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 L2-space.
It extends to a bounded operator on that L2-space,
with operator norm equal to 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 
, with 
and 
, give Bochner-Riesz
means, where 
is the positive part. The symbol 
, with 
and the angular-frequency convention, defines the operator used in a Bessel
potential space on Schwartz functions.
