The Marcinkiewicz multiplier condition for bilinear operators

Abstract

This article is concerned with the question of whether Marcinkiewicz multipliers on ${\mathbb{R}}^{2n}$ give rise to bilinear multipliers on ${\mathbb{R}}^n\times {\mathbb{R}}^n$. We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.

42 pages