An F. and M. Riesz theorem for bounded symmetric domains
Annales de l'Institut Fourier, Volume 37 (1987) no. 2, p. 139-150

We generalize the classical F. and M. Riesz theorem to metrizable compact groups whose center contains a copy of the circle group. Important examples of such groups are the isotropy groups of the bounded symmetric domains.

The proof uses a criterion for absolute continuity involving L p spaces with p<1: A measure μ on a compact metrisable group K is absolutely continuous with respect to Haar measure dk on K if for some p<1 a certain subspace of L p (K,dk) which is related to μ has sufficiently many continuous linear functionals to separate its points. For abelian K this criterion is due to J.H. Shapiro.

Le théorème de F. et M. Riesz classique est étendu aux groupes compacts et métrisables dont le centre contient une copie du groupe du cercle. Des exemples importants de tels groupes sont les groupes d’isotropie des domaines symétriques bornés.

La preuve se sert d’un critère pour la continuité absolue qui emploie les espaces L p avec p<1 : une mesure μ sur un groupe K métrisable et compact est absolument continue par rapport à la mesure de Haar dk de K si, pour un p<1, un certain sous-espace de L p (K,dk), dépendant de μ, a un nombre suffisant de fonctionnelles linéaires continues pour séparer les points. Si K est abélien ce critère est dû à J.H. Shapiro.

@article{AIF_1987__37_2_139_0,
     author = {Brummelhuis, R. G. M.},
     title = {An F. and M. Riesz theorem for bounded symmetric domains},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {37},
     number = {2},
     year = {1987},
     pages = {139-150},
     doi = {10.5802/aif.1090},
     zbl = {0607.43002},
     mrnumber = {89c:43002},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1987__37_2_139_0}
}
Brummelhuis, R. G. M. An F. and M. Riesz theorem for bounded symmetric domains. Annales de l'Institut Fourier, Volume 37 (1987) no. 2, pp. 139-150. doi : 10.5802/aif.1090. http://www.numdam.org/item/AIF_1987__37_2_139_0/

[1] A. B. Aleksandrov, Existence of inner functions in the unit ball, Mat. Sb., 118 (160), N2 (6) (1982), 147-163. | MR 83i:32002 | Zbl 0503.32001

[2] A. B. Aleksandrov, Essays on non locally convex Hardy classes, Complex Analysis and Spectral theory, Seminar, Leningrad 1979/1980, V. P. Havin and N. K. Nikol'skii (ed.), 1-89. | MR 84h:46066 | Zbl 0482.46035

[3] P. L. Duren, Theory of Hp Spaces, Acad. Press, New York, 1970. | MR 42 #3552 | Zbl 0215.20203

[4] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Acad. Press, New York, 1978. | Zbl 0451.53038

[5] Y. Kanjin, A convolution measure algebra on the unit disc, Tohoku Math. J., 28 (1976), 105-115. | MR 53 #1178 | Zbl 0321.43011

[6] A. Koranyi, Holomorphic and harmonic functions on bounded symmetric domains, C.I.M.E. summer course on Geometry of Bounded Homogeneous Domains, Cremonese, Roma, 1968, 125-197. | MR 38 #6098 | Zbl 0167.06702

[7] W. Rudin, Function Theory in the Unit Ball of Cn, Springer Verlag, Berlin, 1980. | MR 82i:32002 | Zbl 0495.32001

[8] W. Rudin, Inner functions in the unit ball of Cn, J. Funct. Anal, 50 (1983), 100-126. | MR 84i:32007 | Zbl 0554.32002

[9] W. Rudin, Fourier Analysis on Groups, Interscience, John Wiley, 1960.

[10] W. Rudin, Trigonometric series with gaps, J. Math. Mech., 9 (1960), 203-228. | MR 22 #6972 | Zbl 0091.05802

[11] J. H. Shapiro, Subspaces of Lp(G) spanned by characters, 0 < p < 1, Israel J. Math., 29, Nos 2-3 (1978), 248-264. | MR 57 #17123 | Zbl 0382.46015

[12] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Raumen, Invent. Math., 9 (1969), 61-80. | MR 41 #3806 | Zbl 0219.32013

[13] E. M. Stein, Note on the boundary values of holomorphic functions, Ann. of Math., 82 (1965), 351-353. | MR 32 #5923 | Zbl 0173.09004

[14] S. Vági, Harmonic analysis on Cartan and Siegel domains, M.A.A. Studies in Math., vol. 13 : Studies in Harmonic Analysis, J. Ash (ed.), 257-309. | MR 57 #16719 | Zbl 0352.32031