Abstract
The main purpose of these lectures is to study questions of elementary analysis on bounded symmetric domains, namely the realization of these domains as generalizations of the unit disc and the upper halfplane, the study of the structure of their boundary and the boundary behaviour of holomorphic functions. This is done in sections 3 to 6 which contain material otherwise available only in journals (mainly [14], [16], [24]). Some slight simplifications and improvements have been made here; it will, by the way, be apparent that the subject still has plenty of open problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. L. Baily and A. Borel, Compactification of arithmetic quotients of bounded symmetric domains, Ann. of Math. 84 (1966), 442–528
S. Bochner, Group invariance of Cauchy's formula in several variables, Ann. of Math. 45 (1944), 686–707.
H. Braun and M. Koecher, Jordan-Algebren, Springer 1966.
E. Cartan, Sur les domaines bornés homogènes de l'espace de n variables, Abh. Math. Sem. Hamburg, 11(1935), 116–162.
C. Chevalley, Lie groups, Vol. I, Princeton University Press, 1946.
S. G. Gindikin, Analysis in homogeneous domains, Uspekhi Mat. Nauk 19 (1964), 3–92 (in Russian).
Harish-Chandra, Representations of semi-simple Lie groups VI, Amer. J. Math 78(1956), 564–628
S. Helgason, Differential geometry and symmetric spaces, Academic Press, 1962.
Ch. Hertneck, Positivitatsbereiche und Jordan-Strukturen, Math. Ann. 146 (1962), 433–455.
UHirzebruchUber Jordan-Algebren und beschränkte symmetrischeGebiete, Math. Z94 (1966), 387–390
G. Hochschild, The structure of Lie groups, Holden-Day, 1965.
L. K. Hua, Harmonie analysis of functions of several complex variables in the classical domains, Amer. Math. Soc. 1963.
N. Jacobson, Lie algebras, Interscience, 1962.
A.Koranyi, The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. of Math 82(1965) 332–350
A. Koranyi, Analytic invariants of bounded symmetric domains, to appear in Proc. Amer. Math. Soc.
A. Koranyi and J. A. Wolf, Realization of Hermitian symmetric domains as generalized half-planes, Ann. of Math. 81 (1965), 265–288.
C. C Moore, Compactifications of symmetric spaces II. The Cartan domains, Amer. J. Math86(1964)358–378
I. I. Pyateckii-Shapiro, Geometry of classical domains and auto-morphic functions, Fizmatgiz 1961, (in Russian).
I. Satake, On representations and compactifications of symmetric Riemannian spaces, Ann. of Math. 71 (1960), 77–110.
E. M. Stein, Note on the boundary values of holomorphic functions, Ann. of Math. 82 (1965), 351–353.
[20a] E. B. Vinberg, The theory of convex homogeneous cones, Trudy Mosk. Mat. Obs., 12 (1963), 303–358 (in Russian); English translation in Trans. Moscow Math. Soc. for the year 1963, 340–403.
N.Weiss, Almost everywhere convergence of Poisson intgrals on tube domains over cones, to appear in Trans. Amer. Math. Soc.
J. A. Wolf, Spaces of constant curvature, Mc Graw-Hill, 1967.
J. A. Wolf, On the classification of Hermitian symmetric spaces, J. of Math., and Mechanics 13 (1964, 489–496
J. A. Wolf and A. Koranyi, Generalized Cayley transformations of bounded symmetric domains, Amer. J. Math. 87(1965), 899–939.
A. Zygmund, Trigonometric series, Cambridge University Press, 1959.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Koranyi, A. (2011). Holomorphic and Harmonic Functions on Bounded Symmetric Domains. In: Vesentini, E. (eds) Geometry of Homogeneous Bounded Domains. C.I.M.E. Summer Schools, vol 45. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11060-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-11060-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-11059-7
Online ISBN: 978-3-642-11060-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)