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

Author
I. E. Chyzhykov, M. A. Voitovych
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), 480499.

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), 501518.

12. C.N. Linden Integral logarithmic means for regular functions, Pacific J. of Math., 138 (1989), 119127.

13. C.N. Linden The characterization of orders for regular functions, Math. Proc. Cambridge Phil. Soc., 111 (1992), 299307.

14. M. Stoll Boundary limits of subharmonic functions in the disc, Proc. Amer. Math. Soc., 93 (1985), 567568.

15. I.E. Chyzhykov Generalization of a Hardy-Littlewood theorem, Math. Methods and Phys.-Mech. Fields, 49 (2006), 2, 7479. (in Ukrainian)

Pages
159-170
Volume
46
Issue
2
Year
2016
Journal
Matematychni Studii
Full text of paper
pdf
Table of content of issue