Consider a general bilinear multiplier operator: $$ T(f,g)(n)=\int_{\Pi}\int_{\Pi}\hat{f}(\xi)\hat{g}(\eta)e^{2\pi i(\xi+\eta)n}m(\xi,\eta)d\xi d\eta, $$ where $\Pi$ is the torus, $n\in\mathbb{Z}$, $m$ is periodic in both variables, $f$ and $g$ are compactly supported functions defined on $\mathbb{Z}$ and $\hat{f}(\xi):=\sum_nf(n)e^{-2\pi in\xi}$.

**Prove or disprove:** $\|T(f,g)\|_{L^2(\mathbb{Z})}\le C\|m\|_{L^2(\Pi^2)}\|f\|_{L^2(\mathbb{Z})}\|g\|_{L^2(\mathbb{Z})}$.

The motivation of my question is the following: It is well-known that in the linear case the $L^2$-norm of a multiplier operator is just the $L^{\infty}$-norm of the multiplier. A bilinear analogue of this fact is that the $L^2\times L^2\to L^2$ norm of a bilinear multiplier operator is at most the $L^{\infty}$-norm of the multiplier (which is sometimes called "symbol"). It is natural to ask if we can get a better estimate, say $L^2$-norm of the multipler (instead of $L^{\infty}$-norm) in the bilinear setting.