Composition operators acting on holomorphic Sobolev spaces

Choe, Boo; Koo, Hyungwoon; Smith, Wayne

doi:10.1090/S0002-9947-03-03273-2

Composition operators acting on holomorphic Sobolev spaces
HTML articles powered by AMS MathViewer

by Boo Rim Choe, Hyungwoon Koo and Wayne Smith PDF

Trans. Amer. Math. Soc. 355 (2003), 2829-2855 Request permission

Abstract:

We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

References

Frank Beatrous and Jacob Burbea, Holomorphic Sobolev spaces on the ball, Dissertationes Math. (Rozprawy Mat.) 276 (1989), 60. MR 1010151
Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
P. W. Jones and Nikolai G. Makarov, Density properties of harmonic measure, Ann. of Math. (2) 142 (1995), no. 3, 427–455. MR 1356778, DOI 10.2307/2118551
Haakan Hedenmalm, Boris Korenblum, and Kehe Zhu, Theory of Bergman spaces, Graduate Texts in Mathematics, vol. 199, Springer-Verlag, New York, 2000. MR 1758653, DOI 10.1007/978-1-4612-0497-8
Daniel H. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, Amer. J. Math. 107 (1985), no. 1, 85–111. MR 778090, DOI 10.2307/2374458
Daniel H. Luecking, Multipliers of Bergman spaces into Lebesgue spaces, Proc. Edinburgh Math. Soc. (2) 29 (1986), no. 1, 125–131. MR 829188, DOI 10.1017/S001309150001748X
Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
Kevin M. Madigan, Composition operators on analytic Lipschitz spaces, Proc. Amer. Math. Soc. 119 (1993), no. 2, 465–473. MR 1152987, DOI 10.1090/S0002-9939-1993-1152987-6
Jie Miao, A property of analytic functions with Hadamard gaps, Bull. Austral. Math. Soc. 45 (1992), no. 1, 105–112. MR 1147249, DOI 10.1017/S0004972700037059
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594, DOI 10.1007/978-1-4613-8098-6
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Joel H. Shapiro, Compact composition operators on spaces of boundary-regular holomorphic functions, Proc. Amer. Math. Soc. 100 (1987), no. 1, 49–57. MR 883400, DOI 10.1090/S0002-9939-1987-0883400-9
Wayne Smith, Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2331–2348. MR 1357404, DOI 10.1090/S0002-9947-96-01647-9
Wayne Smith, Compactness of composition operators on BMOA, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2715–2725. MR 1600145, DOI 10.1090/S0002-9939-99-04856-X
Wayne Smith and Liming Yang, Composition operators that improve integrability on weighted Bergman spaces, Proc. Amer. Math. Soc. 126 (1998), no. 2, 411–420. MR 1443167, DOI 10.1090/S0002-9939-98-04206-3
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