Hardy spaces and twisted sectors for geometric models

Poggi-Corradini, Pietro

doi:10.1090/S0002-9947-96-01564-4

Hardy spaces and twisted sectors for geometric models
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by Pietro Poggi-Corradini PDF

Trans. Amer. Math. Soc. 348 (1996), 2503-2518 Request permission

Abstract:

We study the one-to-one analytic maps $\sigma$ that send the unit disc into a region $G$ with the property that $\lambda G\subset G$ for some complex number $\lambda$, $0<|\lambda |<1$. These functions arise in iteration theory, giving a model for the self-maps of the unit disk into itself, and in the study of composition operators as their eigenfunctions. We show that for such functions there are geometrical conditions on the image region $G$ that characterize their rate of growth, i.e. we prove that $\sigma \in \bigcap _{p<\infty }H^p$ if and only if $G$ does not contain a twisted sector. Then, we examine the connection with composition operators, and further investigate the no twisted sector condition. Finally, in the Appendix, we give a different proof of a result of J. Shapiro about the essential norm of a composition operator.

References

Lars V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0357743
W. J. Trjitzinsky, General theory of singular integral equations with real kernels, Trans. Amer. Math. Soc. 46 (1939), 202–279. MR 92, DOI 10.1090/S0002-9947-1939-0000092-6
P. Duren, Theory of $H^p$ spaces, Academic Press, 1970.
G. Königs, Recherches sur les intégrales de certaines équations functionnelles, Annales Ecole Normale Supérieure (3) 1 (1884), Supplément, 3-41.
J. Garnett, and D. Marshall, Harmonic measure, to appear.
Lowell J. Hansen, Hardy classes and ranges of functions, Michigan Math. J. 17 (1970), 235–248. MR 262512
P. Poggi-Corradini, The Hardy class of geometric models and the essential spectral radius of composition operators, preprint.
Ernest S. Armstrong, On the linear optimal digital servo problem, Internat. J. Control 20 (1974), 347–348. MR 0452999, DOI 10.1080/00207177408932742
J. H. Shapiro, W. Smith, and D. Stegenga, Geometric models and compactness of composition operators, J. Funct. Anal. 127 (1995), 21–62.
J. H. Shapiro and P. D. Taylor, Compact, nuclear, and Hilbert-Schmidt composition operators on $H^{2}$, Indiana Univ. Math. J. 23 (1973/74), 471–496. MR 326472, DOI 10.1512/iumj.1973.23.23041
Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. MR 1237406, DOI 10.1007/978-1-4612-0887-7
Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375–404. MR 881273, DOI 10.2307/1971314
David A. Stegenga and Kenneth Stephenson, A geometric characterization of analytic functions with bounded mean oscillation, J. London Math. Soc. (2) 24 (1981), no. 2, 243–254. MR 631937, DOI 10.1112/jlms/s2-24.2.243