Multiplication and division by inner functions in the space of Bloch functions
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- by Daniel Girela, Cristóbal González and José Ángel Peláez PDF
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Abstract:
A subspace $X$ of the Hardy space $H^1$ is said to have the $f$-property if $h/I \in X$ whenever $h\in X$ and $I$ is an inner function with $h/I \in H^1$. We let $\mathcal B$ denote the space of Bloch functions and $\mathcal B_0$ the little Bloch space. Anderson proved in 1979 that the space $\mathcal B_0\cap H^ \infty$ does not have the $f$-property. However, the question of whether or not $\mathcal B\cap H^ p$ ($1\le p<\infty$) has the $f$-property was open. We prove that for every $p\in [1,\infty )$ the space $\mathcal B\cap H^ p$ does not have the $f$-property.
We also prove that if $B$ is any infinite Blaschke product with positive zeros and $G$ is a Bloch function with $\vert G(z)\vert \to \infty$, as $z\to 1$, then the product $BG$ is not a Bloch function.
References
- J. M. Anderson, On division by inner factors, Comment. Math. Helv. 54 (1979), no. 2, 309–317. MR 535061, DOI 10.1007/BF02566274
- J. M. Anderson, J. Clunie, and Ch. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math. 270 (1974), 12–37. MR 361090
- Christopher J. Bishop, Bounded functions in the little Bloch space, Pacific J. Math. 142 (1990), no. 2, 209–225. MR 1042042, DOI 10.2140/pjm.1990.142.209
- Oscar Blasco, Daniel Girela, and M. Auxiliadora Márquez, Mean growth of the derivative of analytic functions, bounded mean oscillation, and normal functions, Indiana Univ. Math. J. 47 (1998), no. 3, 893–912. MR 1665796, DOI 10.1512/iumj.1998.47.1495
- Douglas M. Campbell, Nonnormal sums and products of unbounded normal functions. II, Proc. Amer. Math. Soc. 74 (1979), no. 1, 202–203. MR 521899, DOI 10.1090/S0002-9939-1979-0521899-3
- Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
- K. M. D′yakonov, Factorization of smooth analytic functions, and Hilbert-Schmidt operators, Algebra i Analiz 8 (1996), no. 4, 1–42 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 4, 543–569. MR 1418253
- Konstantin M. Dyakonov and Daniel Girela, On $Q_p$ spaces and pseudoanalytic extension, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 2, 477–486. MR 1762431
- 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
- Daniel Girela, On a theorem of Privalov and normal functions, Proc. Amer. Math. Soc. 125 (1997), no. 2, 433–442. MR 1363422, DOI 10.1090/S0002-9939-97-03544-2
- V. P. Gurariĭ, The factorization of absolutely convergent Taylor series and Fourier integrals, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 30 (1972), 15–32 (Russian). Investigations on linear operators and the theory of functions, III. MR 0340622
- 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
- Håkan Hedenmalm, On the $f$- and $K$-properties of certain function spaces, Commutative harmonic analysis (Canton, NY, 1987) Contemp. Math., vol. 91, Amer. Math. Soc., Providence, RI, 1989, pp. 89–91. MR 1002590, DOI 10.1090/conm/091/1002590
- B. I. Korenbljum, A certain extremal property of outer functions, Mat. Zametki 10 (1971), 53–56 (Russian). MR 288274
- B. I. Korenbljum and V. M. Faĭvyševskiĭ, A certain class of compression operators that are connected with the divisibility of analytic functions, Ukrain. Mat. Ž. 24 (1972), 692–695, 717 (Russian). MR 0320801
- Peter Lappan, Non-normal sums and products of unbounded normal functions, Michigan Math. J. 8 (1961), 187–192. MR 131554
- Olli Lehto and K. I. Virtanen, Boundary behaviour and normal meromorphic functions, Acta Math. 97 (1957), 47–65. MR 87746, DOI 10.1007/BF02392392
- Kin Y. Li, Interpolating Blaschke products and the left spectrum of multiplication operators on the Bergman space, Hokkaido Math. J. 21 (1992), no. 2, 295–304. MR 1169796, DOI 10.14492/hokmj/1381413684
- M. Rabindranathan, Toeplitz operators and division by inner functions, Indiana Univ. Math. J. 22 (1972/73), 523–529. MR 306964, DOI 10.1512/iumj.1972.22.22044
- V. P. Havin, S. V. Hruščëv, and N. K. Nikol′skiĭ (eds.), Linear and complex analysis problem book, Lecture Notes in Mathematics, vol. 1043, Springer-Verlag, Berlin, 1984. 199 research problems. MR 734178, DOI 10.1007/BFb0072183
- Kenneth Stephenson, Construction of an inner function in the little Bloch space, Trans. Amer. Math. Soc. 308 (1988), no. 2, 713–720. MR 951624, DOI 10.1090/S0002-9947-1988-0951624-3
- Shinji Yamashita, A nonnormal function whose derivative has finite area integral of order $0<p<2$, Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 2, 293–298. MR 565879, DOI 10.5186/aasfm.1978-79.0431
- Shinji Yamashita, A nonnormal function whose derivative is of Hardy class $H^{p}$, $0<p<1$, Canad. Math. Bull. 23 (1980), no. 4, 499–500. MR 602610, DOI 10.4153/CMB-1980-077-1
Additional Information
- Daniel Girela
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
- Email: girela@uma.es
- Cristóbal González
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
- Email: cmge@uma.es
- José Ángel Peláez
- Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Campus de Teatinos, 29071 Málaga, Spain
- Email: pelaez@anamat.cie.uma.es
- Received by editor(s): April 22, 2004
- Received by editor(s) in revised form: November 17, 2004
- Published electronically: October 4, 2005
- Additional Notes: This research has been partially supported by a grant from “La Junta de Andalucía” (FQM-210) and by an MCyT grant BFM2001-1736, Spain.
- Communicated by: Juha M. Heinonen
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 1309-1314
- MSC (2000): Primary 30D45, 30D50, 30D55
- DOI: https://doi.org/10.1090/S0002-9939-05-08049-4
- MathSciNet review: 2199173