Causal compactification and Hardy spaces

Ólafsson, G.; Ørsted, B.

doi:10.1090/S0002-9947-99-02101-7

Causal compactification and Hardy spaces
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by G. Ólafsson and B. Ørsted PDF

Trans. Amer. Math. Soc. 351 (1999), 3771-3792 Request permission

Abstract:

Let $\mathcal {M}=G/H$ be a irreducible symmetric space of Cayley type. Then $\mathcal {M}$ is diffeomorphic to an open and dense $G$-orbit in the Shilov boundary of $G/K\times G/K$. This compactification of $\mathcal {M}$ is causal and can be used to give answers to questions in harmonic analysis on $\mathcal {M}$. In particular we relate the Hardy space of $\mathcal {M}$ to the classical Hardy space on the bounded symmetric domain $G/K\times G/K$. This gives a new formula for the Cauchy-Szegö kernel for $\mathcal {M}$.

References

Mohammed Chadli, Domaine complexe associé à un espace symétrique de type Cayley, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 9, 1157–1162 (French, with English and French summaries). MR 1360776
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
Joachim Hilgert and Gestur Ólafsson, Causal symmetric spaces, Perspectives in Mathematics, vol. 18, Academic Press, Inc., San Diego, CA, 1997. Geometry and harmonic analysis. MR 1407033
J. Hilgert, G. Ólafsson, and B. Ørsted, Hardy spaces on affine symmetric spaces, J. Reine Angew. Math. 415 (1991), 189–218. MR 1096906
Soji Kaneyuki, On orbit structure of compactifications of para-Hermitian symmetric spaces, Japan. J. Math. (N.S.) 13 (1987), no. 2, 333–370. MR 921587, DOI 10.4099/math1924.13.333
Adam Korányi and Joseph A. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math. (2) 81 (1965), 265–288. MR 174787, DOI 10.2307/1970616
Khalid Koufany, Réalisation des espaces symétriques de type Cayley, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 5, 425–428 (French, with English and French summaries). MR 1267820
Khalid Koufany and Bent Ørsted, Espace de Hardy sur le semi-groupe métaplectique, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 2, 113–116 (French, with English and French summaries). MR 1373745
Ottmar Loos, Symmetric spaces. I: General theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0239005
V. F. Molchanov, Holomorphic discrete series for hyperboloids of Hermitian type, J. Funct. Anal. 147 (1997), 26–50.
Gestur Ólafsson, Fourier and Poisson transformation associated to a semisimple symmetric space, Invent. Math. 90 (1987), no. 3, 605–629. MR 914851, DOI 10.1007/BF01389180
G. Ólafsson, Causal symmetric spaces. Mathematica Gottingensis. 15 (1990).
G. Ólafsson, Symmetric spaces of Hermitian type, Differential Geom. Appl. 1 (1991), no. 3, 195–233. MR 1244444, DOI 10.1016/0926-2245(91)90001-P
G. Ólafsson, Spherical Functions and Spherical Laplace Transform on Ordered Symmetric Spaces. Preprint, 1997.
G. Ólafsson and B. Ørsted, The holomorphic discrete series for affine symmetric spaces. I, J. Funct. Anal. 81 (1988), no. 1, 126–159. MR 967894, DOI 10.1016/0022-1236(88)90115-2
G. Ólafsson and B. Ørsted, The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels, Trans. Amer. Math. Soc. 326 (1991), no. 1, 385–405. MR 1002923, DOI 10.1090/S0002-9947-1991-1002923-0
Wilfried Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969/70), 61–80 (German). MR 259164, DOI 10.1007/BF01389889
Robert J. Stanton, Analytic extension of the holomorphic discrete series, Amer. J. Math. 108 (1986), no. 6, 1411–1424. MR 868896, DOI 10.2307/2374530
E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions; Reprint of the fourth (1927) edition. MR 1424469, DOI 10.1017/CBO9780511608759