Growth of p-th means of the Poisson-Stieltjes integrals in
polydisc

I. E. Chyzhykov; O. A. Kutnyak

doi:10.30970/ms.52.1.48-54

We prove a sharp upper estimate of the $p$th means of the Poisson-Stieltjes integrals in the unit polydisc for $p>1$. The estimate is given in terms of the smoothness of a complex-valued Stieltjes measure $\mu$. If the measure $\mu$ is positive, the estimate becomes equivalent to the smoothness condition.

1. Budagov A.A. Moduli of continuity of uniform permutations. Thesis of candidate sciences, Odesa, 1992. — 100 p. (in Russian)

2. Chyzhykov I.E. Generalization of one of the Hardy-Littlewood theorem// Mathematical methods and physical-mechanical fields. – 2006. – V.49, №2 – P. 74–79. (in Ukrainian)

3. Chyzhykov I., Voitovych M. Growth description of pth means of the Green potential in the unit ball// Complex Variables and Elliptic Equations. – 2017. – V.62, №7. – P. 899–913.

4. Chyzhykov I.E., Zolota O.A. Sharp estimates of the growth of the Poisson-Stieltjes integral in the polydisc// Mat. Stud. – 2010. – V.34, №2. – P. 193–196. Corrections in Mat. Stud. – 2012. – V.37, №2. – P. 223–224.

5. M. Djrbashian, Integral Transforms and Representation of Functions in the Complex Domain, Moscow, 1966. (in Russian)

6. W. Rudin, Function Theory in Polydiscs, New York-Amsterdam, 1969.

7. A. Zygmund, Trigonometric Series, Cambridge, 1959.

	Growth of p-th means of the Poisson-Stieltjes integrals in polydisc
Author	I. E. Chyzhykov¹, O. A. Kutnyak² chyzhykov@yahoo.com¹, o.zolota@gmail.com² 1) Faculty of Mechanics and Mathematics Lviv Ivan Franko National University; 2) Institute of Physics, Mathematics, Economics and Innovative Technologies Drohobych Ivan Franko State Pedagogical University
Abstract	We prove a sharp upper estimate of the $p$th means of the Poisson-Stieltjes integrals in the unit polydisc for $p>1$. The estimate is given in terms of the smoothness of a complex-valued Stieltjes measure $\mu$. If the measure $\mu$ is positive, the estimate becomes equivalent to the smoothness condition.
Keywords	Poisson-Stieltjes integral; p-th means; unit polydisc; integral modulus of continuity; complexvalued Stieltjes measure; growth
DOI	doi:10.30970/ms.52.1.48-54
Reference	1. Budagov A.A. Moduli of continuity of uniform permutations. Thesis of candidate sciences, Odesa, 1992. — 100 p. (in Russian) 2. Chyzhykov I.E. Generalization of one of the Hardy-Littlewood theorem// Mathematical methods and physical-mechanical fields. – 2006. – V.49, №2 – P. 74–79. (in Ukrainian) 3. Chyzhykov I., Voitovych M. Growth description of pth means of the Green potential in the unit ball// Complex Variables and Elliptic Equations. – 2017. – V.62, №7. – P. 899–913. 4. Chyzhykov I.E., Zolota O.A. Sharp estimates of the growth of the Poisson-Stieltjes integral in the polydisc// Mat. Stud. – 2010. – V.34, №2. – P. 193–196. Corrections in Mat. Stud. – 2012. – V.37, №2. – P. 223–224. 5. M. Djrbashian, Integral Transforms and Representation of Functions in the Complex Domain, Moscow, 1966. (in Russian) 6. W. Rudin, Function Theory in Polydiscs, New York-Amsterdam, 1969. 7. A. Zygmund, Trigonometric Series, Cambridge, 1959.
Pages	48-54
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Growth of p-th means of the Poisson-Stieltjes integrals in polydisc