Theory of Bergman Spaces in the Unit Ball of ${\mathbb{C}}^n$

Theory of Bergman Spaces in the Unit Ball of

ℂ^{n}

[Théorie des espaces de Bergman dans la boule unité de

ℂ^{n}

]

Zhao, Ruhan ; Zhu, Kehe

Mémoires de la Société Mathématique de France, no. 115 (2008) , 109 p.

Résumé
Abstract

Ces dernières années il y a eu un grand nombre de travaux sur les espaces de Bergman pondérés $A_{α}^{p}$ sur la boule unité $𝔹_{n}$ de $ℂ^{n}$ , où $0 < p < \infty$ et $α > - 1$ . Nous étendons cette étude, de manière très naturelle, au cas où $α$ est un nombre réel quelconque et $0 < p \leq \infty$ . Ce traitement unifié couvre tous les espaces de Bergman classiques, les espaces de Bésov, de Lipschitz, l’espace de Bloch, l’espace $H^{2}$ de Hardy, et celui appelé espace d’Arveson. Certains de nos résultats autour de la représentation entière, de l’interpolation complexe, des multiplicateurs de coefficients et des mesures de Carleson, sont nouveaux, y compris pour les espaces de Bergman ordinaires (non-pondérés) sur le disque unité.

There has been a great deal of work done in recent years on weighted Bergman spaces $A_{α}^{p}$ on the unit ball $𝔹_{n}$ of $ℂ^{n}$ , where $0 < p < \infty$ and $α > - 1$ . We extend this study in a very natural way to the case where $α$ is any real number and $0 < p \leq \infty$ . This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space $H^{2}$ , and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergman spaces of the unit disk.

MR Zbl

DOI : 10.24033/msmf.427

Classification : 32A36, 32A18
Mots clés : Unit ball, Bergman space, Lipschitz space, Bloch space, Arveson space, Besov space, Carleson measure, fractional derivative, integral representation, atomic decomposition, complex interpolation, coefficient multiplier

@book{MSMF_2008_2_115__1_0,
     author = {Zhao, Ruhan and Zhu, Kehe},
     title = {Theory of {Bergman} {Spaces} in the {Unit} {Ball} of ${\mathbb{C}}^n$},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {115},
     year = {2008},
     doi = {10.24033/msmf.427},
     mrnumber = {2537698},
     zbl = {1176.32001},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2008_2_115__1_0/}
}

TY  - BOOK
AU  - Zhao, Ruhan
AU  - Zhu, Kehe
TI  - Theory of Bergman Spaces in the Unit Ball of ${\mathbb{C}}^n$
T3  - Mémoires de la Société Mathématique de France
PY  - 2008
IS  - 115
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/MSMF_2008_2_115__1_0/
DO  - 10.24033/msmf.427
LA  - en
ID  - MSMF_2008_2_115__1_0
ER  -

%0 Book
%A Zhao, Ruhan
%A Zhu, Kehe
%T Theory of Bergman Spaces in the Unit Ball of ${\mathbb{C}}^n$
%S Mémoires de la Société Mathématique de France
%D 2008
%N 115
%I Société mathématique de France
%U http://www.numdam.org/item/MSMF_2008_2_115__1_0/
%R 10.24033/msmf.427
%G en
%F MSMF_2008_2_115__1_0

Zhao, Ruhan; Zhu, Kehe. Theory of Bergman Spaces in the Unit Ball of ${\mathbb{C}}^n$. Mémoires de la Société Mathématique de France, Série 2, no. 115 (2008), 109 p. doi : 10.24033/msmf.427. http://numdam.org/item/MSMF_2008_2_115__1_0/

Bibliographie
Cité par

[1] P. Ahern & W. Cohn – « Besov spaces, Sobolev spaces and Cauchy integrals », Michigan Math. J. 39 (1972), p. 239–261. | MR | Zbl

[2] A. Aleksandrov – Function theory in the unit ball, Several Complex Variables II, G.M. Khenkin and A.G. Vitushkin, ed., Springer, 1994.

