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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 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 $$Tf(x)=\int^x_0f(t) dt$$ 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.

##### MSC:
 4.6e+31 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Zbl 0739.47024
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