Bi-parameter paraproducts

Abstract

In the first part of the paper we prove a bi-parameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the Kato-Ponce inequality in nonlinear PDE, obtaining a fractional Leibnitz rule for derivatives in the $x_1$ and $x_2$ directions simultaneously. Then, we show that the double bilinear Hilbert transform does not satisfy any $L^p$ estimates.

26 pages; no figures. A missing hypothesis on the exponents in some of the estimates, pointed out to us by Loukas Grafakos and Seungly Oh, has been added