Lp multiplier

Remarks

Example: for $h \in \R^n$, let $m(\xi) = e^{2\pi i \xi \cdot h}$, with the convention $\hat f(\xi) = \int f(x) e^{-2\pi i x \cdot \xi} \, dx$. Then $T_m f(x) = f(x + h)$, so $\norm{T_m f}_{L^p} = \norm{f}_{L^p}$ and $m \in \mathcal{M}_p$ with norm $1$ for every $p$.

Non-example: in $\R^1$, $m(\xi) = \xi^2$ is not bounded, so it is not an $L^p$ multiplier. Indeed, for $p = 2$ the Plancherel theorem gives $\norm{T_m f}_{L^2} = \norm{\xi^2 \hat f}_{L^2}$, and its ratio to $\norm{f}_{L^2} = \norm{\hat f}_{L^2}$ is unbounded when $\hat f$ is a fixed bump moved far out.

By the Plancherel theorem, $\mathcal{M}_2 = L^\infty(\R^n)$ with equal norms. By duality, $\mathcal{M}_p = \mathcal{M}_{p'}$ when $1/p + 1/p' = 1$. The space $\mathcal{M}_1 = \mathcal{M}_\infty$ consists of the Fourier transforms of finite Borel measures. By the Riesz–Thorin theorem, $\mathcal{M}_p \subseteq \mathcal{M}_q$ when $q$ lies between $p$ and $2$.