Embedding functions and their role in interpolation theory

Abstract

The embedding functions of an intermediate space $A$ into a Banach couple $\left(A_0,A_1\right)$ are defined as its embedding constants into the couples $\left(\frac{1}{\alpha }{A}_0,\frac{1}{\beta }{A}_1\right)$, $\forall \alpha ,\beta >0$. Using these functions, we study properties and interrelations of different intermediate spaces, give a new description of all real interpolation spaces, and generalize the concept of weak-type interpolation to any Banach couple to obtain new interpolation theorems.