Geodesic
On integral operators of Bergman type on the unit ball of $ \mathbb{R}^n$
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2015), pp. 23-30.

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We prove the boundedness of Bergman type integral operators in mixed norm spaces over the unit ball of $ \mathbb{R}^n$. Bounded harmonic projections are found in the mixed norm and Lipschitz spaces. Corresponding Forelli–Rudin type theorems are proved.
Keywords: unit ball in $ \mathbb{R}^n$, harmonic function, mixed norm space, Bergman space, Bergman operator, projection, Lipschitz space. 
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Ye. G. Tonoyan. On integral operators of Bergman type on the unit ball of $ \mathbb{R}^n$. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 3 (2015), pp. 23-30. http://geodesic.mathdoc.fr/item/UZERU_2015_3_a3/

[1] G.H. Hardy, J.E Littlewood, “Some Properties of Fractional Integrals (II)”, Math. Z., 34 (1932), 403–439 | DOI | MR

[2] G.H. Hardy, J.E Littlewood, “Theorems Concerning Mean Values of Analytic or Harmonic Functions”, Quart. J. Math., 12 (1941), 221–256 | DOI | MR | Zbl

[3] T.M Flett, “The Dual of an Inequality of Hardy and Littlewood and Some Related Inequalities”, J. Math. Anal. Appl., 38 (1972), 746–765 | DOI | MR | Zbl

[4] A.E Djrbashian, F.A. Shamoian, Topics in the Theory of $A^p_{\alpha}$ Spaces, Teubner-Texte zur Math. Leipzig, 105, 1988, 199 pp. | MR | Zbl

[5] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces, Springer-Verlag, New York–Berlin–Heidelberg, 2000 | MR | Zbl

[6] A.E. Djrbashian, “Integral Representations and Continuous Projectors in Some Spaces of Harmonic Functions”, Mat. Sbornik, 121:2 (1983), 259–271 (in Russian) | MR

[7] M. Jevtić, M. Pavlović, “Harmonic Bergman Functions on the Unit Ball in $\mathbb{R}^n$”, Acta Math. Hungar., 85 (1999), 81–96 | DOI | MR | Zbl

[8] C. Liu, J. Shi, G. Ren, “Duality for Harmonic Mixed-Norm Spaces in the Unit Ball of $\mathbb{R}^n$”, Ann. Sci. Math. Quebec, 25 (2001), 179–197 | MR | Zbl

[9] A.I. Petrosyan, “On Weighted Classes of Harmonic Functions in the Unit Ball of $\mathbb{R}^n$”, Complex Variables Theory Appl., 50 (2005), 953–966 | DOI | MR | Zbl

[10] A.I. Petrosyan, “On Weighted Harmonic Bergman Spaces”, Demonstratio Math., 41 (2008), 73–83 | MR | Zbl

[11] R. Coifman, R. Rochberg, “Representation Theorems for Holomorphic and Harmonic Functions in $L^p$”, Asterisque, 77 (1980), 11–66 | MR | Zbl

[12] J. Miao, “Reproducing Kernels for Harmonic Bergman Spaces of the Unit Ball.”, Monatsh. Math., 125 (1998), 25–35 | DOI | MR | Zbl

[13] S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, ed. 2nd, Springer-Verlag, NY, 2001 | MR | Zbl

[14] G. Ren, “Harmonic Bergman Spaces with Small Exponents in the Unit Ball”, Collect. Math., 53 (2002), 83–98 | MR | Zbl

[15] J. Contemp. Math. Anal., 47:5 (2012), 209–220 | DOI | MR | Zbl

[16] J. Contemp. Math. Anal., 50:5 (2015), 236–245 | DOI | MR

[17] K. Avetisyan, “Continuous Inclusions and Bergman Type Operators in $n$-Harmonic Mixed Norm Spaces on the Polydisc”, J. Math. Anal. Appl., 291 (2004), 727–740 | DOI | MR | Zbl

[18] K. Avetisyan, “Weighted Integrals and Bloch Spaces of $n$-Harmonic Functions on the Polydisc”, Potential Analysis, 29 (2008), 49–63 | DOI | MR | Zbl

[19] E. Ligocka, “The Hölder Continuity of the Bergman Projection and Proper Holomorphic Mappings”, Studia Math., 80 (1984), 89–107 | MR | Zbl

[20] A.E. Djrbashian, “Integral Representations for Riesz Systems in the Unit Ball and Some Applications”, Proc. Amer. Math. Soc., 117 (1993), 395–403 | DOI | MR | Zbl