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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Complex interpolation for normed and quasi-normed spaces in several dimensions. III. Regularity results for harmonic interpolation


Author: Zbigniew Slodkowski
Journal: Trans. Amer. Math. Soc. 321 (1990), 305-332
MSC: Primary 46M35; Secondary 32F05, 46B70
DOI: https://doi.org/10.1090/S0002-9947-1990-0991968-1
MathSciNet review: 991968
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Abstract: The paper continues the study of one of the complex interpolation methods for families of finite-dimensional normed spaces $ {\{ {{\mathbf{C}}^n},\vert\vert \cdot \vert{\vert _z}\} _{z \in G}}$, where $ G$ is open and bounded in $ {{\mathbf{C}}^k}$. The main result asserts that (under a mild assumption on the datum) the norm function $ (z,w) \to \vert\vert w\vert\vert _z^2$ belongs to some anisotropic Sobolew class and is characterized by a nonlinear PDE of second order. The proof uses the duality theorem for the harmonic interpolation method (obtained earlier by the author). A new, simpler proof of this duality relation is also presented in the paper.


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DOI: https://doi.org/10.1090/S0002-9947-1990-0991968-1
Keywords: Plurisubharmonic functions, Hessian form, weak derivative, dual norm, interpolation family, harmonic multifunction
Article copyright: © Copyright 1990 American Mathematical Society