Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 1(21). C. 64-77. ISSN 2079-6641

Содержание

DOI: 10.18454/2079-6641-2018-21-1-64-77

MSC 32A07, 432A10, 32A07

ON SOME NEW PROJECTION THEOREMS AND SHARP ESTIMATES IN HERZ TYPE SPACES IN BOUNDED PSEUDOCONVEX DOMAINS

R. F. Shamoyan, A. N. Shipka

Bryansk State University, 241036, Bryansk, Bezhitskaya st., 14, Russia.

E-mail: rsham@mail.ru, shipka.alexandr@yandex.ru

We prove new projection theorems for new Herz type spaces in various domains in Cn in the unit disk, unit ball, bounded pseudoconvex domains and based on these results we provide sharp estimates for distances in such type spaces under one condition on Bergman kernel. Similar type result in such type spaces in tubular domains over symmetric cones will be also provided.

Key words: pseudoconvex and tubular domains,the unit ball, projection theorem, Herz spaces.

УДК 517.53+517.55

О НЕКОТОРЫХ НОВЫХ ТЕОРЕМАХ И ОЦЕНКАХ В ПРОСТРАНСТВАХ ТИПА ГЕРЦА,
ОГРАНИЧЕННЫХ В ПСЕВДОВЫПУКЛЫХ ОБЛАСТЯХ

Р. Ф. Шамоян, А. Н. Щипка

Брянский государственный университет, 241036, Брянск, ул. Бежицкая., 14, Россия

E-mail: rsham@mail.ru, shipka.alexandr@yandex.ru

Мы доказываем новые проекционные теоремы для новых пространств типа Герца в
различных областях в Cn, в единичном круге, единичном шаре, ограниченных псевдовыпуклых областях и на основе этих результатов даем точные оценки расстояний в таких пространствах при одном дополнительном условии на ядро Бергмана. Аналогичный точный результат при одном дополнительном условии будет установлен для пространств подобного типа в трубчатых областях над симметрическими конусами.

Ключевые слова: псевдовыпуклые и трубчатые области, единичный шар, проекционная теорема, пространства Герца

References

  1. Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B., “Hardy-type inequalities and analytic Besov spaces in tube domains over symmetric cones”, J. Reine Angew. Math., 647:25 (2010).
  2. Ortega J., Fabrega J., “Holomorphic Lizorkin-Triebel type spaces”, Journal of Funct. Analysis, 1997, 177-212.
  3. Rudin W., Function theory in the unit ball, Springer-Verlag, New York, 1980.
  4. Shamoyan F., Djrbashian A., Topics in the theory of Apa spaces, Teubner — Text zur Mathematics, Leipzig, 1988.
  5. Bekolle D., Kagou A. T., “Reproducing properties and Lp estimates for Bergman projections in Siegel domains of second type”, Studia Math., 115:3 (1995).
  6. Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B., “Analytic Besov spaces and symmetric cones”, Jour. Fur seine and ang., 647 (2010), 25-56.
  7. Shamoyan R., Mihic O., “On a distance function in some New analytic Bergman type spaces in higher dimension”, Journal of Function spaces, 2014 (2014).
  8. Shamoyan R., “On some extremal problems in certain harmonic function spaces”, Issues of Analysis, 20(I) (2013), 43-58.
  9. Zhu K., Spaces of Holomorphic Functions in the unit ball, Springer-Verlag, New York, 2005, 226 pp.
  10. Xu W., “Distances from Bloch functions to some Mobius invariant function spaces in the unit ball of Cn”, Journal. of Funct. Spaces and Appl., 7:1 (2009), 91-104.
  11. Kurilenko S., Shamoyan R., “On Extremal problems in tubular domains”, Issues of Analysis, 3(21) (2013), 44-65.
  12. Ortega J. M., Fabrega J., “Mixed-norm spaces and interpolation”, Studia Math., 109:3 (1994), 233-254.
  13. Shamoyan R. F., Arsenovi’c M., “Some remarks on extremal problems in weighted Bergman spaces of analytic function”, Commun. Korean Math. Soc., 27:4 (2012), 753-762.
  14. Shamoyan R., Mihic O., “On new estimates for distances in analytic function spaces in higher dimension”, Sib. Elektron. Mat. Izv., 2009, 514-517.
  15. R. Shamoyan and O. Mihic, “On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball”, Bol. Asoc. Mat. Venez., 17:2 (2010), 89-103.
  16. Duren P., Theory of Hp Spaces, Academic Press, 1970.
  17. Beatrous F., “Lp-estimates for extensions of holomorphic function”, Michigan Math. Journal, 32:3 (1985), 361-380.
  18. Ahern P., Schneider R., “Holomorphic Lipschitz function in pseudoconvex domains”, Amer. Journal of Math., 101:3 (1979), 543-565.
  19. Kerzman N. and Stein E. M., “The Szego kernel in terms of Cauchy-Fantappie kernels”, Duke Math. J., 43:2 (1978), 197-223.
  20. Ligocka E., “On the Forelli-Rudin construction and weighted Bergman projection”, Studia Math., 94:3 (1989), 257-272.
  21. Cohn W. S., “Weighted Bergman projection and tangential area integrals”, Studia Math., 106:1 (1993), 59-76.
  22. Abate M., Raissy J., Saracco A., “Toeplitz operator and Carleson measures in strongly pseudoconvex domains”, Journal of. Funct. Anal., 263:11 (2012), 3449-3491.
  23. Arsenovic M., Shamoyan R., “On some sharp estimates for distances in bounded strongly pseudocnvex domains”, Bulletin Korean Math. Society, 1 (2015).

