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

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.