  For $1< p <2$ we obtain sharp inequalities for the supremum of products of
homogeneous polynomials on $L_p(\mu)$, whenever the number of factors is no
greater than the dimension of these Banach spaces (a condition readily
satisfied in the infinite dimensional settings). The results also holds for the
Schatten classes $\mathcal S_p$. For $p>2$ we present some estimates on the
involved constants.