References (GOST)

  1. Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B. Hardy-type inequalities and analytic Besov spaces in tube domains over symmetric cones // J. Reine Angew. Math. 2010. vol. 647. no. 25.
  2. Ortega J., Fabrega J. Holomorphic Lizorkin-Triebel type spaces // Journal of Funct. Analysis. 1997. pp. 177-212.
  3. Rudin W. Function theory in the unit ball. New York: Springer-Verlag, 1980
  4. Shamoyan F., Djrbashian A. Topics in the theory of Apa spaces. Leipzig: Teubner — Text zur Mathematics, 1988.
  5. Bekolle D., Kagou A.T˙ . Reproducing properties and Lp estimates for Bergman projections in Siegel domains of second type // Studia Math., 1995. vol. 115. no. 3.
  6. Bekolle D., Bonami A., Garrigos G., Ricci F., Sehba B. Analytic Besov spaces and symmetric cones // Jour. Fur seine and ang. 2010. vol. 647. pp. 25-56.
  7. Shamoyan R., Mihic O. On a distance function in some New analytic Bergman type spaces in higher dimension // Journal of Function spaces. 2014. vol. 2014.
  8. Shamoyan R. On some extremal problems in certain harmonic function spaces // Issues of Analysis. 2013. vol. 20(I), pp. 43-58.
  9. Zhu K.Spaces of Holomorphic Functions in the unit ball. New York: Springer-Verlag, 2005. 226 p.
  10. Xu W. Distances from Bloch functions to some Mobius invariant function spaces in the unit ball of Cn // Journal. of Funct. Spaces and Appl. 2009. vol.7. no. 1. pp. 91-104.
  11. Kurilenko S., Shamoyan R. On Extremal problems in tubular domains // Issues of Analysis. 2013. vol 3(21). pp. 44-65.
  12. Ortega J. M., Fabrega J. Mixed-norm spaces and interpolation // Studia Math. 1994. vol. 109. no. 3 .pp. 233-254.
  13. Shamoyan R. F., Arsenovi’c M. Some remarks on extremal problems in weighted Bergman spaces of analytic function // Commun. Korean Math. Soc. 2012. vol. 27. no. 4. pp. 753-762.
  14. Shamoyan R., Mihic O. On new estimates for distances in analytic function spaces in higher dimension // Sib. Elektron. Mat. Izv. 2009. pp. 514-517.
  15. Shamoyan R., Mihic O. On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball // Bol. Asoc. Mat. Venez. 2010. vol. 17. no 2. pp. 89-103.
  16.  Duren P. Theory of Hp Spaces. Academic Press, 1970.
  17. Beatrous F. Lp-estimates for extensions of holomorphic function // Michigan Math. Journal. 1985. vol. 32. no. 3. pp. 361-380.
  18. Ahern P., Schneider R. Holomorphic Lipschitz function in pseudoconvex domains // Amer. Journal of Math. 1979. vol. 101. no. 3. pp. 543-565.
  19. Kerzman N., Stein E. M. The Szego kernel in terms of Cauchy-Fantappie kernels // Duke Math. J.1978. vol. 43, no. 2. pp. 197-223.
  20. Ligocka E. On the Forelli-Rudin construction and weighted Bergman projection // Studia Math. 1989. vol. 94. no. 3. pp. 257-272.
  21. Cohn W. S. Weighted Bergman projection and tangential area integrals // Studia Math. 1993. vol. 106. no. 1. pp. 59-76.
  22. Abate M., Raissy J., Saracco A. Toeplitz operator and Carleson measures in strongly pseudoconvex domains // Journal of. Funct. Anal. 2012. vol. 263. no. 11. pp. 3449-3491.
  23. Arsenovic M., Shamoyan R. On some sharp estimates for distances in bounded strongly pseudocnvex domains // Bulletin Korean Math. Society. 2015. vol. 1.

Для цитирования: Shamoyan R. F., Shepka A.V. On some new projection theorems and sharp estimates in Herz type spaces in bounded pseudoconvex domains // Вестник КРАУНЦ. Физ.-мат. науки. 2018. № 1(21). C. 64-77. DOI: 10.18454/2079-6641-2018-21-1-64-77
For citation: Shamoyan R. F., Shepka A.V. On some new projection theorems and sharp estimates in Herz type spaces in bounded pseudoconvex domains, Vestnik KRAUNC. Fiz. mat. nauki. 2018, 21: 1, 64-77. DOI: 10.18454/2079-6641-2018-21-1-64-77

Поступила в редакцию / Original article submitted: 24.12.2017

В окончательном варианте / Revision submitted: 09.02.2018


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.

1

1


ship  Щипка Александр Николаевич – магистрант 1 курса, физико-математического факультета, по направлению «Математика. Комплексный анализ и алгебра», Брянский государственный университет имени академика И. Г. Перовского, Брянск, Россия.
    Shipka Alexandr Nikolaevich –Master of 1 course, Physics and Mathematics Faculty, in the direction of «Mathematics. Complex analysis and algebra», Bryansk State University named after Academician I.G. Perovsky, Bryansk, Russia.

1

1


Скачать статью  R. F. Shamoyan, A. N. Shipka