A note on Malliavin smoothness on the L\'evy space

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.