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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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