Abstract. The goal of this paper is to present an isometric representation of the dual space to Cesàro function space Cp,w, 1 < p < ∞, induced by arbitrary positive weight function w on interval (0, l) where 0 < l ∞. For this purpose given a strictly decreasing nonnegative function Ψ on (0, l), the notion of essential Ψ-concave majorantf of a measurable function f is introduced and investigated. As applications it is shown that every slice of the unit ball of the Cesàro function space has diameter 2. Consequently Cesàro function spaces do not have the Radon-Nikodym property, are not locally uniformly convex and they are not dual spaces.
The condition δ 2 in Cesàro-Orlicz sequence spaces equipped with the Luxemburg norm is discussed. The comparison theorem for these spaces is presented. Some counterexamples are provided.
We investigate the relationships between smooth and strongly smooth points of the unit ball of an order continuous symmetric function space E, and of the unit ball of the space of τ -measurable operators E(M, τ ) associated to a semifinite von Neumann algebra (M, τ ). We prove that x is a smooth point of the unit ball in E(M, τ ) if and only if the decreasing rearrangement μ(x) of the operator x is a smooth point of the unit ball in E, and either μ(∞; f ) = 0, for the function f ∈ S E × supporting μ(x), or s(x * ) = 1. Under the assumption that the trace τ on M is σ -finite, we show that x is strongly smooth point of the unit ball in E(M, τ ) if and only if its decreasing rearrangement μ(x) is a strongly smooth point of the unit ball in E. Consequently, for a symmetric function space E, we obtain corresponding relations between smoothness or strong smoothness of the function f and its decreasing rearrangement μ( f ). Finally, under suitable assumptions, we state results relating the global properties such as smoothness and Fréchet smoothness of the spaces E and E(M, τ ).
Abstract. We show that among all Musielak-Orlicz function spaces on a σ-finite non-atomic complete measure space equipped with either the Luxemburg norm or the Orlicz norm the only spaces with the Daugavet property are L1, L∞, L1 ⊕1 L∞ and L1 ⊕∞ L∞. In particular, we obtain complete characterizations of the Daugavet property in the weighted interpolation spaces, the variable exponent Lebesgue spaces (Nakano spaces) and the Orlicz spaces.
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