Growth of pth means of the PoissonStieltjes integrals in polydisc 

Author 
chyzhykov@yahoo.com^{1}, o.zolota@gmail.com^{2}
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 PoissonStieltjes integrals in the unit polydisc for $p>1$. The estimate is given in terms of the smoothness of a complexvalued Stieltjes measure $\mu$. If the measure $\mu$ is positive, the estimate becomes equivalent to the smoothness condition.

Keywords 
PoissonStieltjes integral; pth means; unit polydisc; integral modulus of continuity; complexvalued
Stieltjes measure; growth

DOI 
doi:10.30970/ms.52.1.4854

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 HardyLittlewood theorem// Mathematical methods and physicalmechanical fields. – 2006. – V.49, ¹2 – 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, ¹7. – P. 899–913. 4. Chyzhykov I.E., Zolota O.A. Sharp estimates of the growth of the PoissonStieltjes integral in the polydisc// Mat. Stud. – 2010. – V.34, ¹2. – P. 193–196. Corrections in Mat. Stud. – 2012. – V.37, ¹2. – 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 YorkAmsterdam, 1969. 7. A. Zygmund, Trigonometric Series, Cambridge, 1959. 
Pages 
4854

Volume 
52

Issue 
1

Year 
2019

Journal 
Matematychni Studii

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