Complex interpolation for \(2^ N\) Banach spaces (Q1822315)

In this paper we study complex interpolation for a family of \(2^ N\) complex Banach spaces. These spaces correspond to the vertices of the cube \([0,1]^ N\), while the interpolation spaces correspond to the interior points of \([0,1]^ N:\) precisely, the interpolation space corresponding to \(t=(t_ 1,...,t_ N)\) is defined as the space of the values at t of the functions which belong to a class \({\mathcal F}\) of holomorphic functions on the open tube \((]0,1[+i{\mathbb{R}})^ N\), having suitable behaviours at infinity and at the boundary of the tube. Here the main idea is to make use, instead of functions which are (in some sense) continuous at the boundary, of holomorphic functions which are Poisson integrals of their values at the boundary. We prove several properties of the interpolation spaces, relating them to the classical complex method of Calderòn (which coincides with ours when \(N=1)\), and give a characterization of the dual spaces.