Henkin measures, Riesz products and singular sets
Annales de l'Institut Fourier, Volume 48 (1998) no. 3, pp. 699-728.

The mutual singularity problem for measures with restrictions on the spectrum is studied. The d-pluriharmonic Riesz product construction on the complex sphere is introduced. Singular pluriharmonic measures supported by sets of maximal Hausdorff dimension are obtained.

On considère le problème de singularité mutuelle pour les mesures ayant des restrictions sur le spectre. On introduit les produits de Riesz pluriharmoniques sur la sphère complexe. On obtient des mesures singulières pluriharmoniques supportées par les ensembles de dimension de Hausdorff maximale.

     author = {Doubtsov, Evgueni},
     title = {Henkin measures, {Riesz} products and singular sets},
     journal = {Annales de l'Institut Fourier},
     pages = {699--728},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {3},
     year = {1998},
     doi = {10.5802/aif.1635},
     zbl = {0913.31003},
     mrnumber = {99k:32020},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1635/}
AU  - Doubtsov, Evgueni
TI  - Henkin measures, Riesz products and singular sets
JO  - Annales de l'Institut Fourier
PY  - 1998
SP  - 699
EP  - 728
VL  - 48
IS  - 3
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1635/
DO  - 10.5802/aif.1635
LA  - en
ID  - AIF_1998__48_3_699_0
ER  - 
%0 Journal Article
%A Doubtsov, Evgueni
%T Henkin measures, Riesz products and singular sets
%J Annales de l'Institut Fourier
%D 1998
%P 699-728
%V 48
%N 3
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1635/
%R 10.5802/aif.1635
%G en
%F AIF_1998__48_3_699_0
Doubtsov, Evgueni. Henkin measures, Riesz products and singular sets. Annales de l'Institut Fourier, Volume 48 (1998) no. 3, pp. 699-728. doi : 10.5802/aif.1635. https://aif.centre-mersenne.org/articles/10.5802/aif.1635/

[A1] A.B. Aleksandrov, Inner functions on compact spaces, Funct. Anal. Appl., 18 (1984), 87-98. | MR | Zbl

[A2] A.B. Aleksandrov, Function theory in the ball, in: G. M. Khenkin and A. G. Vitushkin (Eds.), Encyclopaedia Math. Sci., 8 (Several Complex Variables II), Springer, Berlin, 1994, 107-178. | Zbl

[AAN] A.B. Aleksandrov, J.M. Anderson, and A. Nicolau, Inner functions, Bloch spaces and symmetric measures, preprint, 1997.

[AB] H. Alexander and J. Bruna, Pluriharmonic interpolation and hulls of C1 curves in the unit sphere, Rev. Mat. Iberoamericana, 11 (1995), 547-568. | MR | Zbl

[Bi] P. Billingsley, Ergodic theory and information, Wiley, New York, 1965. | MR | Zbl

[B1] J. Bourgain, The Dunford-Pettis property for the ball-algebras, the polydisc-algebras and the Sobolev spaces, Studia Math., 77 (1984), 245-253. | MR | Zbl

[B2] J. Bourgain, Applications of the spaces of homogeneous polynomials to some problems on the ball algebra, Proc. Amer. Math. Soc., 93 (1985), 277-283. | MR | Zbl

[Br] R.J.M. Brummelhuis, An F. and M. Riesz theorem for bounded symmetric domains, Ann. Inst. Fourier, Grenoble, 37-2 (1987), 139-150. | Numdam | MR | Zbl

[CT] J. Cima and R. Timoney, The Dunford-Pettis property for certain planar uniform algebras, Mich. Math. J., 34 (1987), 99-104. | MR | Zbl

[D1] E.S. Dubtsov (=Doubtsov), Singular parts of pluriharmonic measures, Zap. Nauchn. Sem. S.Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 217 (1994), 54-58 (Russian) ; English transl., J. Math. Sci. (New York), 85-2 (1997), 1790-1793. | EuDML | Zbl

[D2] E.S. Dubtsov, Some questions of the harmonic analysis on a complex sphere, Vestnik St. Petersburg Univ. Math., 28-1 (1995), 12-16. | MR | Zbl

[D3] E. Doubtsov, Regular unitarily invariant spaces on the complex sphere, submitted. | Zbl

[He] B.S. Henriksen, A peak set of Hausdorff dimension 2n-1 for the algebra A(D) in the boundary of a domain D with C∞-boundary in Cn, Math. Ann., 259 (1982), 271-277. | EuDML | MR | Zbl

[HV] S.V. Hruščëv and S.A. Vinogradov, Free interpolation in the space of uniformly convergent Taylor series, Lect. Notes Math., 864 (1981), 171-213. | MR | Zbl

[I] K. Izuchi, Bourgain algebras of the disc, polydisc and ball algebras, Duke Math. J., 66 (1992), 503-519. | MR | Zbl

[M] D.E. Menchoff, Sur l'unicité du développement trigonométrique, C. R. Acad. Sci. Paris, Sér. A-B, 163 (1916), 433-436. | JFM

[N] A. Nagel, Cauchy transforms of measures and a characterization of smooth peak interpolation sets for the ball algebra, Rocky Mountain J. Math., 9 (1979), 299-305. | MR | Zbl

[P] A. Peyrière, Étude de quelques propriétés des produits de Riesz, Ann. Inst. Fourier, Grenoble, 25-2 (1975), 127-169. | EuDML | Numdam | MR | Zbl

[RS] J.-P. Rosay and E.L. Stout, On pluriharmonic interpolation, Math. Scand., 63 (1988), 268-281. | EuDML | MR | Zbl

[Ru] W. Rudin, Function Theory in the Unit Ball of Cn, Grundlehren Math. Wiss., 241, Springer, Berlin, 1980. | MR | Zbl

[RW] J. Ryll and P. Wojtaszczyk, On homogeneous polynomials on a complex ball, Trans. Amer. Math. Soc., 276 (1983), 107-116. | MR | Zbl

[Z] A. Zygmund, Trigonometric Series, vol. 1, Cambridge Univ. Press, 1959. | Zbl

Cited by Sources: