Geodesic
Drop of the smoothness of an outer function compared to the smoothness of its modulus, under restrictions on the size of boundary values
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 101-115 
A. N. Medvedev. Drop of the smoothness of an outer function compared to the smoothness of its modulus, under restrictions on the size of boundary values. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 43, Tome 434 (2015), pp. 101-115. http://geodesic.mathdoc.fr/item/ZNSL_2015_434_a8/
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $F$ be an outer function on the unit disk. It is well known that its smootheness properties may be two times worse that those of the modulus of its boundary values, but under some restrictions on $\log|F|$ this gap becomes smaller. It is shown that the smoothness decay admits a convenient description in terms of a rearrangement invariant Banach function space containing $\log|F|$. All results are of pointwise nature.

[1] C. Bennett, R. Sharpley, Interpolation of operators, Academic Press, 1988 | MR | Zbl

[2] G. Ya. Bomash, “Mnozhestva pika dlya analiticheskikh klassov Geldera”, Zap. nauchn. semin. LOMI, 157, 1987, 129–136 | MR | Zbl

[3] D. W. Boyd, “A class of operators on the Lorentz spaces $M(\Phi)$”, Canad. J. Math., 19 (1967), 839–841 | DOI | MR | Zbl

[4] K. M. Dyakonov, “Equivalent norms on Lipschitz-type spaces of holomorphic functions”, Acta Mathematica, 178:2 (1997), 143–167 | DOI | MR | Zbl

[5] V. P. Khavin,, “Obobschenie teoremy Privalova–Zigmunda o module nepreryvnosti sopryazhennoi funktsii”, Izv. Akad. nauk Armyanskoi SSR, seriya Matematika, 1971, no. 6, 252–258, 265–287 | Zbl

[6] S. G. Krein, Yu. I. Petunin, E. M. Semënov, Interpolyatsiya lineinykh operatorov, M., 1978 | MR

[7] N. A. Shirokov,, “Dostatochnye usloviya dlya gelderovskoi gladkosti funktsii”, Algebra i analiz, 25:3 (2013), 200–206 | MR | Zbl

[8] S. Spanne, “Some function spaces defined using the mean oscillation over cubes”, Annali della Scuola Normale Superiore di Pisa – Classe di Scienze, 19:4 (1965), 593–608 | MR | Zbl

[9] A. V. Vasin, S. V. Kislyakov, A. N. Medvedev, “Lokalnaya gladkost analiticheskoi funktsii v sravnenii s gladkostyu ee modulya”, Algebra i analiz, 25:3 (2013), 52–85 | MR | Zbl