Fine convergence and admissible convergence for symmetric spaces of rank one

Korányi, Adam; Taylor, J. C.

doi:10.1090/S0002-9947-1981-0590418-1

Fine convergence and admissible convergence for symmetric spaces of rank one
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by Adam Korányi and J. C. Taylor

Trans. Amer. Math. Soc. 263 (1981), 169-181

DOI: https://doi.org/10.1090/S0002-9947-1981-0590418-1

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Abstract:

The connections between fine convergence in the sense of potential theory and admissible convergence to the boundary for quotients of eigenfunctions of the Laplace-Beltrami operator are investigated. This leads to a version of the local Fatou theorem on symmetric spaces of rank one which is considerably stronger than previously known results. The appendix establishes the relationship between harmonic measures on the intersection of the Martin boundaries of a domain and a subdomain.

References

N. Bourbaki, Éléments de mathématique. I: Les structures fondamentales de l’analyse. Fascicule VIII. Livre III: Topologie générale. Chapitre 9: Utilisation des nombres réels en topologie générale, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1045, Hermann, Paris, 1958 (French). Deuxième édition revue et augmentée. MR 0173226
M. Brelot and J. L. Doob, Limites angulaires et limites fines, Ann. Inst. Fourier (Grenoble) 13 (1963), no. fasc. 2, 395–415 (French). MR 196107, DOI 10.5802/aif.152
A. P. Calderón, On the behaviour of harmonic functions at the boundary, Trans. Amer. Math. Soc. 68 (1950), 47–54. MR 32863, DOI 10.1090/S0002-9947-1950-0032863-9
Lennart Carleson, On the existence of boundary values for harmonic functions in several variables, Ark. Mat. 4 (1962), 393–399 (1962). MR 159013, DOI 10.1007/BF02591620
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338

Methods of mathematical physics

Amédée Debiard, Comparaison des espaces $H^{p}$ géométrique et probabilistes au-dessus de l’espace hermitien hyperbolique, Bull. Sci. Math. (2) 103 (1979), no. 3, 305–351 (French, with English summary). MR 539352
J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458. MR 109961, DOI 10.24033/bsmf.1494
J. L. Doob, A non-probabilistic proof of the relative Fatou theorem, Ann. Inst. Fourier (Grenoble) 9 (1959), 293–300. MR 117454, DOI 10.5802/aif.93
Mogens Flensted-Jensen, Paley-Wiener type theorems for a differential operator connected with symmetric spaces, Ark. Mat. 10 (1972), 143–162. MR 318974, DOI 10.1007/BF02384806
H. Föllmer, Feine Topologie am Martinrand eines Standardprozesses, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 12 (1969), 127–144 (German, with English summary). MR 245092, DOI 10.1007/BF00531645
Kohur Gowrisankaran, Extreme harmonic functions and boundary value problems, Ann. Inst. Fourier (Grenoble) 13 (1963), no. fasc. 2, 307–356. MR 164051, DOI 10.5802/aif.148
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571 (French). MR 139756, DOI 10.5802/aif.125
F. I. Karpelevič, The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces, Trans. Moscow Math. Soc. 1965 (1967), 51–199. Amer. Math. Soc., Providence, R.I., 1967. MR 0231321
Adam Korányi, Harmonic functions on Hermitian hyperbolic space, Trans. Amer. Math. Soc. 135 (1969), 507–516. MR 277747, DOI 10.1090/S0002-9947-1969-0277747-0
Adam Koranyi, Harmonic functions on symmetric spaces, Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970), Pure and Appl. Math., Vol. 8, Dekker, New York, 1972, pp. 379–412. MR 0407541
Adam Korányi, A survey of harmonic functions on symmetric spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 323–344. MR 545272
A. Korányi and R. B. Putz, Local Fatou theorem and area theorem for symmetric spaces of rank one, Trans. Amer. Math. Soc. 224 (1976), no. 1, 157–168. MR 492068, DOI 10.1090/S0002-9947-1976-0492068-2
Robert S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc. 49 (1941), 137–172. MR 3919, DOI 10.1090/S0002-9947-1941-0003919-6
H. Lee Michelson, Fatou theorems for eigenfunctions of the invariant differential operators on symmetric spaces, Trans. Amer. Math. Soc. 177 (1973), 257–274. MR 319113, DOI 10.1090/S0002-9947-1973-0319113-6
Linda Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier (Grenoble) 7 (1957), 183–281 (French). MR 100174, DOI 10.5802/aif.70
James Serrin, On the Harnack inequality for linear elliptic equations, J. Analyse Math. 4 (1955/56), 292–308. MR 81415, DOI 10.1007/BF02787725
J. C. Taylor, The Martin boundaries of equivalent sheaves, Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 1, 433–456 (English, with French summary). MR 267120, DOI 10.5802/aif.346

Fine and admissible convergence for the unit ball in

Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer-Verlag, New York-Heidelberg, 1972. MR 0498999