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.