The Kakeya maximal function at scale 
averages a function
over thin unit-length tubes and takes the supremum over
their translations in each direction. In 
, for 
and a unit vector 
, put
 |
This cylinder has volume 
. It is defined by
 |
The output is a function of direction on the sphere 
. The analogous definition in 
uses volume 
, where 
is the volume of the unit
ball in 
. It satisfies 
pointwise and 
.
