Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 1(21). C. 48-63. ISSN 2079-6641

Содержание

DOI: 10.18454/2079-6641-2018-21-1-48-63

MSC 32A07, 432A10, 32A07

ON SOME NEW ESTIMATES RELATED WITH BERGMAN BALL AND POISSON INTEGRAL IN TUBULAR DOMAIN AND UNIT BALL

R. F. Shamoyan¹, O. R. Mihi´c²

¹Department of Mathematics, Bryansk State Technical University, Bryansk 241050, Russia
²Department of Mathematics, Fakultet organizacionih nauka, University of Belgrade, Jove Ili´ca 154, Belgrade, Serbia

E-mail: rsham@mail.ru,oliveradj@fon.rs

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.

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

УДК 517.53+517.55

О НЕКОТОРЫХ НОВЫХ ОЦЕНКАХ, СВЯЗАННЫХ С ТЕОРЕМАМИ ОБ ОГРАНИЧЕННОСТИ ПРОЕКТОРОВ ТИПА БЕРГМАНА И ИНТЕГРАЛОМ ПУАССОНА В ТРУБЧАТОЙ ОБЛАСТИ И ЕДИНИЧНОМ ШАРЕ

Р. Ф. Шамоян¹, О.Р. Михич²

¹Брянский государственный технический университет, 241050, г. Брянск, Россия
²Отдел математики, Университет Белграда, 154, Белград, Сербия

E-mail: rsham@mail.ru,oliveradj@fon.rs

Введены новые аналитические пространства типа Герца, основанные на шарах Бергмана в трубчатых областях над симметричными конусами и на декартовом произведении таких областей. Мы приводим для этих пространств новые максимальные теоремы и теоремы вложения, обобщающие известные результаты в единичном круге. Мы определяем новый интеграл типа Пуассона в единичном шаре и на декартовом произведении таких областей и распространяем известную классическую максимальную теорему, связанную с ним. Также буду приведены другие результаты для интегралов подобного типа обобщающие некоторые  известные теоремы об интегралах Пуассона в единичном шаре.

Ключевые слова: трубчатые области над симметричными конусами, пространства типа Герца, интегральные операторы типа Бергмана, максимальные.

References

  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.
  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.
  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.
  4. Chyzhykov I., Zolota O., “Growth of the Poisson — Stieltjes integral in the polydisk”, Zh. Mat. Fiz. Anal. Geom., 7(2) (2011), 141-157.
  5. Flett T., “Inequalities for the pth mean values of harmonic and subharmonic functions with p≤1”, Proc. London Math. Soc., 20 (1970), 249-275.
  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.
  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.
  9. Nagel A., Stein M., “On certain maximal functions and approach regions”, Advances in mathematics, 54 (1984), 83–106.
  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.
  11. Rudin W., Function Theory in the Unit Ball of Cn, Springer, New York, 2008.
  12. Sehba B. F., “Bergman type operators in tube domains over symmetric cones”, Proc. Edinburg. Math. Society, 52 (2009), 529–544.
  13. Sehba B. F., “Hankel operators on Bergman Spaces of tube domains over symmetric cones”, Integral Equations Operator Theory, 62:2 (2008), 233–245.
  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.
  15. Shamoyan R., Mihi´c O., “In search of traces of some holomorphic functions in polyballs”, Revista Notas de MatemAtica, 4 (2008), 1–23.
  16. Shamoyan R., Mihi´c O., “On traces of holomorphic functions on the unit polyball”, Appl. Anal. Discrete Math., 3 (2009), 198–211.
  17. Shvedenko S.V., “Hardy classes and related spaces of analytic functions in the unit disc, polydisc and ball, in Russian”, Itogi nauki i tekhniki, VINITI, 1985, 3–124.
  18. Zhu K., Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics. V. 226, Springer, New York, 2005.

