Wick calculus for the square of a Gaussian random variable with application to Young and hypercontractive inequalities

We investigate a probabilistic interpretation of the Wick product associated to the chisquare distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian7 and Poissonian12 cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.