Estimates for polynomial norms on Banach spaces

Our work is related to problems $73$ and $74$ of Mazur and Orlicz in ``The Scottish Book" (ed. R. D. Mauldin). Let $k_1, \ldots, k_n$ be nonnegative integers such that $\sum_{i=1}^{n} k_{i}=m$, and let $\mathbb{K}(k_1, \ldots, k_n; X)$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, be the smallest number satisfying the property: if $L$ is any symmetric $m$-linear form on a Banach space $X$, then $$\sup_{\|x_{i}\|\leq 1 ,\atop i=1,2,\ldots ,n} |L(x_{1}^{k_1},\ldots ,x_{n}^{k_n})|\leq \mathbb{K}(k_1, \ldots, k_n; X)\sup_{\|x\|\leq 1} |L(x, \ldots ,x)|\,,$$ where the exponents $k_{1}, \ldots, k_{n}$, are as described above, and each $k_{i}$ denotes the number of coordinates in which the corresponding base variable appears. In the case of complex Banach spaces, the problem of optimising the constant $\mathbb{C}(k_1, \ldots, k_n; X)$ is well-studied. In the more challenging case of real Banach spaces much less is known about the estimates for $\mathbb{R}(k_1, \ldots, k_n; X)$. In this work, both real and complex settings are examined using results from the local theory of Banach spaces, as well as from interpolation theory of linear operators. In the particular case of complex $L^{p}(\mu)$ spaces, and for certain values of $p$, our results are optimal. As an application, we prove Markov-type inequalities for homogeneous polynomials on Banach spaces.