论文标题
关于从C（[0,1]）到可变lebesgue空间的连续功能的单数扩展
On singular extensions of continuous functionals from C([0,1]) to variable Lebesgue spaces
论文作者
论文摘要
Valadier和Hensgen独立证明了功能$ ϕ（x）= \ int_ {0}^{1}^{1} x（t）dt，\，\，\，x \ in L^{\ infty}（[0,1] $ $ l^{\ infty}（[0,1]）。$ abramovich和wickstead获得了Banach Lattices的一些一般结果。目前的注释中，我们调查了可变指数lebesgue空间的类似问题，即，如果连续函数的空间$ c（[0,1]）$在$ l^{p（\ cdot）}（[0,1]）中封闭的子空间是封闭的子空间，则$然后在$ c（[0,1]）上的每个有界的线性函数是$ c（[0,1]） $ l^{p（\ cdot）}（[0,1]）$。
Valadier and Hensgen proved independently that the restriction of functional $ϕ(x)=\int_{0}^{1}x(t)dt,\,\,x\in L^{\infty}([0,1])$ on the space of continuous functions $C([0,1])$ admits a singular extension back to the whole space $L^{\infty}([0,1]).$ Some general results in this direction for the Banach lattices were obtained by Abramovich and Wickstead. In present note we investigate analogous problem for variable exponent Lebesgue spaces, namely we prove that if the space of continuous functions $C([0,1])$ is closed subspace in $L^{p(\cdot)}([0,1]),$ then every bounded linear functional on $C([0,1])$ is the restriction of a singular linear functional on $L^{p(\cdot)}([0,1])$.
