A Fourier restriction operator maps a Schwartz function 
on 
to its Fourier transform on a specified surface 
,
 |
Restriction estimates ask for exponents 
for which
 |
where 
is surface measure and 
is a constant independent of 
. When 
,
such an estimate extends uniquely to the Lp-space,
since Schwartz functions are dense
in that space. For such data, the Fourier
transform need not have pointwise values on the surface,
so this bounded operator is the meaning of restriction.
For 
, the dual estimate is for the Fourier
extension operator from 
to 
,
where 
and 
. The Fourier
restriction conjecture specifies expected sharp ranges for curved surfaces.
