Fourier multiplier

Summary

A Fourier multiplier is an operator that acts on a function by multiplying its Fourier transform by a fixed function $m$, called the symbol. Differential operators with constant coefficients, the Hilbert transform, the heat semigroup and frequency filters are all of this form. Every symbol in $L^\infty$ gives a bounded operator on $L^2$, and the bounded operators on $L^2(\R^n)$ that commute with translations are exactly these. Which symbols give bounded operators on $L^p$ for $p \ne 2$ is a central and hard question of harmonic analysis, studied through $L^p$ multipliers. See the definition.