Normed Köthe spaces as intermediate spaces of L₁ and L_∞

Stuart Edward Mills

Vol. 52 (1974), No. 1, 157–173

DOI: 10.2140/pjm.1974.52.157

Abstract

Let (Δ,Σ,μ) be a totally σ-finite measure space and let M(Δ) be the set of all complex-valued μ-measurable functions on Δ. This paper is concerned with determining whether certain classes of normed Köthe spaces (Banach function spaces) are intermediate spaces of L₁ = L₁(μ) and L_∞ = L_∞(μ). It is proven that L₁ ∩L_∞ and L₁ + L_∞ are associate Orlicz spaces and that for every nontrivial Young’s function Φ there is an equivalent Young’s function Φ₁ such that the Orlicz space L_uΦ1 is an intermediate space of L₁ and L_∞. The notion of a universal Köthe space is presented and it is proven that if Λ is a universal Köthe space then L₁ ∩ L_∞⊂ Λ ⊂ L₁ + L_∞. Furthermore, if Λ is normed, in particular Λ = L_ρ, then there is an equivalent universally rearrangement invariant norm ρ₁ for which L_ρ1 is an intermediate space of L₁ and L_∞.

Mathematical Subject Classification 2000

Primary: 46E30

Milestones

Received: 18 January 1973

Revised: 28 November 1973

Published: 1 May 1974