Lorentz Spaces Of Vector-Valued Measures |
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Authors: | Blasco Oscar; Gregori Pablo |
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Institution: | Departamento de Análisis Matemático, Universidad de Valencia Doctor Moliner, 50 E-46100 Burjassot (Valencia), Spain oscar.blasco{at}uv.es
Departamento de Análisis Matemático, Universidad de Valencia Doctor Moliner, 50 E-46100 Burjassot (Valencia), Spain pablo.gregori{at}uv.es |
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Abstract: | Given a non-atomic, finite and complete measure space ( , ,µ)and a Banach space X, the modulus of continuity for a vectormeasure F is defined as the function F(t) = supµ(E) t |F|(E)and the space Vp,q(X) of vector measures such that t1/p' F(t) Lq((0,µ( )],dt/t) is introduced. It is shown thatVp,q(X) contains isometrically Lp,q(X) and that Lp,q(X) = Vp,q(X)if and only if X has the RadonNikodym property. It isalso proved that Vp,q(X) coincides with the space of cone absolutelysumming 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 identifiedwith the interpolation space obtained by the real method (V1(X),V (X))1/p',q. Spaces where the variation of F is replaced bythe semivariation are also considered. |
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