Blaschke products and nonideal ideals in higher order Lipschitz algebras
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- by K. M. Dyakonov
- St. Petersburg Math. J. 21 (2010), 979-993
- DOI: https://doi.org/10.1090/S1061-0022-2010-01127-0
- Published electronically: September 22, 2010
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Abstract:
We investigate certain ideals (associated with Blaschke products) of the analytic Lipschitz algebra $A^\alpha$, with $\alpha >1$, that fail to be “ideal spaces”. The latter means that the ideals in question are not describable by any size condition on the function’s modulus. In the case where $\alpha =n$ is an integer, we study this phenomenon for the algebra $H^\infty _n=\{f : f^{(n)}\in H^\infty \}$ rather than for its more manageable Zygmund-type version. This part is based on a new theorem concerning the canonical factorization in $H^\infty _n$.References
- Lennart Carleson, Sets of uniqueness for functions regular in the unit circle, Acta Math. 87 (1952), 325–345. MR 50011, DOI 10.1007/BF02392289
- K. M. D′yakonov, Smooth functions and co-invariant subspaces of the shift operator, Algebra i Analiz 4 (1992), no. 5, 117–147 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 4 (1993), no. 5, 933–959. MR 1202727
- Konstantin M. Dyakonov, Division and multiplication by inner functions and embedding theorems for star-invariant subspaces, Amer. J. Math. 115 (1993), no. 4, 881–902. MR 1231150, DOI 10.2307/2375016
- Konstantin M. Dyakonov, Multiplication by Blaschke products and stability of ideals in Lipschitz algebras, Math. Scand. 73 (1993), no. 2, 246–258. MR 1269262, DOI 10.7146/math.scand.a-12469
- Konstantin M. Dyakonov, Equivalent norms on Lipschitz-type spaces of holomorphic functions, Acta Math. 178 (1997), no. 2, 143–167. MR 1459259, DOI 10.1007/BF02392692
- Konstantin M. Dyakonov, Holomorphic functions and quasiconformal mappings with smooth moduli, Adv. Math. 187 (2004), no. 1, 146–172. MR 2074174, DOI 10.1016/j.aim.2003.08.008
- Konstantin M. Dyakonov, Self-improving behaviour of inner functions as multipliers, J. Funct. Anal. 240 (2006), no. 2, 429–444. MR 2261690, DOI 10.1016/j.jfa.2006.03.020
- E. M. Dyn′kin, Free interpolation sets for Hölder classes, Mat. Sb. (N.S.) 109(151) (1979), no. 1, 107–128, 166 (Russian). MR 538552
- E. M. Dyn′kin, The pseudoanalytic extension, J. Anal. Math. 60 (1993), 45–70. MR 1253229, DOI 10.1007/BF03341966
- 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
- V. P. Havin, The factorization of analytic functions that are smooth up to the boundary, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22 (1971), 202–205 (Russian). MR 0289783
- N. A. Shirokov, Division and multiplication by inner functions in spaces of analytic functions smooth up to the boundary, Complex analysis and spectral theory (Leningrad, 1979/1980) Lecture Notes in Math., vol. 864, Springer, Berlin-New York, 1981, pp. 413–439. MR 643387
- N. A. Shirokov, Free interpolation in spaces $C^{A}_{r,\omega }$, Mat. Sb. (N.S.) 117(159) (1982), no. 3, 337–358, 431 (Russian). MR 648412
- Nikolai A. Shirokov, Analytic functions smooth up to the boundary, Lecture Notes in Mathematics, vol. 1312, Springer-Verlag, Berlin, 1988. MR 947146, DOI 10.1007/BFb0082810
Bibliographic Information
- K. M. Dyakonov
- Affiliation: ICREA and Universitat de Barcelona, Departament de Matemàtica Aplicada i Anàlisi, Gran Via 585, E-08007 Barcelona, Spain
- Email: dyakonov@mat.ub.es
- Received by editor(s): January 14, 2009
- Published electronically: September 22, 2010
- Additional Notes: Supported in part by grant MTM2008-05561-C02-01 from El Ministerio de Ciencia e Innovación (Spain) and grant 2009-SGR-1303 from AGAUR (Generalitat de Catalunya).
- © Copyright 2010 American Mathematical Society
- Journal: St. Petersburg Math. J. 21 (2010), 979-993
- MSC (2010): Primary 30J10, 30H10, 46J15, 46J20
- DOI: https://doi.org/10.1090/S1061-0022-2010-01127-0
- MathSciNet review: 2604546
Dedicated: To Victor Petrovich Havin, with admiration (best phrased as a palindrome): \Russian{VOT PEDAGOG ADEPTOV}!