Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Complex interpolation of normed and quasinormed spaces in several dimensions. I


Author: Zbigniew Slodkowski
Journal: Trans. Amer. Math. Soc. 308 (1988), 685-711
MSC: Primary 32F05; Secondary 32E30, 32M10, 46M35
DOI: https://doi.org/10.1090/S0002-9947-1988-0951623-1
MathSciNet review: 951623
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A variety of complex interpolation methods for families of normed or quasi-normed spaces, parametrized by points of domains in complex homogeneous spaces, parametrized by points of domains in complex homogeneous spaces, is developed. Results on existence, continuity, uniqueness, reiteration and duality for interpolation are proved, as well as on interpolation of operators. A minimum principle for plurisubharmonic functions is obtained and used as a tool for the duality theorem.


References [Enhancements On Off] (What's this?)

  • [1] R. Coifman, M. Cwikel, R. Rochberg, Y. Sagher and G. Weiss, The complex method for interpolation of operators acting on families of Banach spaces, Lecture Notes in Math., vol. 779, Springer-Verlag, Berlin and New York, 1980, pp. 123-153. MR 576042 (81k:46075)
  • [2] -, A theory of complex interpolation for families of Banach spaces, Adv. in Math. 33 (1982), 203-229. MR 648799 (83j:46084)
  • [3] R. L. Hunt and J. J. Murray, $ q$-plurisubharmonic functions and a generalized Dirichlet problem, Michigan Math. J. 25 (1978), 299-316. MR 512901 (80b:32018)
  • [4] C. O. Kiselman, The partial Legendre transformation for plurisubharmonic functions, Invent. Math. 49 (1978), 137-148. MR 511187 (80d:32014)
  • [5] -, How smooth is the shadow of a smooth convex body? preprint.
  • [6] S. Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975), 153-166. MR 0377126 (51:13299)
  • [7] R. Narasimhan, Analysis on real and complex manifolds, North-Holland, Amsterdam, 1968. MR 0251745 (40:4972)
  • [8] R. Rochberg, Interpolation of Banach spaces and negatively curved vector bundles, Pacific J. Math. 110 (1984), 355-376. MR 726495 (85m:46077)
  • [9] -, Function theoretic results for complex interpolation families of Banach spaces, Trans. Amer. Math. Soc. 284 (1984), 745-758. MR 743742 (85m:46078)
  • [10] R. Rochberg and G. Weiss, Some topics in complex interpolation theory, Topics in Modern Harmonic Analysis, Ist. Naz. Alta Mat. F. Severi, Roma, 1983, pp. 769-818. MR 748883 (86a:46100)
  • [11] W. Rudin, Function theory in the unit ball of $ {C^n}$, Springer-Verlag, New York, 1980. MR 601594 (82i:32002)
  • [12] Z. Slodkowski, The Bremermann-Dirichlet problem for $ q$-plurisubharmonic functions, Ann. Scuola Norm. Sup. Pisa, Cl.-Sci. (4) 11 (1984), 303-326. MR 764948 (86a:32038)
  • [13] -, Local maximum property and $ q$-plurisubharmonic functions in uniform algebras, J. Math. Anal. Appl. 115 (1986), 105-130. MR 835588 (87j:32050)
  • [14] -, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), 255-260. MR 818455 (87c:32023)
  • [15] -, An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra, Trans. Amer. Math. Soc. 294 (1986), 367-377. MR 819954 (87e:32021)
  • [16] -, Complex interpolation families of normed spaces over several-dimensional parameter space. Abstracts of the Special Session in Several Complex Variables, 826th Meeting of the AMS, Indianapolis, April 1986.
  • [17] -, On complex interpolation methods for families of normed spaces over domains in $ {{\mathbf{C}}^k}$, Talk at International Conference on Harmonic Measure, Toldeo, Ohio, July, 1986.
  • [18] -, Pseudoconvex classes of functions I. Pseudoconcave and pseudoconvex sets, Pacific J. Math. (to appear). MR 961240 (89m:32031)
  • [19] -, Pseudoconvex class of functions. II. Affine pseudoconvex classes on $ {R^N}$ (submitted).
  • [20] -, Pseudoconvex classes of functions. III. Characaterizations of dual pseudoconvex classes on complex homogeneous spaces. Trans. Amer. Math. Soc. (to appear). MR 957066 (89m:32032)
  • [21] -, Pseudoconvex classes of functions. IV (in preparation).
  • [22] J. B. Walsh, Continuity of envelopes of plurisubharmonic functions, J. Math. Mech. 18 (1986), 143-148. MR 0227465 (37:3049)
  • [23] T. Wolff, A note on interpolation spaces, Harmonic Analysis, Lecture Notes in Math., vol. 908, Springer-Verlag, Berlin and New York, 1982, pp. 199-204. MR 654187 (83f:46082)
  • [24] R. Coifman and S. Semmes, Interpolation of Banach spaces and non-linear Dirichlet problems, preprint.
  • [25] R. Rochberg, The work of Coifman and Semmes on complex interpolation, several complex variables and $ PDE$'s, US-Swedish Seminar on Function Spaces and Applications, Lund, June, 1986 (to appear).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32F05, 32E30, 32M10, 46M35

Retrieve articles in all journals with MSC: 32F05, 32E30, 32M10, 46M35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1988-0951623-1
Keywords: Interpolation spaces, subinterpolation and superinterpolation families, plurisubharmonic functions, subharmonic functions, pseudoconvex classes, Dirichlet problem, complex homogeneous spaces, strictly pseudoconvex domains
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society