Fiber-wise Calderón–Zygmund decomposition and application to a bi-dimensional paraproduct

We are interested in a new kind of bi-dimensional bilinear paraproducts (appearing in (Amer. J. Math. 132 (2010) 201–256)), which do not fit into the setting of bilinear Calderón–Zygmund operators. In this paper, we propose a fiber-wise Calderón–Zygmund decomposition, which is specially adapted to this kind of bi-dimensional paraproduct.