Strict K-monotonicity and K-order continuity in symmetric spaces

Abstract

This paper is devoted to strict K -monotonicity and K -order continuity in symmetric spaces. Using a local approach to the geometric structure in a symmetric space E we investigate a connection between strict K -monotonicity and global convergence in measure of a sequence of the maximal functions. Next, we solve an essential problem whether an existence of a point of K -order continuity in a symmetric space E on [0, ∞) implies that the embedding E → L 1 [0, ∞) does not hold. We present a complete characterization of an equivalent condition to K -order continuity in a symmetric space E using a notion of order continuity and the fundamental function of E. We also investigate a relationship between strict K -monotonicity and K -order continuity in symmetric spaces and show some examples of Lorentz spaces and Marcinkiewicz spaces having these properties or not. Finally, we discuss a local version of a crucial correspondence between order continuity and the Kadec-Klee property for global convergence in measure in a symmetric space E.

Lower and upper local uniform $K$ -monotonicity in symmetric spaces

“…Hence, the Lorentz space E endowed with the given norm is strictly K-monotone, whence x is an LKM point in E. Assume that (y n ) ⊂ E, y n ≺ x for any n ∈ N and y n E → x E . Then, since x is an LKM point and x * (∞) = 0, by Theorem 3.2 in [9] it follows that y * n → x * globally in measure. Therefore, by property 2.11 in [17] we get y * n (t) → x * (t) for all t ∈ [0, 1].…”

“…The crucial inspiration for our discussion was found in paper [7], where there has been studied an application of strict K-monotonicity and K-order continuity to the best dominated approximation with respect to the Hardy-Littlewood-Pólya relation ≺. It is worth mentioning that in view of the previous result, in [9] there has been investigated, among others, a full criteria for K-order continuity in symmetric spaces.…”

Relationships between K-monotonicity and rotundity properties with application

“…So, in view of Lemma 2.5 in [8] we obtain y * (∞) = 0. In consequence, since y ∈ S E , by (20) and (22) as well as by assumption that E is strictly K-monotone, in view of Theorem 1 in [6] we conclude that (23) y * * n → y * * and x * * n → y * * globally in measure. Now, since E d is compactly fully k-rotund and strictly Kmonotone, by Theorem 4.9 we have E is upper locally uniformly K-monotone.…”

“…Assume that ψ is the fundamental function of the associate space E ′ of a symmetric space E. (i) ⇒ (iii). Immediately, by Lemma 2.5 in[8] and Lemma 3.1 and Remark 3.2 we get E is K-order continuous and φ(∞) = ∞.Hence, by Corollary 2 in[6] and by Remark 3.2 it follows that E is order continuous and E ′ ֒→ {f : f * (∞) = 0}. (ii) ⇒ (i).…”

“…Recently, in the papers [13,18], there have been shown, among others, a complete correspondence between fully k-rotundity properties, rotundity and reflexivity with an application to the approximation theory in Banach spaces. The next interesting results were published in [3], where authors have investigated, inter alia, rotundity properties on E d the positive cone of all nonnegative and decreasing elements of a K-monotone symmetric space E. The further motivation of our investigation can be found in [5,6,7,11], where authors have presented a correspondence and complete criteria for K-order continuity, strict K-monotonicity and uniform K-monotonicity properties with application to the best dominated approximation problems in the sense of the Hardy-Littlewood-Pólya relation. The main idea of this paper is to find a relationship between reflexivity, fully k-rotundity properties and uniform K-monotonicity properties with application to the approximation theory.…”