Unboundedness of the ball bilinear multiplier operator

Abstract

For all $n > 1$, the characteristic function of the unit ball in $\mathbb{R}^{2n}$ is not the symbol of a bounded bilinear multiplier operator from $L^{p}(\mathbb{R}^{n}) \times L^{q}(\mathbb{R}^{n})$ to $L^{r}(\mathbb{R}^{n})$ when $1/p+1/q = 1/r$ and exactly one of $p$, $q$, or $r' = r/(r-1)$ is less than $2$.