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.
