Lp multiplier

Summary

An $L^p$ multiplier is a bounded function $m$ on $\R^n$ such that the operator $f \mapsto (m \hat f)^\vee$ is bounded on the $L^p$ space $L^p(\R^n)$. It is a Fourier multiplier that respects the size of functions measured in the $L^p$ norm. For $p = 2$ every bounded function is an $L^p$ multiplier, but for $p \ne 2$ the question is subtle. Many operators of analysis and PDE are multiplier operators, such as partial sums of Fourier integrals and Riesz transforms. So criteria for $L^p$ multipliers are a central topic of harmonic analysis.