Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cauchy integral equalities and applications


Author: Boo Rim Choe
Journal: Trans. Amer. Math. Soc. 315 (1989), 337-352
MSC: Primary 32A35
DOI: https://doi.org/10.1090/S0002-9947-1989-0935531-9
MathSciNet review: 935531
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study bounded holomorphic functions $ \pi $ on the unit ball $ {B_n}$ of $ {\mathbb{C}^n}$ satisfying the following so-called Cauchy integral equalities:

\begin{displaymath}\begin{array}{*{20}{c}} {C[{\pi ^{m + 1}}\bar \pi ] = {\gamma _m}{\pi ^m}} & {(m = 0,1,2, \ldots)} \\ \end{array} \end{displaymath}

for some sequence $ {\gamma _m}$ depending on $ \pi $. Among the applications are the Ahern-Rudin problem concerning the composition property of holomorphic functions on $ {B_n}$, a projection theorem about the orthogonal projection of $ {H^2}({B_n})$ onto the closed subspace generated by holomorphic polynomials in $ \pi $, and some new information about the inner functions. In particular, it is shown that if we interpret $ {\text{BMOA}}({B_n})$ as the dual of $ {H^1}({B_n})$, then the map $ g \to g \circ \pi $ is a linear isometry of $ {\text{BMOA}}({B_1})$ into $ {\text{BMOA}}({B_n})$ for every inner function $ \pi $ on $ {B_n}$ such that $ \pi (0) = 0$.

References [Enhancements On Off] (What's this?)

  • [A] P. Ahern, On the behavior near torus of functions holomorphic in the ball, Pacific J. Math. 107 (1983), 267-278. MR 705748 (84i:32023)
  • [ACP] J. M. Anderson, J. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12-37. MR 0361090 (50:13536)
  • [AR1] P. Ahern and W. Rudin, Bloch functions, $ BMO$ and boundary zeros, Indiana Math. J. 36 (1987), 131-148. MR 876995 (88d:42036)
  • [AR2] -, Paley-type gap theorems for $ {H^p}$-functions on the ball, Indiana Univ. Math. J. (to appear).
  • [B] K. G. Binmore, Analytic functions with Hadamard gaps, Bull. London Math. Soc. 1 (1969), 211-217. MR 0244473 (39:5787)
  • [C1] B. R. Choe, Composition with bounded holomorphic functions on the ball, Ph.D. thesis, Univ. of Wisconsin-Madison, 1988.
  • [C2] -, Weights induced by homogeneous polynomials, Pacific J. Math. (to appear). MR 1011210 (91e:32004)
  • [CRW] R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. MR 0412721 (54:843)
  • [G] J. B. Garnett, Bounded analytic functions, Academic, New York, 1981. MR 628971 (83g:30037)
  • [K] C. Kolaski, Projections onto spaces of holomorphic functions, Indiana Univ. Math. J. 29 (1980), 769-775. MR 589441 (83g:46025)
  • [RR] W. Ramey and P. Russo, Behavior of functions in $ BMOA$ and $ VMOA$ near $ (2n - 2)$-dimensional submanifold, Indiana Math. J. 37 (1988), 73-81. MR 942095 (89f:32008)
  • [Ru1] W. Rudin, Composition with inner functions, Complex Variables 4 (1984), 7-19. MR 770982 (86j:32010)
  • [Ru2] -, Function theory in the unit ball of $ {\mathbb{C}^n}$, Springer-Verlag, Berlin, Heidelberg and New York, 1980.
  • [Rus] P. Russo, Boundary behavior of $ {\text{BMO}}({B_n})$., Trans. Amer. Math. Soc. 292 (1985), 733-740. MR 808751 (87d:32030)
  • [U] D. Ullrich, Radial divergence in $ BMOA$, preprint.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32A35

Retrieve articles in all journals with MSC: 32A35


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0935531-9
Keywords: Cauchy Integral Equalities, the Ahern-Rudin problem, projection
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society