From weak to strong L1-convergence by an oscillation restriction criterion of BMO type

Abstract

Recently, Girardi gave acharacterization of relative strong L 1 R-compactness in terms of relative weak L 1 R-compactness and the Bocce criterion [18]. Here this result is generalized and extended by presenting a less stringent oscillation restriction condition (ORC) which enforces the transcendence of weak into an appropiately reformulated form of strong convergence in L 1 E , for E a separable re exive Banach space. The proof has a very simple, transparent structure, because it relies on basic, wellknown facts from Young measure theory; this brings the result in line with the current literature.