Pseudoconvex classes of functions. III. Characterization of dual pseudoconvex classes on complex homogeneous spaces

Author:
Zbigniew Slodkowski

Journal:
Trans. Amer. Math. Soc. **309** (1988), 165-189

MSC:
Primary 32F05; Secondary 32M10

DOI:
https://doi.org/10.1090/S0002-9947-1988-0957066-9

MathSciNet review:
957066

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**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), 3-35. 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, Springer-Verlag, Berlin and New York, 1980, pp. 123-153. MR**576042 (81k:46075)****[4]**-,*A theory of complex interpolation of families of Banach spaces*, Adv. in Math.**33**(1982), 203-229. MR**648799 (83j:46084)****[5]**R. L. Hunt and J. J. Murray, -*plurisubharmonic functions and a generalized Dirichlet problem*, Michigan Math. J.**25**(1978), 299-316. 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*, North-Holland, Amsterdam, 1968. MR**0251745 (40:4972)****[9]**Z. Slodkowski,*The Bremermann-Dirichlet problem for*-*plurisubharmonic functions*, Ann. Scuola Norm. Sup. (Pisa) Ser. IV**11**(1984), 303-326. MR**764948 (86a:32038)****[10]**-,*Local maximum property and*-*plurisubharmonic functions in uniform algebras*, J. Math. Anal. Appl.**115**(1986), 105-130. MR**835588 (87j:32050)****[11]**-,*Pseudoconvex classes of functions*. I,*Pseudoconcave and pseudoconvex sets*, Pacific J. Math.**134**(1988), 343-376. MR**961240 (89m:32031)****[12]**-,*Pseudoconvex classes of functions*, II (submitted)**[13]**-,*Complex interpolation of normed and quasi-normed 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. 253-276. MR**740887 (85j:32031)****[15]**R. Coifman and S. Semmes,*Interpolation of Banach spaces and nonlinear Dirichlet problems*.

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1988-0957066-9

Keywords:
Pseudoconvex classes,
subharmonic functions,
-plurisubharmonic functions,
dual classes of functions,
complex homogeneous spaces

Article copyright:
© Copyright 1988
American Mathematical Society