  In this article, we give a new proof of the It\^o formula for some integral
processes related to the space-time L\'evy white noise introduced in Balan
(2015) as an alternative for the Gaussian white noise perturbing an SPDE. We
discuss two applications of this result, which are useful in the study of SPDEs
driven by a space-time L\'evy noise with finite variance: a maximal inequality
for the $p$-th moment of the stochastic integral, and the It\^o representation
theorem leading to a chaos expansion similar to the Gaussian case.
