On asymptotic behavior of the $p$th means of the Green potential for $0<p\leq 1$

I. E. Chyzhykov; M. A. Voitovych

doi:10.15330/ms.46.2.159-170

For $0< p\le 1$ we prove sharp estimates of $p$th means of the invariant Green potentials in the unit ball in $\mathbb{C}^n$ in terms of smoothness properties of a measure.

1. I.E. Chyzhykov, Growth and representation of analytic and harmonic functions in the unit disc, Ukrainian Math. Bulletin, 3 (2006), №1, 31.44.

2. I.E. Chyzhykov, Growth of analytic functions in the unit disc and complete measure in the sense of Grishin, Mat. Stud., 29 (2008), 35.44.

3. I.E. Chyzhykov, Growth of pth means of analytic and subharmonic function in the unit disk and angular distribution of zeros, arXiv:1509.02141v2 [math.CV], (2015), 1.19.

4. I. Chyzhykov, M. Voitovych, On the growth of the Cauchy-Szeg.o transform in the unit ball, J. Math. Phys. Anal, Geom., 11 (2015), №1, 236.244.

5. I. Chyzhykov, M. Voitovych, Growth description of pth means of the Green potential in the unit ball, Complex Variables and Elliptic Equations, http://dx.doi.org/10.1080/17476933.2016.1251423 (2016).

6. S.J. Gardiner, Representation and growth of subharmonic functions in half-space, Proc. London Math. Soc., (3), 48 (1984), 300.318.

7. A. Grishin, Continuity and asymptotic continuity of subharmonic functions, Math. Physics, Analysis, Geometry, ILPTE, 1 (1994), 193.215. (in Russian)

8. W. Rudin, Theory functions in the unit ball in $\mathbb{C}^n$ Berlin-Heidelberg, New York: Springer Verlag, 1980.

9. M. Stoll, Rate of growth of pth means of invariant potentials in the unit ball of $\mathbb{C}^n$, J. Math. Anal. Appl., 143 (1989), 480–499.

10. M. Stoll, Invariant Potential Theory in the Unit Ball of $\mathbb{C}^n$. – Cambridge: Cambridge University Press, 1994.

11. D. Ulrich, Radial limits of M-subharmonic functions, Trans. Amer. Math. Soc., 292 (1985), 501–518.

12. C.N. Linden Integral logarithmic means for regular functions, Pacific J. of Math., 138 (1989), 119–127.

13. C.N. Linden The characterization of orders for regular functions, Math. Proc. Cambridge Phil. Soc., 111 (1992), 299–307.

14. M. Stoll Boundary limits of subharmonic functions in the disc, Proc. Amer. Math. Soc., 93 (1985), 567–568.

15. I.E. Chyzhykov Generalization of a Hardy-Littlewood theorem, Math. Methods and Phys.-Mech. Fields, 49 (2006), №2, 74–79. (in Ukrainian)

	On asymptotic behavior of the $p$th means of the Green potential for $0< p\leq 1$
Author	I. E. Chyzhykov, M. A. Voitovych chyzhykov@yahoo.com, urkevych@gmail.com Ivan Franko National University of Lviv
Abstract	For $0< p\le 1$ we prove sharp estimates of $p$th means of the invariant Green potentials in the unit ball in $\mathbb{C}^n$ in terms of smoothness properties of a measure.
Keywords	Green potential; unit ball; invariant Laplacian; M-subharmonic function; Riesz measure
DOI	doi:10.15330/ms.46.2.159-170
Reference	1. I.E. Chyzhykov, Growth and representation of analytic and harmonic functions in the unit disc, Ukrainian Math. Bulletin, 3 (2006), №1, 31.44. 2. I.E. Chyzhykov, Growth of analytic functions in the unit disc and complete measure in the sense of Grishin, Mat. Stud., 29 (2008), 35.44. 3. I.E. Chyzhykov, Growth of pth means of analytic and subharmonic function in the unit disk and angular distribution of zeros, arXiv:1509.02141v2 [math.CV], (2015), 1.19. 4. I. Chyzhykov, M. Voitovych, On the growth of the Cauchy-Szeg.o transform in the unit ball, J. Math. Phys. Anal, Geom., 11 (2015), №1, 236.244. 5. I. Chyzhykov, M. Voitovych, Growth description of pth means of the Green potential in the unit ball, Complex Variables and Elliptic Equations, http://dx.doi.org/10.1080/17476933.2016.1251423 (2016). 6. S.J. Gardiner, Representation and growth of subharmonic functions in half-space, Proc. London Math. Soc., (3), 48 (1984), 300.318. 7. A. Grishin, Continuity and asymptotic continuity of subharmonic functions, Math. Physics, Analysis, Geometry, ILPTE, 1 (1994), 193.215. (in Russian) 8. W. Rudin, Theory functions in the unit ball in $\mathbb{C}^n$ Berlin-Heidelberg, New York: Springer Verlag, 1980. 9. M. Stoll, Rate of growth of pth means of invariant potentials in the unit ball of $\mathbb{C}^n$, J. Math. Anal. Appl., 143 (1989), 480–499. 10. M. Stoll, Invariant Potential Theory in the Unit Ball of $\mathbb{C}^n$. – Cambridge: Cambridge University Press, 1994. 11. D. Ulrich, Radial limits of M-subharmonic functions, Trans. Amer. Math. Soc., 292 (1985), 501–518. 12. C.N. Linden Integral logarithmic means for regular functions, Pacific J. of Math., 138 (1989), 119–127. 13. C.N. Linden The characterization of orders for regular functions, Math. Proc. Cambridge Phil. Soc., 111 (1992), 299–307. 14. M. Stoll Boundary limits of subharmonic functions in the disc, Proc. Amer. Math. Soc., 93 (1985), 567–568. 15. I.E. Chyzhykov Generalization of a Hardy-Littlewood theorem, Math. Methods and Phys.-Mech. Fields, 49 (2006), №2, 74–79. (in Ukrainian)
Pages	159-170
Volume	46
Issue	2
Year	2016
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

On asymptotic behavior of the $p$th means of the Green potential for $0< p\leq 1$