arXiv 2107.059932021-07-13math.FA

巴拿赫空间上多项式范数的估计

Estimates for polynomial norms on Banach spaces

Marianna Chatzakou, Yannis Sarantopoulos

备注：: The paper will appear in Dolomites Research Notes on Approximation. Number of pages: 19. arXiv admin note: text overlap with arXiv:2003.11002

摘要：

我们的工作与Mazur和Orlicz在“苏格兰书”（R. D. Mauldin）。令$k_1, \ldots, k_n$是非负整数，使得$\sum_{i=1}^{n} k_{i}=m$，并令$\mathbb{K}(k_1, \ldots, k_n; X)$，其中$\mathbb{K}=\mathbb{R}$或$\mathbb{C}$，是满足以下性质的最小数：如果$L$是Banach空间$X$上的任何对称$m$ -线性形式，则\[ \sup_{\|x_{i}\|\leq 1 ,\atop i=1,2,\ldots ,n} |L(x_{1}^{k_1},\ldots ,x_{n}^{k_n})|\leq \mathbb{K}(k_1, \ldots, k_n; X)\sup_{\|x\|\leq 1} |L(x, \ldots ,x)|\,, \]，其中指数$k_{1}, \ldots, k_{n}$如上所述，每个$k_{i}$表示相应基变量出现的坐标数。在复巴拿赫空间的情况下，优化常数$\mathbb{C}(k_1, \ldots, k_n; X)$的问题是很好的研究。在更具挑战性的情况下，真正的巴拿赫空间少得多是已知的估计$\mathbb{R}(k_1, \ldots, k_n; X)$。在这项工作中，无论是真正的和复杂的设置检查使用的结果从局部理论的巴拿赫空间，以及从线性算子的插值理论。在复杂的$L^{p}(μ)$空间的特殊情况下，对于某些$p$值，我们的结果是最优的。作为应用，我们证明了巴拿赫空间上齐次多项式的马氏型不等式。

英文摘要：

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 $L^{p}(μ)$ 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.

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