  We study Daubechies' time-frequency localization operator, which is
characterized by a window and weight function. We consider a Gaussian window
and a spherically symmetric weight as this choice yields explicit formulas for
the eigenvalues, with the Hermite functions as the associated eigenfunctions.
Inspired by the fractal uncertainty principle in the separate time-frequency
representation, we define the $n$-iterate spherically symmetric Cantor set in
the joint representation. For the $n$-iterate Cantor set, precise asymptotic
estimates for the operator norm are then derived up to a multiplicative
constant.
