A note on bi-linear multipliers

Abstract

In this paper we prove that if ${\chi }_{{}_E}(\xi -\eta )-$ the indicator function of measurable set $E\subseteq {\mathbb{R}}^d,$ is a bi-linear multiplier symbol for exponents $p,q,r$ satisfying the Hölder's condition $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and exactly one of $p,q,$ or $r^{\prime }=\frac{r}{r-1}$ is less than $2,$ then $E$ is equivalent to an open subset of ${\mathbb{R}}^d.$