Linear functionals on Orlicz spaces: General theory

Abstract

The purpose of this paper is two-fold. First it contains a complete study centering around the singular linear functionals, analyzing certain factor (or quotient) spaces, of the Orlicz spaces. Second, the so-called 'generalized Young's functions' and the associated Orlicz spaces, and their adjoint spaces, are also considered. (Precise definitions will be given later.) The work here is a continuation of [19] and the notation and terminology of that paper will be maintained. However, the theory presented here subsumes [19], and the exposition is essentially self-contained.If Φ and Ψ are complementary Young's functions (cf. Definition 1 below), let L φ and U be the corresponding Orlicz spaces on a (not necessarily finite or even localizable) measure space (Ω, Σ, μ) which has only the (nonrestrictive) finite subset property. This latter means that every set of positive μ-measure has a subset of positive finite μ-measure. in the sense defined above where Φ is a Young's function. The result is restated precisely in Theorem 1 below for completeness. Moreover, it is shown later (Proposition 2) that (L φ )* admits a direct sum decomposition consisting of the singular and the absolutely continuous elements. Also the generalized (or the extended from the original version of) Young's functions are considered in detail below. This distinction is irrelevant for the elementary theory of Orlicz spaces (such as completeness, etc.), or if the underlying measure μ is finite, but will be relevant for the study of the adjoint spaces and the understanding of their structures when μ is nonfinite. This will be clear from the work of § 3 and § 4 below.The main contribution of this paper is the complete characterization of (I/)*, the adjoint space of L φ , when both Φ and μ are general. This is achieved by considering, somewhat more generally, the properties of certain factor spaces of L φ which in particular illuminate the structure of singular linear functionals. Also the (L φ )*-space when Φ is a 'generalized Young's function' is considered and characterized. Moreover, the general methods presented here are applicable in analyzing the more general Banach function spaces such as the Kδthe-Toeplitz spaces [12] and their extensions [15]. A brief summary of the results is as follows.After preliminaries in the next section, representation theorems, extending the work of [1] and [19], are proved in §3. These results are further extended, when Φ is a generalized Young's function, in § 4. Some miscellaneous results are given in § 5 where the current status of the work, and the representation of the elements of (I/)**, and related problems are discussed. Then \\ \\ φ and N φ ( ) are norm functionals and they define the same topology for L φ in the sense thatWith these norms L φ becomes a Banach (or B-) space. The proofs of these results may be found in ([13] (f).If / = 0, a.e., is false then N φ (f) > 0 and it may be assumed, by a normalization if necessary, that N φ (f) = l( = \\f\\ φ ).Since Φ and Ψ are continuous, a result of ([13], p. 92)...

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