matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/2411-9326-2025-2-56-64matmess-78Research ArticleМАТЕМАТИКАOn new sharp theorems on Toeplits operators in Lizorkin–Triebel spacesOn new sharp theorems on Toeplits operators in Lizorkin–Triebel spacesShamoyanR. F.ShamoyanR. F.

Romy F. Shamoyan

83 Astrakhanskaya Street, Saratov 410012

Romy F. Shamoyan

83 Astrakhanskaya Street, Saratov 410012

rsham@mail.ruTomashevskayaE. B.TomashevskayaE. B.

Elena B. Tomashevskaya

43 Bezhitskaya Street, Bryansk 241050

Elena B. Tomashevskaya

43 Bezhitskaya Street, Bryansk 241050

tomele@mail.ruSaratov State UniversityРоссияSaratov State UniversityRussian FederationBryansk State Technical UniversityРоссияBryansk State Technical UniversityRussian Federation2025300620253225664Copyright © Shamoyan R.F., Tomashevskaya E.B., 20252025Shamoyan R.F., Tomashevskaya E.B.Shamoyan R.F., Tomashevskaya E.B.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://matmess.elpub.ru/jour/article/view/78

In this short note we provided new sharp results concerning the action of so called integral Toeplits operators in holomorphic Lizorkin-Triebel function spaces in the unit polydisk. Holomorphic function spaces of Lizorkin-Triebel type in the unit polydisk extend well known Bergman and Hardy spaces simultuanously. Because of this it is interesting to study these new spaces in higher dimension. Our results may have various interesting applications in complex function theory. Of one and several complex variables. Our proofs in particular are based on several standard estimates of complex function theory of several complex variables. We also mention that Toeplits operators were mainly studied previously only in one dimension namely in the unit disk case. Our results may have also interesting applications in operator theory.

In this short note we provided new sharp results concerning the action of so called integral Toeplits operators in holomorphic Lizorkin-Triebel function spaces in the unit polydisk. Holomorphic function spaces of Lizorkin-Triebel type in the unit polydisk extend well known Bergman and Hardy spaces simultuanously. Because of this it is interesting to study these new spaces in higher dimension. Our results may have various interesting applications in complex function theory. Of one and several complex variables. Our proofs in particular are based on several standard estimates of complex function theory of several complex variables. We also mention that Toeplits operators were mainly studied previously only in one dimension namely in the unit disk case. Our results may have also interesting applications in operator theory.

mixed norm spacesunit diskanalytic functionBergman projectionHardy–Sobolev spacesBergman–Sobolev spacesmixed norm spacesunit diskanalytic functionBergman projectionHardy–Sobolev spacesBergman–Sobolev spacesReferencesAleksandrov A. B. and Peller V. V. “Hankel operators and similarity to a contraction,” Intern. Math. Res. Notices, 6, 263–275 (1996).Aleksandrov A. B. and Peller V. V. “Hankel operators and similarity to a contraction,” Intern. Math. Res. Notices, 6, 263–275 (1996).Arsenovi’c M. and Shamoyan R. F. “On multipliers on holomorphic F p,q α type spaces on the unit polydisc,” J. Sib. Federal Univ. Mathematics and Physics, 5 №. 4, 471–479 (2012).Arsenovi’c M. and Shamoyan R. F. “On multipliers on holomorphic F p,q α type spaces on the unit polydisc,” J. Sib. Federal Univ. Mathematics and Physics, 5 №. 4, 471–479 (2012).Ortega J. and J. F‘abrega, “Holomorphic Triebel-Lizorkin spaces,” J. Funct. Anal. 151 No. 1, 177–212 (1997).Ortega J. and J. F‘abrega, “Holomorphic Triebel-Lizorkin spaces,” J. Funct. Anal. 151 No. 1, 177–212 (1997).Shamoyan R. F., Multipliers, Toeplitz operators and duality in some spaces of analytic functions in the unit polydisk [in Russian], PhD Dissertation, Moscow, (2001).Shamoyan R. F., Multipliers, Toeplitz operators and duality in some spaces of analytic functions in the unit polydisk [in Russian], PhD Dissertation, Moscow, (2001).Shamoyan R. F., On multipliers from Bergman-type spaces to Hardy spaces in the polydisk,” Ukr. Math. J., 10, 1405–1415 (2000).Shamoyan R. F., On multipliers from Bergman-type spaces to Hardy spaces in the polydisk,” Ukr. Math. J., 10, 1405–1415 (2000).Aleksandrov A. B., Function theory in the ball, in several complex variables. II, Springer-Verl., New York (1994).Aleksandrov A. B., Function theory in the ball, in several complex variables. II, Springer-Verl., New York (1994).Rudin W., Function theory in the polydisks, Benjamin, New York (1969).Rudin W., Function theory in the polydisks, Benjamin, New York (1969).Harutyunyan A. and Shamoyan F., “Toeplitz operators in multidimensional spaces Hp (α) of M. M. Djrbashian,” J. Contemp. Math. Anal., National Acad. Sci. Armenia 30 No. 2, 70–78 (1995).Harutyunyan A. and Shamoyan F., “Toeplitz operators in multidimensional spaces Hp (α) of M. M. Djrbashian,” J. Contemp. Math. Anal., National Acad. Sci. Armenia 30 No. 2, 70–78 (1995).Janson S., Peetre J., and Semmes S., “On the action of Hankel and Toeplitz operators on some function spaces,” Duke Math. J. 51 No. 4, 937–958 (1984).Janson S., Peetre J., and Semmes S., “On the action of Hankel and Toeplitz operators on some function spaces,” Duke Math. J. 51 No. 4, 937–958 (1984).Rudin W. Function Theory in the Unit Ball of C n, New York, Springer-Verl. (1980).Rudin W. Function Theory in the Unit Ball of C n, New York, Springer-Verl. (1980).Trigub R. M., “Multipliers of class Hp (Dm) for p ∈ (0, 1) and the approximative properties of summation methods for power series [in Russian],” Dokl. Akad. Nauk [Russian Acad. Sci. Dokl.Math.] 335 No. 6, 697–699 (1994).Trigub R. M., “Multipliers of class Hp (Dm) for p ∈ (0, 1) and the approximative properties of summation methods for power series [in Russian],” Dokl. Akad. Nauk [Russian Acad. Sci. Dokl.Math.] 335 No. 6, 697–699 (1994).Trigub R. M., “Multipliers in the Hardy spaces Hp(Dm) with p ∈ (0, 1] and approximation properties of summability methods for power series [in Russian],” Mat. Sb., 188 No. 4, 145–160 (1997). (English version: Sbornik Mathematics, 188, No. 4, 621–638 (1997).Trigub R. M., “Multipliers in the Hardy spaces Hp(Dm) with p ∈ (0, 1] and approximation properties of summability methods for power series [in Russian],” Mat. Sb., 188 No. 4, 145–160 (1997). (English version: Sbornik Mathematics, 188, No. 4, 621–638 (1997).

The authors declare that there are no conflicts of interest present.