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On the uniform convexity of the Besicovitch–Orlicz space of almost periodic functions with Orlicz norm. (English) Zbl 1112.46013
Let \(\phi\) be an Orlicz function (convex function \(\phi : {\mathbb R}_+ \to {\mathbb R}_+\) such that \(\phi (0)= 0\), \(\phi (u) >0\) for \(u > 0\), \(\phi(u)/u \mathop{\longrightarrow} 0\) for \(u \to 0\) and \(\phi(u)/u \mathop{\longrightarrow} \infty\) for \(u \to \infty\)). The Besicovitch–Orlicz space \(B^\phi ({\mathbb R})\) is the space of measurable functions \(f\) on \({\mathbb R}\) such that \(\rho_{B^\phi} (\lambda f) <+\infty\) for some \(\lambda >0\), where \(\rho_{B^\phi} (f) = \overline{\lim}_{T \to\infty} {1 \over 2T} \int_{-T}^T \phi (| f(t)| )\,dt\), endowed with the Luxemburg norm \(\| f\| _{B^\phi} = \inf\{ k>0 \mid \rho_{B^\phi} (f/k) \leq 1\}\) (it is actually a semi-norm; one considers the associated normed space). The Besicovitch–Orlicz space of almost periodic functions \(B^\phi\)-a.p. is the closure in \(B^\phi ({\mathbb R})\) of the set of the trigonometric polynomials \(\sum_j \alpha_j \exp(i \lambda_j t)\).
In [M. Morsli, Ann. Soc. Math. Pol. (I) Commentat. Math. 34, 137–152 (1994; Zbl 0839.46012); see also Funct. Approximatio, Comment. Math. 22, 95–106 (1993; Zbl 0831.46019)], the first author studied the uniform convexity of \(B^\phi\)-a.p. with the Luxemburg norm. In the paper under review, the present authors do the same with the Orlicz norm (defined by duality: \(||| f||| _{B^\phi} = \sup\{ M(| fg| )\); \(g\in B^\phi\)-a.p., \(\rho_{B^\psi} (g) \leq 1\}\)), where \(\psi\) is the conjugate of the Orlicz function \(\phi\) and \(M(f)=\lim_{T\to \infty} {1 \over 2T} \int_{-T}^T f(t)\,dt\).
They prove that \(B^\phi\)-a.p. is uniformly convex with the Orlicz norm if and only if \(\phi\) satisfies the usual \(\Delta_2\) condition and is uniformly convex, meaning that for every \(a \in ]0,1[\), there exists a \(\delta (a) \in ]0,1[\) such that \(\phi \big({u + au \over 2}\big) \leq \big(1 - \delta (a)\big) {\phi (u) + \phi (au) \over 2}\) for \(u\) large enough. In order to prove the necessity, they show that there is a natural isometry from \(E^\phi ([0,1])\) into \(B^\phi\)-a.p. (for both the Luxemburg and Orlicz norms), where \(E^\phi ([0,1]) =\{f\mid\int_0^1 \phi\big(\lambda | f(t)| \big)\,dt < +\infty\,\;\forall \lambda >0\}\) is the Morse–Transue space.
Reviewer: Daniel Li (Lens)
MSC:
46B20 Geometry and structure of normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42A75 Classical almost periodic functions, mean periodic functions
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