In lecture notes I have, it is mentioned that the characteristic function $\chi_I$ of an interval in $\mathbb{R}$ is an $L^p$ Fourier multiplier for $1 < p < \infty$. I thought this would be easy to check, but I'm simply not seeing it.
I need to show that for any Schwartz $f$, we have $$ \| \mathcal{F}^{-1}[\chi_I \mathcal{F}f] \|_p \leq C||f||_p.$$ Since $f \in \mathcal{S}$, there's nothing too strange happening here -- the transforms are still functions.
Using duality, density, and the Fourier multiplication formula, it suffices to show that
$$ \left| \int_I [\mathcal{F}f(\xi)]\; [\mathcal{F}^{-1}g(\xi)] \; d\xi \right| \leq C \|f\|_p\|g\|_{p'}$$ for any $g \in \mathcal{S}$, where $p'$ is the Holder conjugate of $p$. I don't know how to show this though. Any guidance would be welcome.
The only way to do this is to go via the Hilbert Transform $H$, whose multiplier is some multiple of $i(I_{(0,\infty)} - I_{(-\infty,0)})$. The $L^p$ boundedness of the Hilbert transform was proved by M. Riesz Theorem, and it was considered quite a breakthrough when it was first proven.