Admissibility of majorants in certain model subspaces: Necessary conditions

Belov, Yu.

doi:10.1090/S1061-0022-09-01059-0

Admissibility of majorants in certain model subspaces: Necessary conditions
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by Yu. S. Belov
Translated by: S. Kislyakov

St. Petersburg Math. J. 20 (2009), 507-525

DOI: https://doi.org/10.1090/S1061-0022-09-01059-0

Published electronically: June 1, 2009

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Abstract:

A nonnegative function $\omega$ on $\mathbb {R}$ is called an admissible majorant for an inner function $\Theta$ if there is a nonzero function $f\in H^2\ominus \Theta H^2$ such that $|f|\le \omega$. Some conditions necessary for admissibility are presented in the case where $\Theta$ is meromorphic.

References

Anton D. Baranov, Polynomials in the de Branges spaces of entire functions, Ark. Mat. 44 (2006), no. 1, 16–38. MR 2237209, DOI 10.1007/s11512-005-0006-1
N. K. Bari, Trigonometricheskie ryady, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1961 (Russian). With the editorial collaboration of P. L. Ul′janov. MR 0126115
A. D. Baranov and A. A. Borichev, Entire functions of exponential type with prescribed modulus on the real axis (unpublished).
A. D. Baranov, A. A. Borichev, and V. P. Havin, Majorants of meromorphic functions with fixed poles, Indiana Univ. Math. J. 56 (2007), no. 4, 1595–1628. MR 2354693, DOI 10.1512/iumj.2007.56.3045
A. D. Baranov and V. P. Khavin, Admissible majorants for model subspaces and arguments of inner functions, Funktsional. Anal. i Prilozhen. 40 (2006), no. 4, 3–21, 111 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 40 (2006), no. 4, 249–263. MR 2307699, DOI 10.1007/s10688-006-0042-z
Yu. S. Belov and V. P. Khavin, On a theorem of I. I. Privalov on the Hilbert transform of Lipschitz functions, Mat. Fiz. Anal. Geom. 11 (2004), no. 4, 380–407 (Russian, with English, Russian and Ukrainian summaries). MR 2114001
Yu. S. Belov, Criteria for the admissibility of majorants for model subspaces with a rapidly increasing argument of the generating inner function, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 345 (2007), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 34, 55–84, 141 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 148 (2008), no. 6, 813–829. MR 2432176, DOI 10.1007/s10958-008-0028-x
Yu. S. Belov, Model functions with an almost prescribed modulus, Algebra i Analiz 20 (2008), no. 2, 3–18 (Russian); English transl., St. Petersburg Math. J. 20 (2009), no. 2, 163–174. MR 2423994, DOI 10.1090/S1061-0022-09-01042-5
K. M. D′yakonov, Moduli and arguments of analytic functions from subspaces in $H^p$ that are invariant under the backward shift operator, Sibirsk. Mat. Zh. 31 (1990), no. 6, 64–79 (Russian); English transl., Siberian Math. J. 31 (1990), no. 6, 926–939 (1991). MR 1097956, DOI 10.1007/BF00970058
Victor Havin and Burglind Jöricke, The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 28, Springer-Verlag, Berlin, 1994. MR 1303780, DOI 10.1007/978-3-642-78377-7
Victor Havin and Javad Mashreghi, Admissible majorants for model subspaces of $H^2$. I. Slow winding of the generating inner function, Canad. J. Math. 55 (2003), no. 6, 1231–1263. MR 2016246, DOI 10.4153/CJM-2003-048-8
Victor Havin and Javad Mashreghi, Admissible majorants for model subspaces of $H^2$. II. Fast winding of the generating inner function, Canad. J. Math. 55 (2003), no. 6, 1264–1301. MR 2016247, DOI 10.4153/CJM-2003-049-5
Dzh. Mashregi, F. L. Nazarov, and V. P. Khavin, The Beurling-Malliavin multiplier theorem: the seventh proof, Algebra i Analiz 17 (2005), no. 5, 3–68 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 17 (2006), no. 5, 699–744. MR 2241422, DOI 10.1090/S1061-0022-06-00926-5
N. K. Nikol′skiĭ, Lektsii ob operatore sdviga, “Nauka”, Moscow, 1980 (Russian). MR 575166