[3] J. Anderson, J. Clunie & C. Pommerenke – « On Bloch functions and normal functions », J. reine angew. Math. 270, p. 12–37. | MR | EuDML | Zbl

[4] J. Arazy, S. Fisher, S. Janson & J. Peetre – « Membership of Hankel operators on the ball in unitary ideals », J. London Math. Soc. 43 (1991), p. 485–508. | MR | Zbl

[5] N. Arcozzi – « Carleson measures for analytic Besov spaces: the upper triangle case »,, J. Inequal. Pure Appl. Math., 6 (2005) no. 1, Art. 13. | MR | EuDML | Zbl

[6] N. Arcozzi, R. Rochberg & E. Sawyer – « Carleson measures for analytic Besov spaces », Rev. Mat. Iberoamericana 18 (2002), p. 443–510. | MR | EuDML | Zbl

[7] —, Carleson measures and interpolating sequences for Besov spaces on complex balls, Memoirs Amer. Math. Soc., vol. 859, 2006. | Zbl

[8] W. Arveson – « Subalgebras of $C^{*}$ -algebras III, multivariable operator theory », Acta Math. 181 (1998), p. 159–228. | MR | Zbl

[9] F. Beatrous – « Estimates for derivatives of holomorphic functions in pseudoconvex domains », Math. Z. 191 (1986), p. 91–116. | MR | EuDML | Zbl

[10] F. Beatrous & J. Burbea – « Characterizations of spaces of holomorphic functions in the ball », Kodai Math. J. 8 (1985), p. 36–51. | MR | Zbl

[11] —, Holomorphic Sobolev spaces on the ball, Dissertationes Math., Warszawa, vol. 276, 1989.

[12] J. Bennet, D. Stegenga & R. Timoney – « Coefficients of Bloch and Lipschitz functions », Illinois J. Math. 25 (1981), p. 520–531. | MR | Zbl

[13] C. Bennett & R. Sharpley – Interpolation of Operators, Academic Press, New York, 1988. | MR | Zbl

[14] J. Bergh & J. Löfström – Interpolation Spaces: An Introduction, Grundlehrem, vol. 223, Springer, Berlin, 1976. | MR | Zbl

[15] L. Carleson – « An interpolation problem for bounded analytic functions », Amer. J. Math. 80 (1958), p. 921–930. | MR | Zbl

[16] —, « Interpolation by analytic functions and the corona problem », Ann. Math. 76 (1962), p. 547–559. | Zbl

[17] X. Chen & K. Guo – Analytic Hilbert Modules, Chapman Hall/CRC Press, Boca Raton, 2003. | MR

[18] B. R. Choe – « Projections, the weighted Bergman spaces and the Bloch space », Proc. Amer. Math. Soc. 108 (1990), p. 127–136. | MR | Zbl

[19] B. R. Choe, H. Koo & H. Yi – « Positive Toeplitz operators between harmonic Bergman spaces », Potential Anal. 17 (2002), p. 307–335. | MR | Zbl

[20] J. Cima & W. Wogen – « A Carleson measure theorem for the Bergman space of the ball », J. Operator Theory 7 (1982), p. 157–165. | MR | Zbl

[21] R. Coifman & R. Rochberg – « Representation theorems for holomorphic and harmonic functions », Astérisque 77 (1980), p. 11–66. | MR | Zbl | Numdam

[22] R. Coifman, R. Rochberg & G. Weiss – « Factorization theorems for Hardy spaces of several complex variables », Ann. Math. 103 (1976), p. 611–635. | MR | Zbl

[23] P. Duren – Theory of $H^{p}$ Spaces, Academic Press, New York, 1970. | MR

[24] P. Duren, B. Romberg & A. Shields – « Linear functionals on $H^{p}$ spaces with $0 < p < 1$ », J. reine angew. Math. 238 (1969), p. 32–60. | MR | EuDML | Zbl

[25] F. Forelli & W. Rudin – « Projections on spaces of holomorphic functions on balls », Indiana Univ. Math. J. 24 (1974), p. 593–602. | MR | Zbl

