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On the Bernstein-Nikolsky inequality. II. (English) Zbl 0848.46014
There are considered Orlicz spaces $$L_\Phi (\mathbb{R}^n)$$ with the Luxemburg norm $$|\cdot |_\Phi$$ and spaces $$M_{\sigma, \Phi}$$ of all entire functions of exponential type $$\sigma$$ which as functions of real $$x\in \mathbb{R}^n$$ belong to $$L_\Phi (\mathbb{R}^n)$$. The author considers a connection between the spaces $${\mathcal S}'$$ and $$M_{\sigma, \Phi}$$ (Theorems 1 and 2, respectively) and properties of the elements from $$M_{\sigma, \Phi}$$ and $$L_\Phi (\mathbb{R}^n)$$ (Theorems 3 and 4, respectively). Very important is Theorem 5 with a nice proof which says that if $$f$$ is such that $$D^\alpha f(x)\in L_\Phi (\mathbb{R}^n)$$ (where $$\alpha$$ is a multi-index) then $\liminf_{|\alpha |\to \infty} (|\xi^{-\alpha} |\;|D^\alpha f|_\Phi )^{1/ |\alpha |}\geq 1.$ [For part I see M. Morimoto and the author, same J. 14,No. 1, 231-238 (1991; Zbl 0803.41017)].
Reviewer: A.Waszak (Poznań)

MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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