Interpolation of Vector Measures

• Published in: Acta mathematica Sinica. English series Vol. 27; no. 1; pp. 119 - 134
• Main Author: Ricardo del CAMPO Antonio FERNANDEZ Fernando MAYORAL Francisco NARANJO Enrique A. SANCHEZ-PEREZ
• Format: Journal Article
• Language: English
• Published: Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 2011; Springer Nature B.V
• Subjects: Banach; Equality; Function space; Interpolation; Lattices; Mathematical analysis; Mathematical functions; Mathematics; Mathematics and Statistics; Norms; Representations; Vectors (mathematics); 可测空间; 可积函数空间; 插值公式; 空间分析; 空间测量; Interpolation; 46G10; vector measure; Banach function space; 46E30; 46B70
• ISSN: 1439-8516; 1439-7617
• DOI: 10.1007/s10114-011-9542-8

Let （Ω, ∑） be a measurable space and mo ： E→ Xo and m1 ： E → X1 be positive vector measures with values in the Banach KSthe function spaces Xo and X1. If 0 〈 a 〈 1, we define a X01-ax1a new vector measure [m0, m]a with values in the Calderdn lattice interpolation space and we analyze the space of integrable functions with respect to measure [m0, m1]a 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.