论文标题
可变lebesgue空间中的固定点特性和反射性
Fixed Point Properties and reflexivity in Variable Lebesgue Spaces
论文作者
论文摘要
在本文中，研究了弱的固定点属性（$ W $ -FPP）和可变lebesgue空间中的固定点属性（FPP）。给定$（ω，σ，μ）$ a $σ$ - finite度量和$ p（\ cdot）$一个可变指数函数，$ w $ -fpp在可变的lebesgue space $ l^{p（\ cdot）}（ω）（ω）$ pectements $ p（\ cdot $ p（\ cdot）$ 1 $ a的$ 1 $ a的$ pppection $ l^{p（\ cdot）}（ω）$ a的$ 1 $ a $ avemant $ a的$ 1的compoce $ a impection $ a $ pppe $ 1 $ a $ a $ a $ a $ a $ a的coplence [特别是，每个反身$ l^{p（\ cdot）}（ω）$都具有FPP，我们的结果揭示了满足$ W $ -FPP的某些非反射变量Lebesgue空间的存在，与经典的Lebesgue $ L^P $ -Spaces形成了鲜明的对比。与FPP相关，我们证明莫里（Maurey）的$ l^1 $ - 空格的结果可以扩展到具有连续规范的较大的可变$ l^{p（\ cdot）}（ω）$ space，也就是说，每个反射性子空间的$ l^{p（\ cdot）}（\ cdot）}（\ cdot）}（ω（ω（ω））$。从来没有\ - 但是，由于可以找到满足FPP满足的$ l^{p（\ cdot）} $的某些非反射子空间，因此莫里的匡威不再存在。结果，我们发现几种非反射性中nakano序列空间$ \ ell^{p_n} $确实具有卢森堡标准的FPP。就作者而言，这个序列空间家族产生了第一个已知的非反射经典Banach空间，享受FPP，而无需进行任何重生的程序。还分析了$ \ ell_1 $ in $ l^{p（\ cdot）}（ω）$的渐近\ thine Isemetric副本的失败。
In this paper the weak fixed point property ($w$-FPP) and the fixed point property (FPP) in Variable Lebesgue Spaces are studied. Given $(Ω,Σ,μ)$ a $σ$-finite measure and $p(\cdot)$ a variable exponent function, the $w$-FPP is completely characterized for the variable Lebesgue space $L^{p(\cdot)}(Ω)$ in terms of the exponent function $p(\cdot)$ and the absence of an isometric copy of $L_1[0,1]$. In particular, every reflexive $L^{p(\cdot)}(Ω)$ has the FPP and our results bring to light the existence of some nonreflexive variable Lebesgue spaces satisfying the $w$-FPP, in sharp contrast with the classic Lebesgue $L^p$-spaces. In connection with the FPP, we prove that Maurey's result for $L^1$-spaces can be extended to the larger class of variable $L^{p(\cdot)}(Ω)$ spaces with order continuous norm, that is, every reflexive subspace of $L^{p(\cdot)}(Ω)$ has the FPP. Never\-theless, Maurey's converse does not longer hold in the variable setting, since some nonreflexive subspaces of $L^{p(\cdot)}(Ω)$ satisfying the FPP can be found. As a consequence, we discover that several nonreflexive Nakano sequence spaces $\ell^{p_n}$ do have the FPP endowed with the Luxemburg norm. As far as the authors are concerned, this family of sequence spaces gives rise to the first known nonreflexive classic Banach spaces enjoying the FPP without requiring of any renorming procedure. The failure of asympto\-tically isometric copies of $\ell_1$ in $L^{p(\cdot)}(Ω)$ is also analyzed.
