上范数与系数范数：齐次多项式的等价常数
The sup-norm vs. the norm of the coefficients: equivalence constants for homogeneous polynomials
摘要：
令$A_{p,r}^m(n)$为满足以下不等式的最佳常数：对于每个$m$-$n$复变量中的齐次多项式$P(z) = \sum_{|α|=m} a_α z^α$，$$\big( \sum_{|α|=m} |a_α|^{r} \big)^{1/r} \leq A_{p,r}^m(n) \sup_{z \in B_{\ell_p^n}} \big|P(z) \big| .$$对于每个度$m$，以及$ p,r \in [1,\infty]$的宽范围值（包括$p \in [1,2]$病例中的任何$r$，以及2-同质病例中的任何$r$和$p$），我们给出了这些常数在$n$（变量数）趋于无穷大时的正确渐近行为。值得注意的是，在许多情况下，这些不等式的极值多项式不是（如传统预期的那样）使用经典的随机幺模多项式找到的，并且需要单项式的特殊组合配置。也就是说，我们表明，斯坦纳多项式（即，$m$ -齐次多项式，使得对应于非零系数的多个索引形成部分Steiner系统）对$p,r$的值的一定范围进行工作。作为一个副产品，我们提出了一些应用这些估计的Banach空间的张量积的插值，研究（混合）无条件的多项式空间和多变量冯诺依曼不等式。
英文摘要：
Let $A_{p,r}^m(n)$ be the best constant that fulfills the following inequality: for every $m$-homogeneous polynomial $P(z) = \sum_{|α|=m} a_α z^α$ in $n$ complex variables, $$\big( \sum_{|α|=m} |a_α|^{r} \big)^{1/r} \leq A_{p,r}^m(n) \sup_{z \in B_{\ell_p^n}} \big|P(z) \big| .$$ 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.
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