Complex interpolation for normed and quasi-normed spaces in several dimensions. III. Regularity results for harmonic interpolation
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Abstract:
The paper continues the study of one of the complex interpolation methods for families of finite-dimensional normed spaces ${\{ {{\mathbf {C}}^n},|| \cdot |{|_z}\} _{z \in G}}$, 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 $(z,w) \to ||w||_z^2$ 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.References
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Additional Information
- © Copyright 1990 American Mathematical Society
- 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