Pseudoconvex classes of functions. III. Characterization of dual pseudoconvex classes on complex homogeneous spaces
Author:
Zbigniew Slodkowski
Journal:
Trans. Amer. Math. Soc. 309 (1988), 165189
MSC:
Primary 32F05; Secondary 32M10
MathSciNet review:
957066
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Abstract: Invariant classes of functions on complex homogeneous spaces, with properties similar to those of the class of plurisubharmonic functions, are studied. The main tool is a regularization method for these classes, and the main theorem characterizes dual classes of functions (where duality is defined in terms of the local maximum property). These results are crucial in proving a duality theorem for complex interpolation of normed spaces, which is given elsewhere.
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 A. D. Alexandrov, Almost everywhere existence of the second differential of a convex function and properties of convex surfaces connected with it, Leningrad State Univ. Ann. Math. Ser. 6 (1939), 335. Russian
 [2]
 H. Buseman, Convex surfaces, Interscience, New York, 1958. MR 0105155 (21:3900)
 [3]
 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, SpringerVerlag, Berlin and New York, 1980, pp. 123153. MR 576042 (81k:46075)
 [4]
 , A theory of complex interpolation of families of Banach spaces, Adv. in Math. 33 (1982), 203229. MR 648799 (83j:46084)
 [5]
 R. L. Hunt and J. J. Murray, plurisubharmonic functions and a generalized Dirichlet problem, Michigan Math. J. 25 (1978), 299316. MR 512901 (80b:32018)
 [6]
 S. Kobayashi and K. Nomizu, Foundations of differential geometry, Interscience, New York, 1969.
 [7]
 S. G. Krantz, Function theory of several complex variables, Wiley, New York, 1982. MR 635928 (84c:32001)
 [8]
 R. Narasimhan, Analysis on real and complex manifolds, NorthHolland, Amsterdam, 1968. MR 0251745 (40:4972)
 [9]
 Z. Slodkowski, The BremermannDirichlet problem for plurisubharmonic functions, Ann. Scuola Norm. Sup. (Pisa) Ser. IV 11 (1984), 303326. MR 764948 (86a:32038)
 [10]
 , Local maximum property and plurisubharmonic functions in uniform algebras, J. Math. Anal. Appl. 115 (1986), 105130. MR 835588 (87j:32050)
 [11]
 , Pseudoconvex classes of functions. I, Pseudoconcave and pseudoconvex sets, Pacific J. Math. 134 (1988), 343376. MR 961240 (89m:32031)
 [12]
 , Pseudoconvex classes of functions, II (submitted)
 [13]
 , Complex interpolation of normed and quasinormed spaces in several dimensions. I, Trans. Amer. Math. Soc. (to appear).
 [14]
 H. Wu, On certain Kähler manifolds which are complete, Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, R.I., 1984, pp. 253276. MR 740887 (85j:32031)
 [15]
 R. Coifman and S. Semmes, Interpolation of Banach spaces and nonlinear Dirichlet problems.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198809570669
PII:
S 00029947(1988)09570669
Keywords:
Pseudoconvex classes,
subharmonic functions,
plurisubharmonic functions,
dual classes of functions,
complex homogeneous spaces
Article copyright:
© Copyright 1988
American Mathematical Society
