Fourier multiplier

Remarks

By Plancherel, $T_m$ is a bounded linear operator on $L^2(\R^n)$ with operator norm equal to $\norm{m}_{L^\infty}$. It commutes with every translation, and conversely every bounded operator on $L^2(\R^n)$ that commutes with all translations is $T_m$ for a unique $m \in L^\infty(\R^n)$.

Example: on $\R$, the symbol $m(\xi) = -i \operatorname{sgn}(\xi)$ is bounded, and $T_m$ is the Hilbert transform. Example: for $t > 0$, the symbol $m(\xi) = e^{-4\pi^2 t \abs{\xi}^2}$ gives the heat semigroup: $u(t, \cdot) = T_m f$ solves $\partial_t u = \Delta u$ with $u(0, \cdot) = f$, because $\Delta$ acts on the Fourier side as multiplication by $-4\pi^2 \abs{\xi}^2$.

Non-example: the symbol $m(\xi) = 2\pi i \xi$ on $\R$ is not bounded; the corresponding operator is differentiation, which is not defined on all of $L^2(\R)$. It is still a Fourier multiplier in the wider sense of an operator on the Schwartz space or on tempered distributions.

For the torus $\mathbb{T}^n$ the same definition uses a bounded sequence $(m(k))_{k \in \Z^n}$ that multiplies the Fourier coefficients. The symbol $m$ is called an $L^p$ multiplier when $T_m$, restricted to $L^2 \cap L^p$, extends to a bounded operator on $L^p$.