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On some equivalent geometric properties in the Besicovitch-Orlicz space of almost periodic functions with Luxemburg norm. (English) Zbl 1224.46018
Let $$\varphi \:\mathbb R\longrightarrow [0, \infty )$$ be a Young function, i.e., $$\varphi$$ is even, convex, $$\varphi (0)=0,\;\varphi (u)>0$$ if $$u>0$$ and $$\varphi (u)\longrightarrow \infty$$ as $$u\to \infty$$. Let $$M( \mathbb R)$$ be the space of Lebesgue locally integrable functions on $$\mathbb R$$. For $$f\in M( \mathbb R)$$, put $$\rho _{\varphi }(f)=\limsup _{T\to \infty }\frac {1}{2T}\int _{-T}^T \varphi (| f(t)| )\, dt$$. Define then the Besicovitch-Orlicz space $$B^{\varphi }=\{f\in M(\mathbb R)\,:\, \rho _{\varphi }(\lambda f)< \infty$$ for some $$\lambda >0\}$$. Consider the Luxemburg norm, i.e., $$\| f\| _{B^{\varphi }}= \inf \{k>0\,:\, \rho _{\varphi }(f/k)\leq 1,\;f\in B^{\varphi }\}.$$ Then $$\big (B^{\varphi },\| \cdot \| _{B^{\varphi }}\big )$$ is a Banach space. Finally, define the Besicovitch-Orlicz space of almost periodic functions $$\widetilde {B}^\varphi _{a.p.}$$ as the closure of the class of Bohr’s almost periodic functions in the topology of the so called modular convergence; thus, $$\widetilde {B}^{\varphi }_{a.p.}\subset B^{\varphi }$$. Theorem: The following assertions are mutually equivalent: (i) $$\big (\widetilde {B}^{\varphi }_{a.p.},\| \cdot \| _{B^{\varphi }}\big )$$ is locally uniformly convex; (ii) on the unit sphere of $$\widetilde {B}^{\varphi }_{a.p.}$$, every weak convergent sequence converges in norm; (iii) $$\big (\widetilde {B}^{\varphi }_{a.p.},\| \cdot \| _{B^{\varphi }}\big )$$ is strictly convex; and (iv) the function $$\varphi$$ is strictly convex and satisfies the $$\Delta _2$$-condition, i.e., there are $$K>2$$ and $$u_0\geq 0$$ such that $$\varphi (2u)\leq K\varphi (u)$$ whenever $$u>u_0$$. The implication (i)$$\Rightarrow$$(ii) is valid in every Banach space. The equivalence (iii)$$\Leftrightarrow$$(iv) is taken from [M. Morsli, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 34, 137–152 (1994; Zbl 0839.46012)]. (ii)$$\Rightarrow$$(iv) is derived, with some extra effort, from [R. Płuciennik, T. F. Wang and Y. L. Zhang , Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 33, 135–151 (1993; Zbl 0793.46012)]. Finally, (iv)$$\Rightarrow$$(i) is done via several lemmas.
##### MSC:
 46B20 Geometry and structure of normed linear spaces 42A75 Classical almost periodic functions, mean periodic functions 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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