Relative Rearrangement Method for Estimating Dual Norms

The Generalized-Γ-Space GΓ(p, m, w) contains many classical rearrangement invariant spaces. Here we shall study its associate space and we shall estimate its associate norm. In particular, we characterize all optimal functions u achieving the associate norm of Generalized Γ-space GΓ(p, m, w) when it is reflexive. For the purpose, we use the notion of relative rearrangement and new additional results on this concept. Moreover, we prove that the space GΓ(p, m, w) is reflexive under the conditions that m > 1 and p ≥ 2.