A Fourier extension operator associated with a surface 
and surface measure 
is the operator
 |
For integrable 
,
this is the inverse Fourier transform of the
measure 
.
It is dual to the Fourier restriction
operator under the integral pairing.
For the unit sphere 
with surface measure 
, the constant
function 
gives
 |
for 
, with value 
at 
.
This example belongs to 
for 
but fails to belong to 
. It explains the strict output exponent in the three-dimensional
Fourier restriction conjecture.
