On the moments of a multiple Wiener-Ito integral and the space induced by the polynomials of the integral

In the first part of this note we derive a lower bound for E\Ip(f)\m where Ip(f) is the multiple Wiener-Ito integral of the kernel f(t1,-.,tp), t∈T and T=[0, ∈). In the second part we consider the linear space H{Ip(f)) of the L2 functionals induced by the random variables {(Ip(f))m,meN}. For ≥ it is not known whether H{Ipf} is the same space of L2 functionals which are measurable with respect to the subsigma field induced by the random variable Ip(f). In the final part of this note some special results concerning the conditional expectation on the Wiener space are derived. In particular, it is shown that the conditional expectation of a random variable belonging to an odd chaos given the even chaos is zero.