Geodesic
On Grand and Small Bergman Spaces
Matematičeskie zametki, Tome 104 (2018) no. 3, pp. 439-446.

Voir la notice de l'article provenant de la source Math-Net.Ru

Grand and small Bergman spaces of functions holomorphic in the unit disc are introduced. The boundedness of the Bergman projection operator on grand Bergman spaces is proved. The main result consists of estimates for functions in grand and small Bergman spaces near the boundary, which differ from those in the case of the classical Bergman space by a logarithmic multiplier with positive (for grand spaces) or negative (for small spaces) exponent.
Keywords: Bergman space, Bergman projection, grand Bergman space, small Bergman space, boundary estimates.
Mots-clés : grand Lebesgue space, small Lebesgue space 
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A. N. Karapetyants; S. G. Samko. On Grand and Small Bergman Spaces. Matematičeskie zametki, Tome 104 (2018) no. 3, pp. 439-446. http://geodesic.mathdoc.fr/item/MZM_2018_104_3_a8/

[1] S. Bergman, The Kernel Function and Conformal Mapping, American Math. Soc., Providence, RI, 1970 | MR | Zbl

[2] A. E. Djrbashian, F. A. Shamoian, Topics in the Theory of $A^p_\alpha$ Spaces, Teubner-Texte Math., 105, BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1988 | MR | Zbl

[3] N. L. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space, Oper. Theory Adv. Appl., 185, Birkhäuser Verlag, Basel, 2008 | MR | Zbl

[4] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Grad. Texts in Math., 199, Springer, New York, 2000 | MR | Zbl

[5] P. Duren, A. Schuster, Bergman Spaces, Math. Surveys Monogr., 100, Amer. Math. Soc., Providence, RI, 2004 | DOI | MR | Zbl

[6] K. Zhu, “Operator Theory in Function Spaces”, Math. Surveys Monogr., 138, Amer. Math. Soc., Providence, RI, 2007 | DOI | MR | Zbl

[7] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Grad. Texts in Math., 226, Springer, New York, 2004 | MR

[8] T. Iwaniec, C. Sbordone, “On the integrability of the Jacobian under minimal hypotheses”, Arch. Rational Mech. Anal., 119:2 (1992), 129–143 | DOI | MR | Zbl

[9] L. Greco, T. Iwaniec, C. Sbordone, “Inverting the $p$-harmonic operator”, Manuscripta Math., 92:2 (1997), 249–258 | DOI | MR | Zbl

[10] G. Di Fratta, A. Fiorenza, “A direct approach to the duality of grand and small Lebesgue spaces”, Nonlinear Anal., 70:7 (2009), 2582–2592 | DOI | MR | Zbl

[11] A. Fiorenza, G. E. Karadzhov, “Grand and small Lebesgue spaces and their analogs”, Z. Anal. Anwendungen, 23:4 (2004), 657–681 | DOI | MR | Zbl

[12] A. Fiorenza, “Duality and reflexivity in grand Lebesgue spaces”, Collect. Math., 51:2 (2000), 131–148 | MR | Zbl

[13] C. Capone, A. Fiorenza, “On small Lebesgue spaces”, J. Funct. Spaces Appl., 3:1 (2005), 73–89 | DOI | MR | Zbl

[14] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces Vol. 2. Variable Exponent Hölder Morrey–Campanato and Grand Spaces, Oper. Theory Adv. Appl., 249, Birkhäuser, Cham, 2016 | MR

[15] N. G. de Bruijn, Asymptotic Methods in Analysis, North-Holland Publ., Amsterdam, 1961 | MR

[16] R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey, D. E. Knuth, “On the Lambert $W$ function”, Adv. Comput. Math., 5:4 (1996), 329–359 | DOI | MR | Zbl

[17] S. M. Umarkhadzhiev, “Obobschenie ponyatiya grand-prostranstva Lebega”, Izv. vuzov. Matem., 2014, no. 4, 42–51

[18] D. Kruz-Uribe, A. Fiorenza, O. M. Guzman, “Vlozheniya grand-prostranstv Lebega, malykh prostranstv Lebega, i prostranstv Lebega s peremennym pokazatelem”, Matem. zametki, 102:5 (2017), 736–748 | DOI | MR

[19] A. Karapetyants, H. Rafeiro, S. Samko, “Boundedness of the Bergman projection and some properties of Bergman type spaces”, Complex Anal. Oper. Theory, 2018 | DOI