The sup-norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials

TL;DR

This paper investigates the asymptotic behavior of constants relating the sup-norm and coefficient norms of homogeneous polynomials, revealing new extremal polynomial configurations and applications in Banach space theory.

Contribution

It provides the precise asymptotic behavior of these constants for various p, r, and degree m, introducing Steiner polynomials as extremal examples in certain cases.

Findings

Asymptotic behavior of constants is characterized for large n.

Steiner polynomials serve as extremal polynomials in some parameter ranges.

Applications include tensor product interpolation and multivariable von Neumann's inequality.

Abstract

Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|\alpha|=m} a_{\alpha} z^{\alpha}$ in $n$ complex variables, $$\big( \sum_{|\alpha|=m} |a_{\alpha}|^{r} \big)^{1/r} \leq A_{p,r}^m(n) \sup_{z \in B_{\ell_p^n}} \big|P(z) \big| .$$ For every degree $m$, and a wide range of values of $ p,r \in [1,\infty]$ (including any $r$ in the case $p \in [1,2]$, and any $r$ and $p$ for the 2-homogeneous case), we give the correct asymptotic behavior of these constants as $n$ (the number of variables) tends to infinity. Remarkably, in many cases, extremal polynomials for these inequalities are not (as traditionally expected) found using classical random unimodular polynomials, and special combinatorial configurations of monomials are needed. Namely, we show that Steiner polynomials (i.e., $m$-homogeneous polynomials such that the multi-indices corresponding to the nonzero coefficients form partial Steiner systems), do the work for certain range of values of $p,r$. As a byproduct, we present some applications of these estimates to the interpolation of tensor products of Banach spaces, to the study of (mixed) unconditionality in spaces of polynomials and to the multivariable von Neumann's inequality.