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On L p(x) norms. (English) Zbl 0953.46018
Let X=(X(Ω),·) be a Banach function space and let fX 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 G 2 G 3 , n=1 G n =, the norms fχ G n tend to zero as n. We say that f has the continuous norm in X if lim r0+ fχ (x-r,x+r)Ω =0 for every xΩ ¯ and lim r fχ Ω(-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)= 0 x f(t)dt is a compact mapping from a Banach function space (X,v) into L 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:
46E30Spaces of measurable functions