Abstract
Let (Ω, Σ) be a measurable space and m 0: Σ → X 0 and m 1: Σ → X 1 be positive vector measures with values in the Banach Köthe function spaces X 0 and X 1. If 0 < α < 1, we define a new vector measure [m 0, m 1] α with values in the Calderón lattice interpolation space X 1−ga0 X α1 and we analyze the space of integrable functions with respect to measure [m 0, m 1] α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of L p-spaces. Since each p-convex order continuous Köthe function space with weak order unit can be represented as a space of p-integrable functions with respect to a vector measure, we provide in this way a technique to obtain representations of the corresponding complex interpolation spaces. As applications, we provide a Riesz-Thorin theorem for spaces of p-integrable functions with respect to vector measures and a formula for representing the interpolation of the injective tensor product of such spaces.
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Supported by La Junta de Andalucía, D.G.I. under projects MTM2006-11690-C02, MTM2009-14483-C02 (M.E.C. Spain) and FEDER
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del Campo, R., Fernández, A., Mayoral, F. et al. Interpolation of vector measures. Acta. Math. Sin.-English Ser. 27, 119–134 (2011). https://doi.org/10.1007/s10114-011-9542-8
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DOI: https://doi.org/10.1007/s10114-011-9542-8