We consider Malliavin calculus based on the Itô chaos decomposition of square integrable random variables on the Lévy 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 Lévy process is a compound Poisson process on a finite time interval.
"A note on Malliavin smoothness on the Lévy space." Electron. Commun. Probab. 22 1 - 12, 2017. https://doi.org/10.1214/17-ECP65