Interpolation of vector measures

Abstract

Let (Ω, Σ) be a measurable space and m0: Σ → X0 and m1: Σ → X1 be positive vector measures with values in the Banach Kothe function spaces X0 and X1. If 0 < α < 1, we define a new vector measure [m0, m1]α with values in the Calderon lattice interpolation space X01−gaX1α and we analyze the space of integrable functions with respect to measure [m0, m1]α in order to prove suitable extensions of the classical Stein-Weiss formulas that hold for the complex interpolation of Lp-spaces. Since each p-convex order continuous Kothe 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.