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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Complex interpolation for normed and quasi-normed spaces in several dimensions. III. Regularity results for harmonic interpolation
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by Zbigniew Slodkowski PDF
Trans. Amer. Math. Soc. 321 (1990), 305-332 Request permission

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.
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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