On some new estimates related with Bergman ball and Poisson integral in tubular domain and unit ball

R. F. Shamoyan ; O. Mihić

Vestnik KRAUNC. Fiziko-matematičeskie nauki, no. 1 (2018), pp. 48-63

Résumé

We introduce new Herz type analytic spaces based on Bergman balls in tubular domains over symmetric cones and in products of such type domains. We provide for these Herz type spaces new maximal and embedding theorems extending known results in the unit disk. In addition we define new Poisson-type integral in the unit ball and extend a known classical maximal theorem related with it. Related results for such type integrals will be given.

Détail

Keywords: tubular domains over symmetric cones, Herz type spaces, Bergman type integral operators, maximal theorems, embedding theorems, unit ball.
Mots-clés : Poisson-type integral

Bibliographie

[1] Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B., “Analytic Besov spaces and Hardy type inequalities in tube domains over symmetric cones”, J. Reine Angew. Math., 647 (2010), 25–56 | MR

[2] Bekolle D., Bonami A., Garrigos G., Nana C., Peloso M., Ricci F., “Lecture notes on Bergman projectors in tube domain over cones, an analytic and geometric viewpoint”, Proceeding of the International Workshop on Classical Analysis, Yaounde, 2001 | MR

[3] Bekolle D., Bonami A., Garrigos G., Ricci F., “Littlewood – Paley decomposition and Besov spaces related to symmetric cones and Bergman Projections in Tube Domains”, Proc. Lond. Math. Soc., 89:3 (2004), 317–360 | DOI | MR

[4] Chyzhykov I., Zolota O., “Growth of the Poisson-Stieltjes integral in the polydisk”, Zh. Mat. Fiz. Anal. Geom., 7(2) (2011), 141–157 | MR

[5] Flett T., “Inequalities for the $p$th mean values of harmonic and subharmonic functions with $p\leq 1$”, Proc. London Math. Soc., 20 (1970), 249–275 | DOI | MR

[6] Henkin G. M., Chirka E. M., “Boundary properties of holomorphic functions of several complex variables”, Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat., 4, 1975, 13–142

[7] Krotov V. G., “Estimates of maximal operators connected with the boundary behaviour and their applications”, Trudy Mat. Inst. Steklov, 190, 1989, 117–138 | MR

[8] Nagel A., Rudin W., Shapiro J. H., “Tangential Boundary Behavior of Function in Dirichlet-Type Spaces”, Annals of Mathematics Second Series, 116:2 (1982), 331–360 | DOI | MR

[9] Nagel A., Stein M., “On certain maximal functions and approach regions”, Advances in mathematics, 54 (1984), 83–106 | DOI | MR

[10] Nana C., Sehba B., “Carleson Embeddings and Two Operators on Bergman Spaces of Tube Domains over Symmetric Cones”, Integral Equations and Operator Theory, 83 (2015), 151–178 | DOI | MR

[11] Rudin W., Function Theory in the Unit Ball of $C^{n}$, Springer, New York, 2008 | MR

[12] Sehba B. F., “Bergman type operators in tube domains over symmetric cones”, Proc. Edinburg. Math. Society, 52 (2009), 529–544 | DOI | MR

[13] Sehba B. F., “Hankel operators on Bergman Spaces of tube domains over symmetric cones”, Integral Equations Operator Theory, 62:2 (2008), 233–245 | DOI | MR

[14] Shamoyan R., “A note on Poisson type integrals in pseudoconvex and convex domains of finite type and some related results”, Acta Universitatis Apulensis, 40 (2017), 19–27 | MR

[15] Shamoyan R., Mihić O., “In search of traces of some holomorphic functions in polyballs”, Revista Notas de MatemAtica, 4 (2008), 1–23 | MR

[16] Shamoyan R., Mihić O., “On traces of holomorphic functions on the unit polyball”, Appl. Anal. Discrete Math., 3 (2009), 198–211 | DOI | MR

[17] Shvedenko S. V., “Hardy classes and related spaces of analytic functions in the unit disc, polydisc and ball”, Itogi nauki i tekhniki, VINITI, 23, 3–124 | MR

[18] Zhu K., Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, 226, Springer, New York, 2005 | MR