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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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