Interpolation of Vector Measures

Abstract

[EN] Let (Omega, Sigma) be a measurable space and m(0) : Sigma -> X-0 and m(1) : Sigma -> X-1 be positive vector measures with values in the Banach Kothe function spaces X-0 and X-1. If 0 < alpha < 1, we define a new vector measure [m(0), m(1)](alpha) with values in the Calderon lattice interpolation space (X01-alpha X1 alpha) and we analyze the space of integrable functions with respect to measure [m(0), m(1)](alpha) 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 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.