A Fourier multiplier is an operator 
defined by multiplying the Fourier
transform by a prescribed symbol 
,
 |
For a bounded measurable function 
and a Schwartz function 
, the inverse Fourier transform
defines 
in 
.
Its 
operator norm is the essential
supremum of 
by Plancherel's theorem. Boundedness on other
Lp-spaces is a separate question.
The symbol 
gives the identity operator. The symbols 
give Bochner-Riesz
means, and 
,
with the angular-frequency convention, defines the operator
used in a Bessel potential space.
