Title: Relative rearrangement method for estimating dual norms

Authors: A. Fiorenza, J. M. Rakotoson and L. Zitouni

Issue: Volume 58 (2009), Issue 3, 1127-1150

Abstract: The Generalized-$\Gamma$-Space $\G\Gamma(p,m,w)$ 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 $\Gamma$-space $\G\Gamma(p,m,w)$ 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 $\G\Gamma(p,m,w)$ is reflexive under the conditions that $m>1$ and $p\geq2$.