Let $M_p$ be the class of Fourier multipliers with $L^2$-Boundedness, i.e.
$M_p := \{ m:\mathbb{R}^N \rightarrow \mathbb{R} \ | \ m \ \text{ is measurable}, \ \exists \ C>0 \ \text{such that} \ ||T_m f||_p \leq C||f||_p \ \forall f \in \mathscr{S}(\mathbb{R}^N) \} $,
where $T_m = \mathcal{F}^{-1} [m \hat{f}] $ is a Fourier Multiplier, $\mathscr{S}$ is the Schwartz class of functions, and $||\cdot||_p$ is the $L^p$-Norm.
We are asked to prove that $M_2 = L^\infty$.
It is easy to prove that $L^\infty \subseteq M_2 $. For $m \in L^\infty$, and $f \in \mathscr{S}$, $||T_m f||_2 = || \mathcal{F}[T_m f] ||_2 = ||m \hat{f}||_2 \leq ||m||_\infty ||\hat{f}||_2 = ||m||_\infty ||f||_2 $.
Here, we have simply used Holder's Inequality and the equivalence of $L^2$-Norms under Fourier Transformation.
I am however having trouble proving the inverse, $M_2 \subseteq L^\infty $.
We let $m \in M_2$. Then we want to show that $\exists ||m||_\infty<\infty$.
We know that there exists a constant $C>0$, such that, for all $f \in \mathscr{S}$, $||T_m f ||_2 = ||m \hat{f}||_2 \leq C||f||_2 $.
It is easy to show that, if $m$ is a constant function, then $||m||_\infty \leq C $, however I cannot generalise this result to just measurable functions.
Any hints/sources are appreciated. Thank you