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On ${L}^{p\left(x\right)}$ norms. (English) Zbl 0953.46018
Let $X=\left(X\left({\Omega }\right),\parallel ·\parallel \right)$ be a Banach function space and let $f\in X$ be an arbitrary function. We say that $f$ has an absolutely continuous norm in $X$ if for any sequence of open sets ${G}_{n}$, ${G}_{1}\supset {G}_{2}\supset {G}_{3}\supset \cdots$, ${\bigcap }_{n=1}^{\infty }{G}_{n}=\varnothing$, the norms $\parallel f{\chi }_{{G}_{n}}\parallel$ tend to zero as $n\to \infty$. We say that $f$ has the continuous norm in $X$ if ${lim}_{r\to 0+}\parallel f{\chi }_{\left(x-r,x+r\right)\cap {\Omega }}\parallel =0$ for every $x\in \overline{{\Omega }}$ and ${lim}_{r\to \infty }\parallel f{\chi }_{{\Omega }\setminus \left(-r,r\right)}\parallel =0$. Denote the set of all functions with the continuous norm by ${X}_{c}$. The concept of absolutely continuous norm plays a very important role in characterization of classes of reflexive Banach function spaces and of separable Banach function spaces. For instance, it is known that a Banach function space $X$ is reflexive if and only if both $X$ and its associate space ${X}^{\text{'}}$ have absolutely continuous norm. The concept of continuous norm was introduced by Q. Lai and L. Pick in [J. London Math. Soc. 48, No. 1, 167-177 (1993; Zbl 0739.47024)]. They proved that the Hardy operator $Tf\left(x\right)={\int }_{0}^{x}f\left(t\right)dt$ is a compact mapping from a Banach function space $\left(X,v\right)$ into ${L}^{\infty }$ if and only if the function $1/v$ has a continuous norm in the associate space $\left({X}^{\text{'}},v\right)$. The authors consider the generalized Lebesgue space ${L}^{p\left(x\right)}$ of functions integrable with variable $p$ and study relations between its subspaces of functions with continuous norms, subspaces of functions with absolutely continuous norms and subspaces of bounded functions.
##### MSC:
 4.6e+31 Spaces of measurable functions