Functional properties of rearrangement invariant spaces defined in terms of oscillations

Function spaces whose definition involves the quantity f** - f*, which measures the oscillation of f*, have recently attracted plenty of interest and proved to have many applications in various, quite diverse fields. Primary role is played by the spaces S p (w), with 0 < p < ∞ and w a weight function on (0, ∞), defined as the set of Lebesgue-measurable functions on ℝ such that f* (∞) = 0 and ∥f∥ S p (w):= (∫ 0∞ (f**(s) - f*(s)) p w(s)ds) 1/p < ∞. Some of the main open questions concerning these spaces relate to their functional properties, such as their lattice property, normability and linearity. We study these properties in this paper. © 2005 Elsevier Inc. All rights reserved.