  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.
