Asymptotic behaviour of means of nonpositive M-subharmonic functions

M. A. Voitovych

doi:10.15330/ms.47.1.20-26

We describe growth and decrease of $p$th means, $1\leq p\leq\frac{2n-1}{2(n-1)}$, of nonpositive $\mathcal{M}$-subharmonic functions in the unit ball in $\mathbb{C}^n$ in terms of smoothness properties of a measure. As consequence we obtain a haracterization of asumptotic behaviour for means of Poisson integrals in the unit ball defined by a positive measure.

1. I. Chyzhykov, M. Voitovych, Growth description of pth means of the Green potential in the unit ball, Complex Variables and Elliptic Equations, 52 (2017), №7, 899-913.

2. I. Chyzhykov, M. Voitovych, On the growth of the Cauchy-Szego transform in the unit ball, J. Math. Phys. Anal, Geom., 11 (2015), №3, 236-244.

3. I. Chyzhykov, Growth of analytic functions in the unit disc and complete measure in the sense of Grishin. Mat. Stud., 29 (2008), 35-44.

4. I. 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.

5. S.J. Gardiner, Growth properties of pth means of potentials in the unit ball, Proc. Amer. Math. Soc., 103 (1988), 861-869.

6. A. Grishin, Continuity and asymptotic continuity of subharmonic functions, Math. Physics, Analysis, Geometry, ILPTE, 1 (1994), 193-215. (in Russian)

7. K.T. Khan, J. Mitchell, Greens function on the classical Cartan domains, MRC Technical Summary Report, (1967), №500.

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

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

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

11. M. Stoll, Rate of growth of p-th means of invariant potentials in the unit ball of $\mathbb{C}^n$, II, J. Math. Anal. Appl., 165 (1992), 374-398.

12. M. Stoll, On the rate of growth of the means $M_p$ of holomorphic and pluriharmonic functions on the ball, J. Math. Anal. Appl., 93 (1983), 109-127.

13. M. Stoll, Harmonic and Subharmonic Function Theory on the Hyperbolic Ball, London Mathematical Society Lecture Note Series, V.431, 2016.

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

	Asymptotic behaviour of means of nonpositive M-subharmonic functions
Author	M. A. Voitovych chyzhykov@yahoo.com; urkevych@gmail.com Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	We describe growth and decrease of $p$th means, $1\leq p\leq\frac{2n-1}{2(n-1)}$, of nonpositive $\mathcal{M}$-subharmonic functions in the unit ball in $\mathbb{C}^n$ in terms of smoothness properties of a measure. As consequence we obtain a haracterization of asumptotic behaviour for means of Poisson integrals in the unit ball defined by a positive measure.
Keywords	M-subharmonic function; M-harmonic function; Green potential; unit ball; invariant Laplacian; Riesz measure
DOI	doi:10.15330/ms.47.1.20-26
Reference	1. I. Chyzhykov, M. Voitovych, Growth description of pth means of the Green potential in the unit ball, Complex Variables and Elliptic Equations, 52 (2017), №7, 899-913. 2. I. Chyzhykov, M. Voitovych, On the growth of the Cauchy-Szego transform in the unit ball, J. Math. Phys. Anal, Geom., 11 (2015), №3, 236-244. 3. I. Chyzhykov, Growth of analytic functions in the unit disc and complete measure in the sense of Grishin. Mat. Stud., 29 (2008), 35-44. 4. I. 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. 5. S.J. Gardiner, Growth properties of pth means of potentials in the unit ball, Proc. Amer. Math. Soc., 103 (1988), 861-869. 6. A. Grishin, Continuity and asymptotic continuity of subharmonic functions, Math. Physics, Analysis, Geometry, ILPTE, 1 (1994), 193-215. (in Russian) 7. K.T. Khan, J. Mitchell, Greens function on the classical Cartan domains, MRC Technical Summary Report, (1967), №500. 8. W. Rudin, Theory functions in the unit ball in $\mathbb{C}^n$, Berlin-Heidelberg, New York: Springer Verlag (1980). 9. M. Stoll, Invariant Potential Theory in the Unit Ball of $\mathbb{C}^n$, Cambridge University Press, 1994. 10. M. Stoll, Rate of growth of p-th means of invariant potentials in the unit ball of $\mathbb{C}^n$, J. Math. Anal. Appl., 143 (1989), 480-499. 11. M. Stoll, Rate of growth of p-th means of invariant potentials in the unit ball of $\mathbb{C}^n$, II, J. Math. Anal. Appl., 165 (1992), 374-398. 12. M. Stoll, On the rate of growth of the means $M_p$ of holomorphic and pluriharmonic functions on the ball, J. Math. Anal. Appl., 93 (1983), 109-127. 13. M. Stoll, Harmonic and Subharmonic Function Theory on the Hyperbolic Ball, London Mathematical Society Lecture Note Series, V.431, 2016. 14. D. Ulrich, Radial limits of M-subharmonic functions, Trans. Amer. Math. Soc., 292 (1985), 501-518.
Pages	20-26
Volume	47
Issue	1
Year	2017
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html