LORENTZ SPACES OF VECTOR-VALUED MEASURES

Abstract

Given a non-atomic, finite and complete measure space (Ω,Σ,μ) and a Banach space X, the modulus of continuity for a vector measure F is defined as the function ωF(t) = supμ(E)⩽t |F|(E) and the space Vp,q(X) of vector measures such that t−1/p′ ωF(t)∈ Lq((0,μ(Ω)],dt/t) is introduced. It is shown that Vp,q(X) contains isometrically Lp,q(X) and that Lp,q(X) = Vp,q(X) if and only if X has the Radon–Nikodym property. It is also proved that Vp,q(X) coincides with the space of cone absolutely summing operators from Lp′,q′ into X and the duality Vp,q(X*)=(Lp′,q′(X))* where 1/p+1/p′= 1/q+1/q′ = 1. Finally, Vp,q(X) is identified with the interpolation space obtained by the real method (V1(X), V∞(X))1/p′,q. Spaces where the variation of F is replaced by the semivariation are also considered.