Interpolation of Subcouples, New Results and Applications

Abstract: Suppose that X and Y are Banach couples and suppose that there is a bounded linear couple map Q from Y to X which has the property that Q restricted to the endpoint spaces is injective and the images of the endpointspaces of Y are closed in the endpoint spaces of X, then we say that Y is a subcouple of X. If F is an interpolation functor we want to know how F(Y) is related to F(X). In particular we want to know for which F it holds that Q is an injection that maps F(Y) onto a closed subspace of F(X). In recent years interest has been paid to subcouples of finite codimension and in particular to subcouples of codimension one. We will in this thesis present an interpolation theory for subcouples of codimension one and then generalize it to finite codimension. Our theory will include both a larger class of couples and a larger class of interpolation functors than earlier results.The interpolation method that will be considered is the regular real method. Our general theory will imply older results by Kalton, Ivanov and Löfström. We will use the theory to answer questions about Hardy-type inequalities that were raised by Krugljak, Maligranda and Persson in 1999 and our new theory will also answer a question concerning interpolation of Banach algebras.