We define the set $M_p$, for $1 \leq p \leq \infty$, as follows:
$M_p := \{ m : \mathbb{R}^N \rightarrow \mathbb{R}, \ m \ \text{measurable} \ | \ \exists C > 0, \ \text{such that } ||T_mf||_p \leq C ||f||_p, \ \forall f \in \mathscr{S}(\mathbb{R}^N) \} $,
where $\mathscr{S}(\mathbb{R}^N)$ is the class of Schwartz functions, and $T_m f := \mathcal{F}^{-1}[m \hat{f}]$ is a Fourier Multiplier. ($\mathcal{F}$ denotes the Fourier Transform).
We wish to prove that $M_p = M_{q}$, where $\frac{1}{p} + \frac{1}{q} = 1 $.
I have no idea where to start with this. It doesn't seem like we can apply Holder's inequality to $||T_mf||_p$, as $T_m$ is not a function or distribution itself.
I am aware that you can define a Fourier Multiplier $T_m$ using Calderon-Zygmund Operator $K$, giving $T_mf := K \ast f $. But is it the case that all Fourier Multipliers can be written in this fashion? And if so, what can we say about $K$? Do we know it is integrable/ regular/ nice in any way?
All hints/sources appreciated. Thank you.