Lower bounds for norms of products of polynomials on $L_p$ spaces

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.