Weighted rearrangement-invariant function spaces

In this thesis we focus on generalized Gamma spaces GΓ(p, m, v) and classify some of their intrinsic properties. In an article called Relative Re- arrangement Methods for Estimating Dual Norm (for details see references), the authors attempted to characterize their associate norms but obtained only several one-sided estimates. Equipped with these, they further showed reflexivity of gener- alized Gamma spaces for p ≥ 2 and m > 1 under an additional restriction that the underlying measure space is of finite measure. However, the full characterization of the associate norm and of the reflexivity of such spaces for 2 > p > 1 remained an open problem. In this thesis we shall fill this gap. We extend the theory to a σ-finite measure space. We present a complete characterization of the associate norm, and we find necessary and sufficient conditions for the reflexivity of such spaces.