  We consider Malliavin calculus based on the It\^o chaos decomposition of
square integrable random variables on the L\'evy space. We show that when a
random variable satisfies a certain measurability condition, its
differentiability and fractional differentiability can be determined by
weighted Lebesgue spaces. The measurability condition is satisfied for all
random variables if the underlying L\'evy process is a compound Poisson process
on a finite time interval.
