Relationships between combinatorial measurements and Orlicz norms (II)
Copyright policy )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).]
- University of Connecticut United States
- University of Connecticut United States
\(n\)-dimensional Khintchin inequalities, Sub-α-Orlicz function, Kahane–Khintchin inequality, combinatorial measurement, \(\alpha \)-Orlicz function, n-Dimensional Khintchin inequalities, Littlewood 2n/(n+1)-inequalities, Nontrigonometric harmonic analysis involving wavelets and other special systems, Rademacher system, Combinatorial measurement, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), sub-\(\alpha \)-Orlicz function, α-Orlicz function, Littlewood \(2n/(n+1)\)-inequalities, Orlicz norm, Quasi-asymptotic, Kahane-Khintchin inequality, Analysis, quasi-asymptotic
\(n\)-dimensional Khintchin inequalities, Sub-α-Orlicz function, Kahane–Khintchin inequality, combinatorial measurement, \(\alpha \)-Orlicz function, n-Dimensional Khintchin inequalities, Littlewood 2n/(n+1)-inequalities, Nontrigonometric harmonic analysis involving wavelets and other special systems, Rademacher system, Combinatorial measurement, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.), sub-\(\alpha \)-Orlicz function, α-Orlicz function, Littlewood \(2n/(n+1)\)-inequalities, Orlicz norm, Quasi-asymptotic, Kahane-Khintchin inequality, Analysis, quasi-asymptotic
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