Norm estimates for random polynomials on the scale of Orlicz spaces
The authors prove an upper bound for the uniform norm of homogeneous Bernoulli polynomials on the unit ball of finite-dimensional complex Banach spaces. This result is inspired by the famous Kahane-Salem-Zygmund inequality, see Theorem 4, Chapter 6 in [\textit{J.-P. Kahane}, Some random series of functions. 2nd ed. Cambridge: Cambridge University Press (1985; Zbl 0571.60002)] and its extension in [\textit{F. Bayart}, Q. J. Math. 63, No.\,1, 21--39 (2012; Zbl 1248.46033)]. The main idea is the following definition: Let \(\psi : \mathbb N_{>1} \times \mathbb N \rightarrow [1,\infty )\) and \(\phi : \mathbb N \rightarrow [1,\infty )\) be increasing functions. Then, a Banach sequence space \(E\) (modelled on \(\mathbb N\)) is said to be \((\psi,\phi)\)-admissible, provided that there is a constant \(C>0\) such that, for every choice of scalars \((c_\mathbf{j})_{\mathbf{j} \in \mathcal{J}(m,n)}\) with \(m>1\) and \(n \geq 1\), and functionals \(x_1^\ast, \dots, x_n^\ast \in X^\ast\) on an \(n\)-dimensional Banach space \(X\), there exists a choice of signs \(\varepsilon _\mathbf{j} = \pm 1\), \({\mathbf{j}\in \mathcal{J}(m,n)}\), such that \[ \sup_{z \in B_X} \left|\sum_{\mathbf{j}\in \mathcal{J}(m,n)} \varepsilon _\mathbf{j} c_\mathbf{j} x_\mathbf{j}^\ast(z) \right| \leq C \psi(m,n) \sup_{\mathbf{j} \in \mathcal{J}(m,n)} \frac{|c_{\mathbf{j}}|}{\phi(|[\mathbf{j}]|)} \sup_{z \in B_X} \left\|\sum_{k=1}^n x_k^\ast(z)e_k\right\|_E^m. \] Here, \[ \begin{aligned} \mathcal{M}(m,n) &= \{\mathbf{i} = (i_1,\dots,i_m): 1\leq i_k \leq n\} = \{1,\dots,n\}^m,\\ \mathcal{J}(m,n) &= \{\mathbf{j} = (j_1,\dots,j_m) \in \mathcal{M}(m,n): 1\leq j_1 \leq j_2 \leq \cdots \leq j_m \leq n\}.\end{aligned} \] Moreover, indices \(\mathbf{i},\mathbf{j} \in \mathcal{M}(m,n)\) are called equivalent if there is a permutation \(\sigma \) such that \((i_1,\dots,i_m)=(j_{\sigma(1)},\dots,j_{\sigma(m)})\). The equivalence class of \(\mathbf{i}\) is denoted by \([\mathbf{i}]\) and \(|[\mathbf{i}]|\) denotes the cardinality of \([\mathbf{i}]\). Finally, \(x_{\mathbf{j}}^\ast(z) := x_{j_1}(z) \cdots x_{j_m}(z)\) for \(z \in X\) and \(\{e_1,\dots,e_n\}\) denotes the canonical basis in \(X=(\mathbb{C}^n,\|\cdot\|)\). In this way, the authors get the upper bounds for Orlicz sequence spaces with ``good'' Orlicz functions using tensor product techniques and interpolation theory. These results contain as special case those of Bayart [loc. cit.]. As an application, the authors exhibit some lower estimates of the unconditional basis constant \(\chi(\mathcal{P}(^mX))\) of the monomials \(z^\alpha\), \(\alpha \in \mathbb{N}_0^n\), where \(\mathcal{P}(^mX)\) is the space of all \(m\)-homogeneous scalar-valued polynomials equipped with the usual sup-norm on a Banach space \(X=(\mathbb{C}^n,\|\cdot\|)\). From this, lower and upper estimates are derived for multidimensional Bohr radii of certain Reinhardt domains. The paper is self-contained and pleasant to read.
scientific article; zbMATH DE number 6694357