m, respectively.

This research was supported by National Natural Science Foundation of China (Grant No. 11626213, 11771399, 11671357) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ17A010002).