A H\"older--Young--Lieb inequality for norms of Gaussian Wick products

An important connection between the finite dimensional Gaussian Wick product and Lebesgue convolution product will be proven first. Then this connection will be used to prove an important H\"older inequality for the norms of Gaussian Wick products, reprove Nelson hypercontractivity inequality, and prove a more general inequality whose marginal cases are the H\"older and Nelson inequalities mentioned before. We will show that there is a deep connection between the Gaussian H\"older inequality and classic H\"older inequality, between the Nelson hypercontractivity and classic Young inequality with the sharp constant, and between the third more general inequality and an extension by Lieb of the Young inequality with the best constant. Since the Gaussian probability measure exists even in the infinite dimensional case, the above three inequalities can be extended, via a classic Fatou's lemma argument, to the infinite dimensional framework.