[26] M. Frazier & B. Jawerth – « Decomposition of Besov spaces », Indiana Univ. Math. J. 34 (1985), p. 777–799. | MR | Zbl

[27] J. Garnett – Bounded Analytic Functions, Academic Press, New York, 1981. | MR | Zbl

[28] I. Graham – « The radial derivative, fractional integrals and the comparative growth of means of holomorphic functions on the unit ball in $ℂ^{n}$ », in Recent Developments in Several Complex Variables, vol. 100, Ann. Math. Studies, 1981, p. 171–178. | MR

[29] K. T. Hahn & E. H. Youssfi – « M-harmonic Besov spaces and Hankel operators on the Bergman space on ball of $ℂ^{n}$ », Manuscripta Math. 71 (1991), p. 67–81. | MR | EuDML | Zbl

[30] —, « Möbius invariant Besov spaces and Hankel operators on the Bergman spaces on the unit ball », Complex Variables 17 (1991), p. 89–104. | Zbl

[31] W. Hastings – « A Carleson measure theorem for Bergman spaces », Proc. Amer. Math. Soc. 52 (1975), p. 237–241. | MR | Zbl

[32] L. Hörmander – « $L^{p}$ estimates for (pluri-)subharmonic functions », Math. Scand. 20 (1967), p. 65–78. | MR | EuDML | Zbl

[33] T. Kaptanoglu – « Besov spaces and Bergman projections on the ball », C.R. Acad. Sci. Paris, Sér. I 335 (2002), p. 729–732. | MR | Zbl

[34] —, « Bergman projections on Besov spaces on balls », Illinois J. Math. 49 (2005), p. 385–403. | MR | Zbl

[35] —, « Carleson measures for Besov spaces on the ball », J. Funct. Anal. 250 (2007), p. 483–520. | MR | Zbl

[36] O. Kures & K. Zhu – « A class of integral operators on the unit ball of $ℂ^{n}$ », Integr. Equ. Oper. Theory 56 (2006), p. 71–82. | MR | Zbl

[37] S. Li, H. Wulan, R. Zhao & K. Zhu – « A characterization of Bergman spaces on the unit ball of $ℂ^{n}$ », 2007, to appear in Glasgow Math J.

[38] D. Luecking – « A technique for characterizing Carleson measures on Bergman spaces », Proc. Amer. Math. Soc. 87 (1983), p. 656–660. | MR | Zbl

[39] —, « Embedding theorems for spaces of analytic functions via Khinchine’s inequality », Michigan Math. J. 40 (1993), p. 333–358. | MR | Zbl

[40] M. Nowark – « Bloch and Möbius invariant Besov spaces on the unit ball of $ℂ^{n}$ », Complex Variables 44 (2001), p. 1–12. | MR

[41] C. Ouyang, W. Yang & R. Zhao – « Characterizations of Bergman spaces and the Bloch space in the unit ball of $ℂ^{n}$ », Trans. Amer. Math. Soc. 374 (1995), p. 4301–4312. | MR | Zbl

[42] M. Pavlovic – « Inequalities for the gradient of eigenfunctions of the invariant Laplacian in the unit ball », Indag. Math. 2 (1991), p. 89–98. | MR | Zbl

[43] M. Pavlovic & K. Zhu – « New characterizations of Bergman spaces », Ann. Acad. Sci. Fen. 33 (2008), p. 87–99. | MR | EuDML | Zbl

[44] M. Peloso – « Möbius invariant spaces on the unit ball », Michigan Math. J. 39 (1992), p. 509–536. | MR | Zbl

[45] S. Power – « Hörmander’s Carleson theorem for the ball », Glasg. Math. J. 26 (1985), p. 13–17. | MR | Zbl

[46] R. Rochberg – « Decomposition theorems for Bergman spaces and their applications », in Operators and Function Theory, D. Reidel, 1985, p. 225–277. | MR

[47] W. Rudin – Function Theory in the Unit Ball of $ℂ^{n}$ , Springer, New York, 1980. | MR

