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Lorentz Spaces Of Vector‐Valued Measures
Author(s) -
Blasco Oscar,
Gregori Pablo
Publication year - 2003
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610703004198
Subject(s) - measure (data warehouse) , banach space , mathematics , function space , space (punctuation) , combinatorics , lorentz space , duality (order theory) , vector space , measurable function , physics , lorentz transformation , mathematical analysis , pure mathematics , quantum mechanics , bounded function , linguistics , philosophy , database , computer science
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 ω F ( t ) = sup μ( E )⩽ t | F |( E ) and the space V p,q ( X ) of vector measures such that t −1/p ′ ω F ( t )∈ L q ((0,μ(Ω)], dt/t ) is introduced. It is shown that V p,q ( X ) contains isometrically L p,q ( X ) and that L p,q ( X ) = V p,q ( X ) if and only if X has the Radon–Nikodym property. It is also proved that V p,q ( X ) coincides with the space of cone absolutely summing operators from L p ′, q ′ into X and the duality V p,q ( X * )=( L p ′, q ′( X )) * where 1/p+1/p′= 1/q+1/q′ = 1. Finally, V p,q ( X ) is identified with the interpolation space obtained by the real method ( V 1 ( X ), V ∞ ( X )) 1/p′, q . Spaces where the variation of F is replaced by the semivariation are also considered.