matmessМатематические заметки СВФУMathematical notes of NEFU2411-93262587-876XСеверо-Восточный федеральный университет имени М.К. Аммосова10.25587/SVFU.2020.15.36.006matmess-235Research ArticleМАТЕМАТИКАON SOME NEW ESTIMATES FOR INTEGRALS OF THE LUSIN’S SQUARE FUNCTION IN THE UNIT POLYDISKON SOME NEW ESTIMATES FOR INTEGRALS OF THE LUSIN’S SQUARE FUNCTION IN THE UNIT POLYDISKShamoyanR. F.ShamoyanR. F.

Shamoyan Romi Fayzoevich

90-18 Fokina street, Bryansk, 241050, Russia 

Shamoyan Romi Fayzoevich

90-18 Fokina street, Bryansk, 241050, Russia 

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

Tomashevskaya Elena Bronislavovna

Bryansk State Technical University, bul. 50-letiya Oktyabrya, 7, Bryansk, 241035, Russia 

Tomashevskaya Elena Bronislavovna

Bryansk State Technical University, bul. 50-letiya Oktyabrya, 7, Bryansk, 241035, Russia 

tomele@mail.ru-Россия-Russian FederationBronislavovna Bryansk State Technical UniversityРоссияBryansk State Technical UniversityRussian Federation2020130420262736676Copyright © Shamoyan R.F., Tomashevskaya E.B., 20262026Shamoyan 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/235

The purpose of the note is to obtain new estimates for the quasinorm of Hardy’s analytic classes of in the polydisk. We extend some classical onedimensional assertions to the case of several complex variables.

Our results more precisely provide direct new extention of some known one variable theorems concerning area integral to the case of simplest product domains namely the unit polydisk in Cn. Let further D be a bounded or unbounded domain in Cn. For example, tubular domain over symmetic cone or bounded pseudoconvex domain with smooth boundary. Our results can be probably extended to the case of products of such type complicated domains, namely even to D × ··· × D. This can be probably done based on some approaches we suggested and used in this paper. On the other hand our results in simpler case namely in the unit polydisk may also have various interesting applications in complex function theory in the unit polydisk. 

The purpose of the note is to obtain new estimates for the quasinorm of Hardy’s analytic classes of in the polydisk. We extend some classical onedimensional assertions to the case of several complex variables.

Our results more precisely provide direct new extention of some known one variable theorems concerning area integral to the case of simplest product domains namely the unit polydisk in Cn. Let further D be a bounded or unbounded domain in Cn. For example, tubular domain over symmetic cone or bounded pseudoconvex domain with smooth boundary. Our results can be probably extended to the case of products of such type complicated domains, namely even to D × ··· × D. This can be probably done based on some approaches we suggested and used in this paper. On the other hand our results in simpler case namely in the unit polydisk may also have various interesting applications in complex function theory in the unit polydisk. 

polydisksquare functionanalytic functionHardy spacespolydisksquare functionanalytic functionHardy spacesReferencesTriebel Kh. Theoriya funktsionalnykh prostranstv (The theory of functional spaces), Moscow, (1986).Triebel Kh. Theoriya funktsionalnykh prostranstv (The theory of functional spaces), Moscow, (1986).Dyakonov K. M. Besov spaces and outer functions. Michigan Math. J. 45 (1998), No. 1, 143–157.Dyakonov K. M. Besov spaces and outer functions. Michigan Math. J. 45 (1998), No. 1, 143–157.Dyakonov K. M. Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math., 178 (1997), 143–167.Dyakonov K. M. Equivalent norms on Lipschitz-type spaces of holomorphic functions. Acta Math., 178 (1997), 143–167.Yoneda R. Characterizations of Bloch space and Besov spaces by oscillations. Hokkaido Math. J. 29 (2000), no. 2, 409–451.Yoneda R. Characterizations of Bloch space and Besov spaces by oscillations. Hokkaido Math. J. 29 (2000), no. 2, 409–451.Zhu K. Operator theory in function spaces. New York: Springer, (1990).Zhu K. Operator theory in function spaces. New York: Springer, (1990).Shamoyan R. F. O prostranstvakh golomorfnykh v polikruge funktsiy tipa Lizorkina–Tribelya (On spaces of holomorphic functions in polydisk Lizorkin–Tribel type ), Izv. NAN Arm., 3 (2002), 57–78.Shamoyan R. F. O prostranstvakh golomorfnykh v polikruge funktsiy tipa Lizorkina–Tribelya (On spaces of holomorphic functions in polydisk Lizorkin–Tribel type ), Izv. NAN Arm., 3 (2002), 57–78.Shamoyan R. F. On the quasinorm of holomorphic functions from classes Lizorkin-Tribel in subostov, Symposium ”Fourier Series and their applications” (in Russian), (2002), 54–55.Shamoyan R. F. On the quasinorm of holomorphic functions from classes Lizorkin-Tribel in subostov, Symposium ”Fourier Series and their applications” (in Russian), (2002), 54–55.Guliyev V. S., Lizorkin P. I. Classes of holomorphic and harmonic functions in a polycircle in connection with their boundary values, Research on the theory of differentiable functions of many variables and its applications. Proc. Steklov Inst. Math., 204 (1994), 117–135.Guliyev V. S., Lizorkin P. I. Classes of holomorphic and harmonic functions in a polycircle in connection with their boundary values, Research on the theory of differentiable functions of many variables and its applications. Proc. Steklov Inst. Math., 204 (1994), 117–135.Coifman R., Meyer Y., Stein E. Some new functional spaces and their application to harmonic analysis. Journal of Functional Analysis, 62:2 (1985), 304–335.Coifman R., Meyer Y., Stein E. Some new functional spaces and their application to harmonic analysis. Journal of Functional Analysis, 62:2 (1985), 304–335.Cohn W., Verbitsky I. Factorization of Tent spaces and Hankel operators. Journal of Functional Analysis, 175 (2000), 308–329.Cohn W., Verbitsky I. Factorization of Tent spaces and Hankel operators. Journal of Functional Analysis, 175 (2000), 308–329.Djrbashian M. M., Shamoian F. A. Topics in the theory of Apα spaces, Leipzig, Teubner-Verl., (1988).Djrbashian M. M., Shamoian F. A. Topics in the theory of Apα spaces, Leipzig, Teubner-Verl., (1988).Zygmund A. Trigonometric series Vols. I, II (2nd ed.), Cambridge University Press,(1959).Zygmund A. Trigonometric series Vols. I, II (2nd ed.), Cambridge University Press,(1959).Cohn W. Bergman projections and operators on Hardy spaces. Funct. Anal. J., 144 (1997), 1–19.Cohn W. Bergman projections and operators on Hardy spaces. Funct. Anal. J., 144 (1997), 1–19.Gvaradze M. Mnozhiteli odnogo klassa analiticheskikh funktsiy, opredelennykh na polidiske, Tr. Tbil. Mat. In-ta (In Russian), 66 (1980), 15–21.Gvaradze M. Mnozhiteli odnogo klassa analiticheskikh funktsiy, opredelennykh na polidiske, Tr. Tbil. Mat. In-ta (In Russian), 66 (1980), 15–21.Koosis J.B. Bounded analytic functions. Academic Press, Orlando, Fla, (1981).Koosis J.B. Bounded analytic functions. Academic Press, Orlando, Fla, (1981).Cohn W. A factorization theorem for the derivative of a function in Hp. Proc. Amer. Math.Cohn W. A factorization theorem for the derivative of a function in Hp. Proc. Amer. Math.Soc. 127:2 (1999), 507–517.Soc. 127:2 (1999), 507–517.

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