Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 3(19). C. 25-49. ISSN 2079-6641

Содержание выпуска

DOI: 10.18454/2079-6641-2017-19-3-25-49

MSC 42B15+42B30

ON SOME SHARP THEOREMS ON DISTANCE FUNCTION IN HARDY TYPE, BERGMAN TYPE AND HERZ TYPE ANALYTIC CLASSES

R. F. Shamoyan, S.P. Maksakov

Bryansk State University, 241036, Bryansk, Bezhitskaya st., 14, Russia.
E-mail: rsham@mail.ru, msp222@mail.ru

We present some new sharp estimates concerning distance function in some new mixed norm and Lizorkin-Triebel type spaces in the unit ball.This leads at the same time to direct generalizations of our recent results on extremal problems in such Bergman type spaces. In addition new sharp results in this direction in Hardy-Morrey, and some new weighted Hardy classes in the unit ball and mixed norm Hardy type spaces and Herz type spaces in polyball will be provided and discussed.

Key words: extremal problems, analytic functions, Bergman type spaces, unit ball, Hardy type spaces, Herz type spaces, Mockenhoupt type weights, Hardy-Morrey type spaces.

References

  1. Shamoyan R. F., Mihic O. R., “On a new space of analytic function spaces with mixed norm in the unit polyball”, Palestine Math. Journal, 2015, №4(1), 105–107.
  2. Ortega J., Fabrega J., “Holomorphic Lizorkin-Triebel type spaces”, Journal of Funct. Analysis, 1997, 177–212.
  3. Cascante C., Ortega J., Fabrega J., “The corona theorem for weighted Hardy and Morrey spaces”, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2014, №13(5), 579–607.
  4. Rudin W., Function theory in the unit ball, Springer-Verlag, New York, 1980.
  5. Shamoyan F. A., Djrbashian A., Topics in the theory of Apa spaces, Teubner -Text zur Mathematics, Leipzig, 1988.
  6. Shamoyan R. F., Mihic O. R., “On extremal problems in analytic spaces in the unit ball and polydisk”, Siberian Electronic Mathematical Reports, 2009, №6, 514–517.
  7. Shamoyan R. F., Mihic O. R., “On new estimates for distances in analytic function spaces in the unit disk,polydisk and unit ball”, Bulletin Assoc.Math.Venezolana, 2010, №17, 89–105.
  8. Shamoyan R. F., Mihic O. R., “On a distance function in some New analytic Bergman type spaces in higher dimension”, Journal of Function spaces, 2014, 43–52.
  9. Shamoyan R. F., “On some extremal problems in certain harmonic function spaces”, Issues of Analysis, 2013, №20(I), 43–58.
  10. Zhu K., Spaces of Holomorphic Functions in the unit ball, Springer-Verlag, New York, 2005, 226 pp.
  11. Ortega J., Fabrega J., “Hardy’s inequality and embeddings in holomorphic Triebel-Lizorkin spaces”, Illinois Journal Math., 1999, №43, 733–751.
  12. Xu W. Distances from Bloch functions to some Mobius invariant function spaces in the unit ball of Cn, Journal of Funct. Spaces Appl., 2009, №7(1), 91–104.
  13. Shamoyan R. F., Kurilenko S., “On Extremal problems in tubular domains”, Issues of Analysis, 2013, №3(21), 44–65.
  14. Shamoyan R. F., Arsenovic M., “Some remarks on extremal problems in weighted Bergman spaces of analytic functions”, Commun. Korean Math. Soc., 2012, №27(4), 753–762.
  15. Shamoyan R. F., Mihic O. R., “On new estimates for distances in analytic function spaces in higher dimension”, Sib. Elektron. Mat. Izv., 2009, №6, 514–517.
  16. Shamoyan R. F., Mihic O. R., “On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball”, Bol. Asoc. Mat. Venez., 2010, №17(2), 89–103.
  17. Duren P. L., Theory of Hp Spaces, Academic Press, 1970.
  18. Besov O.V., Il‘in V. P., Nicol‘skij S. M., Integral representation of functions and embedding theorems, 2nd Edition, Fizmatlit, Nauka, 1996.
  19. Ligocka E., “On the Forelli-Rudin construction and weighted Bergman projections”, Studia Math, 1989, №94(3), 257–272.
  20. Abate M., Raissy J., Saracco A., “Toeplitz operators and Carleson measures in strongly pseudoconvex domains”, Journal Funct. Anal., 2012, №263(11), 3449–3491.
  21. Shvedenko S.V., “Hardy classes and related spaces of analytik functions in the init disk, polydisc and unit ball”, Seria Matematica, VINITI, 1985, 3–124.
  22. Shamoyan F. A., Yaroslavceva O.V., “Continuous projections, duality and the diagonal mapping in weighted spaces of holomorphic functions with mixed norm”, Zapiski Nauchnykh Seminarov POMI., 1997, №247, 268–275.
  23. Shamoyan F. A., Antonenkova O. E., “Description of linear continuous functionals in weighted spaces of functions analytic in the ball with mixed norm”, Russian Math. Surveys, 2005, №60(4), 794–795.

