Weighted Boundedness of a Composition of Paraproducts

In this paper we offer alternate upper bound for the operator ${\mathrm{\Pi }}_b^{\ast }{\mathrm{\Pi }}_d$ to the ones present in literature, thus extending the known upper bounds from the $L^2(\mathbb{R})$ setting to $L^p(w)$, for $1<p<\mathrm{\infty },$ and a Muckenhoupt weight $w$. In the $L^2(w)$ setting, we fully characterize the boundedness of the operator.