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Bloch-to-BMOA pullbacks on the disk

Authors: Boo Rim Choe, Wade Ramey and David Ullrich
Journal: Proc. Amer. Math. Soc. 125 (1997), 2987-2996
MSC (1991): Primary 30D45, 47B38
MathSciNet review: 1396971
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Abstract: For a given holomorphic self map $\varphi $ of the unit disk, we consider the Bloch-to-$BMOA$ composition property (pullback property) of $\varphi $. Our results are $(1)$ $\varphi $ cannot have the pullback property if $\varphi $ touches the boundary too smoothly, $(2)$ while $\varphi $ has the pullback property if $\varphi $ touches the boundary rather sharply. One of these results yields an interesting consequence completely contrary to a higher dimensional result which has been known. These results resemble known results concerning the compactness of composition operators on the Hardy spaces. Some remarks in that direction are included.

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  • [1] P. Ahern, On the behavior near torus of functions holomorphic in the ball, Pacific J. Math. 107 (1983), 267-278. MR 84i:32023
  • [2] P. Ahern and W. Rudin, Bloch functions, BMO and boundary zeros, Indiana Univ. Math. J. 36 (1987), 131-148. MR 88d:42036
  • [3] R. B. Burckel, Iterating analytic self maps of the disc, Amer. Math. Monthly 88 (1981), 396-407. MR 82g:30046
  • [4] C. Carathéodory, Theory of functions of a complex variable, Vol. II, 2nd English edition, Chelsea, New York, 1960. MR 16:346c
  • [5] J. S. Choa and B. R. Choe, Composition with a homogeneous polynomial, Bull. Korean Math. Soc. 29 (1992), 57-63. MR 94d:32008
  • [6] B. R. Choe, Cauchy integral equalities and applications, Trans. Amer. Math. Soc. 315 (1989), 337-352. MR 89m:32010
  • [7] P. L. Duren, Theory of $H^{p}$ spaces, Academic, New York, 1970. MR 42:3552
  • [8] J. Garnett, Bounded analytic functions, Academic, New York, 1981. MR 83g:30037
  • [9] B. D. MacCluer and J. H. Shapiro, Angular derivatives and compact composition operators on the Hardy and Bergman spaces, Canad. J. Math. 38 (1986), 878-906. MR 87h:47048
  • [10] W. Ramey and D. Ullrich, Bounded mean oscillations of Bloch pull-backs, Math. Ann. 291 (1991), 590-606. MR 92i:32004
  • [11] P. A. Russo, Boundary behavior of $BMO(B_{n})$, Trans. Amer. Math. Soc. 292 (1985), 733-740. MR 87d:32030
  • [12] J. H. Shapiro and C. Sundberg, Isolation amongst the composition operators, Pacific J. Math. 145 (1990), 117-152. MR 92g:47041
  • [13] J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on $H^{2}$, Indiana Univ. Math. J. 23 (1973), 471-496. MR 48:4816
  • [14] K. Zhu, Operator theory in function spaces, Dekker, New York, 1990. MR 92c:47031
  • [15] C. S. Stanton, $H^{p}$ and $BMOA$ pullback properties of smooth maps, Indiana Univ. Math. J. 40 (1991), 1251-1265. MR 93c:32006

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Additional Information

Boo Rim Choe
Affiliation: Department of Mathematics, Korea University, Seoul, Korea

Wade Ramey
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan

David Ullrich
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma

Keywords: Bloch space, BMOA, pullback property
Received by editor(s): September 22, 1995
Received by editor(s) in revised form: May 17, 1996
Additional Notes: The first author is supported in part by BSRI (96-1407) and GARC (96) of Korea.
Communicated by: Theodore Gamelin
Article copyright: © Copyright 1997 American Mathematical Society

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