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On $L^{p(x)}$ norms. (English) Zbl 0953.46018
Let $X=(X(\Omega),\|. \|)$ 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 $G_n$, $G_1\supset G_2\supset G_3\supset \dots$, $\bigcap_{n=1}^\infty G_n=\emptyset$, the norms $\|f\chi_{G_n}\|$ tend to zero as $n\to\infty$. We say that $f$ has the continuous norm in $X$ if $\lim_{r\to 0+}\|f\chi_{(x-r,x+r)\cap\Omega}\|=0$ for every $x\in\overline\Omega$ and $\lim_{r\to\infty}\|f\chi_{\Omega\setminus(-r,r)}\|=0$. 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 $X'$ have absolutely continuous norm. The concept of continuous norm was introduced by {\it Q. Lai} and {\it L. Pick} in [J. London Math. Soc. 48, No. 1, 167-177 (1993; Zbl 0739.47024)]. They proved that the Hardy operator $Tf(x)=\int^x_0f(t) dt$ is a compact mapping from a Banach function space $(X,v)$ into $L^\infty$ if and only if the function $1/v$ has a continuous norm in the associate space $(X',v)$. The authors consider the generalized Lebesgue space $L^{p(x)}$ 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.

46E30Spaces of measurable functions
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