[48] J. Ryll & P. Wojtaszczyk – « On homogeneous polynomials on a complex ball », Trans. Amer. Math. Soc. 276 (1983), p. 107–116. | MR | Zbl

[49] K. Seip – « Beurling type density theorems in the unit disk », Invent. Math. 113 (1993), p. 21–39. | MR | EuDML | Zbl

[50] —, « Regular sets of sampling and interpolation for weighted Bergman spaces », Proc. Amer. Math. Soc. 117 (1993), p. 213–220. | MR | Zbl

[51] J. Shapiro – « Macey topologies, reproducing kernels and diagonal maps on Hardy and Bergman spaces », Duke Math. J. 43 (1976), p. 187–202. | MR | Zbl

[52] J. Shi – « Inequalities for integral means of holomorphic functions and their derivatives in the unit ball of $ℂ^{n}$ », Trans. Amer. Math. Soc. 328 (1991), p. 619–632.

[53] A. Siskakis – « Weighted integrals of analytic functions », Acta Sci. Math. 66 (2000), p. 651–664. | MR | Zbl

[54] D. Stegenga – « Multipliers of the Dirichlet space », Illinois J. Math. 24 (1980), p. 113–139. | MR | Zbl

[55] E. Stein & G. Weiss – « Interpolation of operators with change of measures », Trans. Amer. Math. Soc. 87 (1958), p. 159–172. | MR | Zbl

[56] S. Stević – « A generalization of a result of Choa on analytic functions with Hadamard gaps », J. Korean Math. Soc. 43 (2006), p. 579–591. | MR | Zbl

[57] M. Stoll – Invariant Potential Theory in the Unit Ball of $ℂ^{n}$ , Cambridge Univ. Press, London, 1994. | MR

[58] F. G. Tricomi & A. Erdelyi – « The asymptotic expansion of a ratio of gamma functions », Pacific J. Math. 1 (1951), p. 133–142. | MR | Zbl

[59] D. Ullrich – « Radial divergence in BMOA », Proc. London Math. Soc. 68 (1994), p. 145–160. | MR | Zbl

[60] D. Vukotić – « A sharp estimate for $A_{α}^{p}$ functions in $ℂ^{n}$ », Proc. Amer. Math. Soc. 117 (1993), p. 753–756. | MR | Zbl

[61] Z. Wu – « Carleson measures and multipliers for the Dirichlet space », J. Funct. Anal. 169 (1999), p. 148–163. | MR | Zbl

[62] H. Wulan & K. Zhu – « Bloch and BMO functions in the unit ball », 53 (2008), p. 1009–1019, Complex Variables. | MR | Zbl

[63] —, « Lipschitz type characterizations of Bergman spaces », to appear in Bull. Canadian. Math. Soc. | Zbl

[64] W. Yang & C. Ouyang – « Exact location of $α$ -Bloch spaces in $L_{a}^{p}$ and $H^{p}$ of a complex unit ball », Rocky Mountain J. Math. 30 (2000), p. 1151–1169. | MR | Zbl

[65] R. Zhao – On a general family of function spaces, vol. 105, Ann. Acad. Sci. Fenn. Math. Dissertationes, 1996, 56pp. | MR

[66] K. Zhu – « Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains », J. Operator Theory 20 (1988), p. 329–357. | MR | Zbl

[67] —, « Möbius invariant Hilbert spaces of holomorphic functions in the unit ball of $ℂ^{n}$ », Trans. Amer. Math. Soc. 323 (1991), 823-842). | MR | Zbl

[68] —, « Bergman and Hardy spaces with small exponents », Pacific J. Math. 162 (1994), p. 189–199. | MR | Zbl

[69] —, « Holomorphic Besov spaces on bounded symmetric domains », Quart. J. Math. Oxford 46 (1995), p. 239–256. | MR | Zbl

[70] —, « Holomorphic Besov spaces on bounded symmetric domains II », Indiana Univ. Math. J. 44 (1995), p. 239–256. | Zbl

[71] —, Spaces of Holomorphic Functions in the Unit Ball, Springer, New York, 2005.

Cité par Sources :