 

References (GOST)

  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. 2010. vol 647. pp. 25–56.
  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
  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. 2004. vol. 89. no. 3. pp. 317–360
  4. Chyzhykov I., Zolota O. Growth of the Poisson — Stieltjes integral in the polydisk // Zh. Mat. Fiz. Anal. Geom. 2011. vol. 7(2). pp. 141-157
  5. Flett T. Inequalities for the pth mean values of harmonic and subharmonic functions with p ≤ 1 // Proc. London Math. Soc. 1978. vol. 20. pp. 249-275
  6. Henkin G. M., Chirka E. M. Boundary properties of holomorphic functions of several complex variables // Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. 1975. vol. 4. pp. 13-142.
  7. Krotov V. G. Estimates of maximal operators connected with the boundary behaviour and their applications // Trudy Mat. Inst. Steklov. 1989. vol. 190. pp. 117-138
  8. Nagel A., Rudin W. , Shapiro J. H. Tangential Boundary Behavior of Function in Dirichlet-Type Spaces // Annals of Mathematics Second Series. 1982. vol. 116(2). pp. 331–360
  9. Nagel A., Stein M. On certain maximal functions and approach regions // Advances in mathematics. 1984. vol. 54. pp. 83–106
  10. Nana C., Sehba B. Carleson Embeddings and Two Operators on Bergman Spaces of Tube Domains over Symmetric Cones // Integral Equations and Operator Theory. 2015. vol. 83. pp. 151–178
  11. Rudin W. Function Theory in the Unit Ball of Cn. New York: Springer, 2008.
  12. Sehba B. F. Bergman type operators in tube domains over symmetric cones // Proc. Edinburg. Math. Society. 2009. vol. 52. pp. 529–544.
  13. Sehba B. F. Hankel operators on Bergman Spaces of tube domains over symmetric cones // Integral Equations Operator Theory. 2008. vol. 62. no. 2. pp. 233–245.
  14. Shamoyan R. A note on Poisson type integrals in pseudoconvex and convex domains of finite type and some related results // Acta Universitatis Apulensis. 2017. vol. 40. pp. 19–27.
  15. Shamoyan R., Mihi´c O. In search of traces of some holomorphic functions in polyballs // Revista Notas de Matematica. 2008. vol. 4. pp. 1–23.
  16. Shamoyan R., Mihi´c O. On traces of holomorphic functions on the unit polyball // Appl. Anal. Discrete Math. 2009. vol. 3. pp. 198–211.
  17. Shvedenko S.V. Hardy classes and related spaces of analytic functions in the unit disc, polydisc and ball Itogi nauki i tekhniki, 1985. vol. 23. pp. 3–124.
  18. Zhu K. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics. vol. 226. New York: Springer, 2005.

Для цитирования: Shamoyan R. F., Mihic O. R. On some new estimates related with Bergman ball and Poisson integral in tubular domain and unit ball // Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 1(21). C. 48-63. DOI: 10.18454/2079-6641-2018-21-1-48-63
For citation: Shamoyan R. F., Mihic O. R. On some new estimates related with Bergman ball and Poisson integral in tubular domain and unit ball, Vestnik KRAUNC. Fiz.-mat. nauki. 2018, 21: 1, 48-63. DOI: 10.18454/2079-6641-2018-21-1-48-63

Поступила в редакцию / Original article submitted: 24.12.2017

В окончательном варианте / Revision submitted: 09.03.2018

Shamoyan 

 Шамоян Роми Файзович – кандидат физико-математических наук, старший научный сотрудник кафедры математического анализа, Брянский государственный университет имени академика И. Г. Перовского, Брянск, Россия.
  Shamoyan Romi Fayzovich – Ph.D. (Phys. & Math.), Senior Researcher, Department of Mathematical Analysis, Bryansk State University named after Academician I. G. Perovsky, Bryansk, Russia.

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mih  Михич Оливера – кандидат физико-математических наук, доцент кафедры математики, Белградский университет, г. Белград, Республика Сербия.
  Mihic Olivera – Ph.D. (Phys. & Math.), Associate professor, Department of Mathematics, University of Belgrade, Belgrade, Republic of Serbia.

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