Relationships between combinatorial measurements and Orlicz norms. II
A sub-\(\alpha\)-Orlicz function \(\Phi\), \(\alpha> 1\), is defined and an Orlicz function \(M_\phi\), generated by \(\Phi\). For any infinite set \(F\subset Y^n\) \((n\geq 1)\), let \(\Psi_F:\mathbb{N}\to\mathbb{N}\) be defined by \[ \Psi_F(s)= \max\{|F\cap(A_1\times\cdots\times A_n)|: A_j\subset Y,\,|A_j|\leq s,\,j= 1,\dots, n\}. \] Let \(d_F(\Phi)= \sup\{\Psi_F(s)/\Phi(s): s\in\mathbb{N}\}\) and \(\zeta_F(\Phi)= \sup\{\|\widehat f\|_{M_\phi}: f\in B_{C_F(\Omega^n)}\}\), where \(\|\cdot\|_{M_\phi}\) is the Orlicz norm. The main result states that for \(n\in\mathbb{N}\) there exist positive constants \(C_n\) and \(D_n\) such that, for all \(F\subset\mathbb{R}^n\) and sub-\(\alpha\)-Orlicz functions \(\Phi\), the inequalities \[ C_n(d_F(\phi))^{\lambda(\alpha)}\leq \zeta_F(\Phi)\leq D_n(d_F(\phi))^{\theta(\alpha)} \] hold with specified powers \(\lambda(\alpha)\) and \(\delta(\alpha)\). This is applied to some extensions of the Kahane-Khintchin inequality. [For part I, see J. Funct. Anal. 257, No.~3, 683--720 (2009; Zbl 1182.46017).]
\(n\)-dimensional Khintchin inequalities
scientific article; zbMATH DE number 5649983