Complex interpolation of normed and quasinormed spaces in several dimensions. II. Properties of harmonic interpolation

Slodkowski, Zbigniew

doi:10.1090/S0002-9947-1990-0949900-2

Complex interpolation of normed and quasinormed spaces in several dimensions. II. Properties of harmonic interpolation
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Trans. Amer. Math. Soc. 317 (1990), 255-285 Request permission

Abstract:

This paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in ${{\mathbf {C}}^k}$. It is shown, among other things, that each of the following properties holds for all the intermediate quasinormed spaces, if it holds for all given boundary spaces: (1) being a normed space; (2) being a Hilbert space; (3) satisfying the triangle inequality by the $r$th power of the quasinorm; (4) being uniformly convex; and (5) being uniformly smooth. As a principal tool, the notion of a harmonic set valued function (a generalization of analytic multifunction) is introduced and studied.

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