A note on bi-linear multipliers

In this paper we prove that if $χ_{_E}(ξ-η)-$ 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 $r'=\frac{r}{r-1}$ is less than $2,$ then $E$ is equivalent to an open subset of $\mathbb{R}^d.$

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