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

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 .