 

References (GOST)

  1. Shamoyan R.F., Mihic O. R. On a new space of analytic function spaces with mixed norm in the unit polyball // Palestine Math. Journal. 2015. 4(1). pp. 105–107.
  2. Ortega J., Fabrega J. Holomorphic Lizorkin-Triebel type spaces // Journal of Funct. Analysis. 1997. pp. 177–212.
  3. Cascante C., Ortega J., Fabrega J. The corona theorem for weighted Hardy and Morrey spaces // Ann. Sc. Norm. Super. Pisa Cl. Sci., 2014. 13(5). pp. 579-607.
  4. Rudin W. Function theory in the unit ball New York: Springer-Verlag, 1980.
  5. Shamoyan F.A., Djrbashian A. Topics in the theory of Apa spaces, Leipzig: Teubner -Text zur Mathematics, 1988.
  6. Shamoyan R.F., Mihic O. R. On extremal problems in analytic spaces in the unit ball and polydisk // Siberian Electronic Mathematical Reports. 2009. 6. pp. 514–517.
  7. Shamoyan R.F., Mihic O. R. On new estimates for distances in analytic function spaces in the unit disk,polydisk and unit ball // Bulletin Assoc.Math.Venezolana. 2010. 17. pp. 89–105.
  8. Shamoyan R.F., Mihic O. R. On a distance function in some New analytic Bergman type spaces in higher dimension // Journal of Function spaces. 2014. pp. 43–52.
  9. Shamoyan R.F. On some extremal problems in certain harmonic function spaces // Issues of Analysis. 2013. 20(I). pp. 43–58.
  10. Zhu K. Spaces of Holomorphic Functions in the unit ball. New York: Springer-Verlag, 2005. 226 p.
  11. Ortega J., Fabrega J. Hardy’s inequality and embeddings in holomorphic Triebel-Lizorkin spaces // Illinois Journal Math. 1999. 43. pp. 733–751.
  12. Xu W. Distances from Bloch functions to some Mobius invariant function spaces in the unit ball of Cn // Journal of Funct. Spaces Appl. 2009. 7(1). pp. 91–104.
  13. Shamoyan R.F., Kurilenko S. On Extremal problems in tubular domains // Issues of Analysis. 2013. 3(21). pp. 44–65.
  14. Shamoyan R.F., Arsenovic M. Some remarks on extremal problems in weighted Bergman spaces of analytic functions // Commun. Korean Math. Soc. 2012. 27(4). pp. 753–762.
  15. Shamoyan R.F., Mihic O. R. On new estimates for distances in analytic function spaces in higher dimension // Sib. Elektron. Mat. Izv. 2009. 6. pp. 514–517.
  16. Shamoyan R.F., Mihic O. R. On new estimates for distances in analytic function spaces in the unit disk, the polydisk and the unit ball // Bol. Asoc. Mat. Venez. 2010. 17(2). pp. 89–103.
  17. Duren P.L. Theory of Hp Spaces. Academic Press, 1970.
  18. Besov O.V., Il‘in V.P. and Nicol‘skij S.M. Integral representation of functions and embedding theorems (Russian), 2nd Edition. Moscow: Fizmatlit, Nauka, 1996.
  19. Ligocka E. On the Forelli-Rudin construction and weighted Bergman projections // Studia Math. 1989. 94(3). pp. 257–272.
  20. Abate M., Raissy J., Saracco A. Toeplitz operators and Carleson measures in strongly pseudoconvex domains // Journal Funct. Anal. 2012. 263(11). pp. 3449–3491.
  21. Shvedenko S.V. Hardy classes and related spaces of analytik functions in the init disk, polydisc and unit ball // Seria Matematica, VINITI, 1985. pp. 3–124.
  22. Shamoyan F.A., Yaroslavceva O.V. Continuous projections, duality and the diagonal mapping in weighted spaces of holomorphic functions with mixed norm // Zapiski Nauchnykh Seminarov POMI., 1997. 247. pp. 268–275.
  23. Shamoyan F.A., Antonenkova O.E. Description of linear continuous functionals in weighted spaces of functions analytic in the ball with mixed norm // Russian Math. Surveys. 2005. 60(4), pp. 794–795.

Для цитирования: Shamoyan R. F., Maksakov S. P. On some sharp theorems on distance function in Hardy type, Bergman type and Herz type analytic classes // Вестник КРАУНЦ. Физ.-мат. науки. 2017. № 3(19). C. 25-49. DOI: 10.18454/2079-6641-2017-19-3-25-49.
For citation: Shamoyan R. F., Maksakov S. P. On some sharp theorems on distance function in Hardy type, Bergman type and Herz type analytic classes, Vestnik KRAUNC. Fiz.-mat. nauki. 2017, 19: 3, 25-49. DOI: 10.18454/2079-6641-2017-19-3-25-49.

Поступила в редакцию / Original article submitted: 06.05.2017

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 IGPerovsky, Bryansk, Russia.

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Maksakov

  Максаков Серафим Павлович – магистрант кафедры математического анализа, Брянский государственный университет имени академика И. Г. Перовского, Брянск, Россия.
      Maksakov Serafim Pavlovich — Master of the Department of Mathematical Analysis, Bryansk State University named after Academician I.G. Perovsky, Bryansk, Russia.

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