Interpolation of vector measures
Let \((\Omega,\Sigma)\) be a measurable space and \(m_{0}: \Sigma \rightarrow X_{0}\) and \(m_{1}: \Sigma \rightarrow X_{1}\) be positive vector measures with values in the Banach Köthe function spaces \(X _{0}\) and \(X _{1}\). If \(0 < \alpha < 1\), the authors define a new vector measure \([m_{0}, m_{1}]_{\alpha}\) with values in the Calderón lattice interpolation space \(X_{0}^{1-\alpha} X_{1}^{\alpha}\). They analyse the space of integrable functions with respect to the 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. More precisely, their aim is to study under which assumptions on the vector measures and the indices involved the formula \([L^{p_0}(m_0), L^{p_1}(m_1)]_\theta=L^p([m_0,m_1]_\alpha)\) holds isometrically, where \(1/p=(1-\theta)/p_0 + \theta/p_1\) and \(\alpha=p \theta/p_1\). For the case \(m_0=m_1\), this is essentially known to be true by a result of the authors (minus the first named author) [Collect.\ Math.\ 61, No.\ 3, 241--252 (2010; Zbl 1228.46017)]. However, in the more general case, matters turn out to be more delicate (the authors provide a counterexample). Eventually, the authors prove that the above formula (where the complex interpolation space may have to be replaced by the Calderón lattice interpolation space to avoid more restrictions) holds whenever the same formula holds for the case \(p_0=p_1=1\) and \(\theta=\alpha\) (the authors state some affirmative examples). As applications, they 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.
scientific article; zbMATH DE number 5912873
scientific article