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

TL;DR

This paper establishes sharp bounds for the norms of products of homogeneous polynomials on $L_p$ spaces for $1<p<2$, with additional estimates for $p>2$, advancing understanding of polynomial behavior in these Banach spaces.

Contribution

It provides the first sharp inequalities for products of polynomials on $L_p$ spaces for $1<p<2$, and extends results to Schatten classes, with new estimates for $p>2$.

Findings

Sharp inequalities for $1<p<2$ on $L_p$ spaces.

Results also apply to Schatten classes $\\mathcal{S}_p$.

Estimates on constants for $p>2$.

Abstract

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.