We prove that the Lebesgue space of variable exponent $L^{p(·)}\left(\Omega \right)$ is modularly uniformly convex in every direction provided the exponent p is finite a.e. and different from 1 a.e. The notion of uniform convexity in every direction was first introduced by Garkavi for the case of a norm. The contribution made in this work lies in the discovery of a modular, uniform-convexity-like structure of $L^{p(·)}\left(\Omega \right)$ , which holds even when the behavior of the exponent $p(·)$ precludes uniform convexity of the Luxembourg norm. Specifically, we show that the modular $\rho \left(u\right)=\underset{\Omega }{\int }\left|u\left(x\right)\right|dx$ possesses a uniform-convexity-like structure even if the variable exponent is not bounded away from 1 or ∞. Our result is new and we present an application to fixed point theory.

中文翻译：

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Lp（·）各个方向的模块化一致凸性及其应用

我们证明了可变指数的Lebesgue空间 ${\mathrm{大号}}^{p（·）}（\Omega ）$ 在指数p为有限ae且不同于1 ae的情况下，在每个方向上均模地均匀凸。在范数情况下，Garkavi首先引入了在每个方向上均一凸的概念。这项工作的贡献在于发现了模块化，均匀凸性的结构 ${\mathrm{大号}}^{p（·）}（\Omega ）$ ，即使指数行为也成立 $p（·）$ 排除了卢森堡规范的统一凸性。具体来说，我们证明了模块化 $\rho （ü）=\underset{\Omega }{\int }\left|ü（X）\right|dX$ 即使变量指数不受1或∞的限制，它也具有类似均匀凸的结构。我们的结果是新的，我们将其应用于定点理论。