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Duality theorems for Hardy and Bergman spaces on convex domains of finite type in $${\mathbb{C}}^ n$$. (English) Zbl 0835.32004
Summary: We study Hardy, Bergman, Bloch, and BMO spaces on convex domains of finite type in $$n$$-dimensional complex space. Duals of these spaces are computed. The essential features of complex domains of finite type, that make these theorems possible, are isolated.

##### MSC:
 32A35 $$H^p$$-spaces, Nevanlinna spaces of functions in several complex variables 32C37 Duality theorems for analytic spaces 42B30 $$H^p$$-spaces
##### Keywords:
Bloch; BMO; convex domains of finite type
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##### References:
 [1] S. ROSS BARKER, Two theorems on boundary values of analytic functions, Proc. A.M.S., 68 (1978), 54-58. · Zbl 0378.32011 [2] F. BEATROUS, Lp estimates for extensions of holomorphic functions, Michigan Math. J., 32 (1985), 361-380. · Zbl 0584.32024 [3] F. BEATROUS and S.-Y. LI, On the boundedness and compactness of operators of Hankel type, J. Funct. Anal., vol. 111 (1993), 350-379. · Zbl 0793.47022 [4] H. P. BOAS, The szegö projection, Sobolev estimates in the regular domain, Trans. A.M.S., 300 (1987), 109-132. · Zbl 0622.32006 [5] S. BELL, Extendibility of Bergman kernel function, Complex analysis II, Lecture Notes in Math., 1276, 33-41, Berlin-Heidelberg-New York. · Zbl 0626.32028 [6] D. CATLIN, Subelliptic estimates for the ∂-Neumann problem, Ann. Math., 126 (1987), 131-192. · Zbl 0627.32013 [7] R. COIFMAN and G. WEISS, Extensions of Hardy spaces and their use in analysis, Bulletin A.M.S., 83 (1977), 569-643. · Zbl 0358.30023 [8] L. CHEN, Ph.D. Thesis, Univ. of California at irvine, 1994. [9] M. CHRIST, Lectures on singular integral operators, Conference Board of Mathematical Sciences, American Mathematical Society, Providence, 1990. · Zbl 0745.42008 [10] B. COUPET, Régularité d’applications holomorphes sur des variétés totalement réelles, Thèse, Université de Provence, 1987. [11] R. COIFMAN, R. ROCHBERG, and G. WEISS, Factorization theorems for Hardy spaces in several variables, Ann. Math., 103 (1976), 611-635. · Zbl 0326.32011 [12] G. DAFNI, Hardy spaces on some pseudoconvex domains, Jour. Geometric Analysis, (1995). · Zbl 0802.32012 [13] C. FEFFERMAN, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., 26 (1974), 1-65. · Zbl 0289.32012 [14] C. FEFFERMAN and E. M. STEIN, Hp spaces of several variables, Acta Math., 129 (1972), 137-193. · Zbl 0257.46078 [15] L. HÖRMANDER, Lp estimates for pluri-subharmonic functions, Math. Scand., 20 (1967), 65-78. · Zbl 0156.12201 [16] N. KERZMAN, The Bergman kernel function. Differentiability at the boundary, Math. Ann., 195 (1972), 149-158. [17] S. KRANTZ, Function theory of several complex variables, 2nd. ed., Wadsworth, Belmont, 1992. · Zbl 0776.32001 [18] S. KRANTZ, Invariant metrics and the boundary behavior of holomorphic functions, Jour. of Geometric Analysis, 1 (1991), 71-97. · Zbl 0728.32002 [19] S. KRANTZ, Holomorphic functions of bounded mean oscillation and mapping properties of the szegö projection, Duke Math. J., 47 (1980), 743-761. · Zbl 0456.32004 [20] S. KRANTZ and S.-Y. LI, A note on Hardy spaces and functions of bounded mean oscillation on domains in ℂn, Michigan Math. Jour., 41 (1994), 51-72. · Zbl 0802.32013 [21] S. KRANTZ and S.-Y. LI, On the decomposition theorems for Hardy spaces in domains in ℂn and applications, J. of Fourier Anal. and Appl., to appear. · Zbl 0886.32003 [22] J. MCNEAL, Convex domains of finite type, Jour. Funct. Anal., 108 (1992), 361-373. · Zbl 0777.31007 [23] J. MCNEAL, Estimates on the Bergman kernels of convex domains, Advances in Math., 109 (1994), 108-139. · Zbl 0816.32018 [24] J. D. MCNEAL and E. M. STEIN, Mapping properties of the Bergman projection on convex domains of finite type, Duke Math. J., 73 (1994), 177-199. · Zbl 0801.32008 [25] J. D. MCNEAL and E. M. STEIN, The szegö projection on convex domains, preprint. · Zbl 0948.32004 [26] A. NAGEL, E. M. STEIN, and S. WAINGER, Balls and metrics defined by vector fields. I. Basic properties, Acta Math., 155 (1985), 103-147. · Zbl 0578.32044 [27] A. NAGEL, J.P. ROSAY, E.M. STEIN, and S. WAINGER, Estimates for the Bergman and szegö kernels in ℂ2, Ann. Math., 129 (1989), 113-149. · Zbl 0667.32016 [28] E.M. STEIN, Singular integral and differentiability properties of functions, Princeton University Press, 1970. · Zbl 0207.13501 [29] E. M. STEIN, Boundary behavior of holomorphic functions of several complex variables, Princeton University Press, Princeton, 1972. · Zbl 0242.32005
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