Multilinear paraproducts revisited

Abstract

We prove that multilinear paraproducts are bounded from products of Lebesgue spaces $L^{p_1}\times \cdots \times L^{p_{m+1}}$ to $L^{p,\infty }$, when $1\le p_1,\dots ,p_m$, $p_{m+1}<\infty$, $1/p_1+\cdots +1/p_{m+1}=1/p$. We focus on the endpoint case when some indices $p_j$ are equal to $1$, in particular we obtain a new proof of the estimate $L^1\times \cdots \times L^1\to L^{1/(m+1),\infty }$.