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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.

46E30 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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