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Carleson measures for the generalized Bergman spaces via a \(T(1)\)-type theorem. (English) Zbl 1159.32004

Let \(\alpha\in\mathbb{R}\), \(\alpha>-n-1\). The generalized Bergman space \(A_\alpha^2\) is defined to be the space of all holomorphic functions \(f\) in the unit ball \(\mathbb{D}\) of \(\mathbb{C}^n\) with the property \[ \|f\|_\alpha^2=\sum\limits_{m\in\mathbb{N}^n}|c(m)|^2 \frac{\Gamma(n+1+\alpha)m!}{\Gamma(n+1+|m|+\alpha)}<\infty, \] where \(f(z)=\sum_{m\in\mathbb{N}^n}c(m)z^m\) is the Taylor expansion of \(f\).
In this paper, the author gives a new characterization of Carleson measures for the spaces \(A_\alpha^2\). In order to do so, the author shows first that the Carleson measures problem for \(A_\alpha^2\) is equivalent to the \(T(1)\)-type problem associated with the Calderón-Zygmund kernel \(K_\alpha\). Then, by introducing a sort of curvature in the unit ball adapted to the kernel \(K_\alpha\), the author establishes a good \(\lambda\) inequality which yields the solution of the \(T(1)\)-type problem.

MSC:

32A36 Bergman spaces of functions in several complex variables
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E15 Banach spaces of continuous, differentiable or analytic functions
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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[1] Ahern, P. and Cohn, W., Exceptional sets for Hardy Sobolev functions, p>1, Indiana Univ. Math. J. 38 (1989), 417–453. · Zbl 0691.46022 · doi:10.1512/iumj.1989.38.38020
[2] Arcozzi, N., Rochberg, R. and Sawyer, E., Carleson measures for the Drury–Arveson Hardy space and other Besov–Sobolev spaces on complex balls, to appear in Adv. Math. · Zbl 1167.32003
[3] Bekollé, D., Inégalité à poids pour le projecteur de Bergman dans la boule unité de C n , Studia Math. 71 (1981/82), 305–323. · Zbl 0516.47016
[4] Carleson, L., Interpolations by bounded analytic functions and the corona problem, Ann. of Math. 76 (1962), 547–559. · Zbl 0112.29702 · doi:10.2307/1970375
[5] Cascante, C. and Ortega, J. M., Carleson measures on spaces of Hardy–Sobolev type, Canad. J. Math. 47 (1995), 1177–1200. · Zbl 0845.46027 · doi:10.4153/CJM-1995-060-4
[6] Cima, J. A. and Wogen, W. R., A Carleson measure theorem for the Bergman space on the ball, J. Operator Theory 7 (1982), 157–165. · Zbl 0499.42011
[7] Cohn, W. S. and Verbitsky, I. E., On the trace inequalities for Hardy–Sobolev functions in the unit ball of C n , Indiana Univ. Math. J. 43 (1994), 1079–1097. · Zbl 0846.32007 · doi:10.1512/iumj.1994.43.43047
[8] Coifman, R. R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes, Springer, Berlin–Heidelberg, 1971. · Zbl 0224.43006
[9] David, G. and Journé, J. L., A boundedness criterion for generalized Calderón–Zygmund operators, Ann. of Math. 120 (1984), 371–397. · Zbl 0567.47025 · doi:10.2307/2006946
[10] Drury, S. W., A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), 300–304. · Zbl 0377.47016
[11] Fazio, G. D., Gutiérrez, C. and Lanconelli, E., Covering theorems, inequalities on metric spaces and applications to PDEs, to appear in Math. Ann. · Zbl 1149.46029
[12] Hörmander, L., L p estimates for (pluri)-subharmonic functions, Math. Scand. 20 (1967), 65–78. · Zbl 0156.12201
[13] Korányi, A. and Reimann, H. M., Foundations for the theory of quasiconformal mappings on the Heisenberg group, Adv. Math. 111 (1995), 1–87. · Zbl 0876.30019 · doi:10.1006/aima.1995.1017
[14] Nazarov, F., Treil, S. and Volberg, A., Cauchy integral and Calderón–Zygmund operators on nonhomogeneous spaces, Int. Math. Res. Notices 1997:15 (1997), 703–726. · Zbl 0889.42013
[15] Rudin, W., Function Theory in the Unit Ball of C n , Grundlehren der Mathematischen Wissenschaften 241, Springer, New York, 1980. · Zbl 0495.32001
[16] Stegenga, D. A., Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), 113–139. · Zbl 0432.30016
[17] Stein, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series 43, Princeton University Press, Princeton, NJ, 1993. · Zbl 0821.42001
[18] Tolsa, X., L 2-boundedness of the Cauchy integral operator for continuous measures, Duke Math. J. 98 (1999), 269–304. · Zbl 0945.30032 · doi:10.1215/S0012-7094-99-09808-3
[19] Tolsa, X., A T(1)-theorem for non-doubling measures with atoms, Proc. London Math. Soc. 82 (2001), 195–228. · Zbl 1033.42011 · doi:10.1112/S0024611500012703
[20] Verdera, J., On the T(1)-theorem for the Cauchy integral, Ark. Mat. 38 (2000), 183–199. · Zbl 1039.42011 · doi:10.1007/BF02384497
[21] Zhao, R. and Zhu, K., Theory of Bergman spaces in the unit ball of C n , to appear in Mem. Soc. Math. France.
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