Euclidean L2 Fourier multiplier

For $m\in L^\infty(\mathbb R^n)$ , the $L^2$ Fourier multiplier with symbol $m$ is the operator $T_m$ determined by

\widehat{T_mf}=m\widehat f\qquad(f\in L^2(\mathbb R^n)).

Plancherel's theorem makes it a bounded operator, with $\|T_m\|_{L^2\to L^2}=\|m\|_\infty$ , where the latter is the essential supremum.

Scope

The upper bound follows by multiplying inside the Fourier $L^2$ norm. The reverse bound follows by taking a nonzero Fourier function supported on a finite-measure subset where $|m|$ is close to its essential supremum. Boundedness of $m$ alone does not imply $L^p$ boundedness for other $p$ . Nor does it define multiplication of arbitrary tempered distributions by $m$ .