Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences
Abstract
In this paper, we define a new capacity Δm on the class of shm functions, which is defined by Laplace operator. We prove that Δm-capacity satisfies Choquet’s axioms of measurability. Moreover, we compare our capacity with Sadullaev-Abdullaev capacities. In particular, it implies that Δm-capacity of a set E is zero if and only if E is a m-polar set.
First Page
156
Last Page
165
References
1. Sadullaev A., Abdullaev B. Potential theory in the class of m-subharmonic functions. Trudy Mat. Inst. Steklova. Vol. 279, pp. 166-192 (2012). (in Russian)
2. Sadullaev A. Pluripotential theory. Applications. Palmarium Academic Publishing (2012). (in Russian)
3. Sadullaev A. Plurisubharmonic measures and capacities on complex manifolds. Uspekhi mat. nauk. T. 36, No. 4, pp. 53-105 (1981). (in Russian)
4. Sadullaev A., Rakhimov K. Capacity Dimension of the Brjuno Set. Indiana University Mathematics Journal. Vol. 64, No. 6, pp. 1829-1834 (2015).
5. Blocki Z. Weak solutions to the complex Hessian equation. Ann. Inst. Fourier. Vol. 55, No. 5, pp. 1735–1756 (2005).
6. Bedford E. and Taylor B. A new capacity for plurisubharmonic functions. Acta Math. Vol. 149, pp. 1-40 (1982).
7. Brelo M. Fundamentals of classical potential theory. Moscow, Mir (1964). (in Russian)
8. Rakhimov K. ℂn-capacity, defined by Laplacian. Uzbek Mathematical Journal. No. 2, pp. 99-105 (2012). (in Russian)
9. Rakhimov K., Capacity dimension of Perez-Marco set. Contemporary Mathematics. Vol. 662, pp. 131-137 (2016).
10. Beckenbach E., Bellman R. Inequalities. Moscow, Mir (1961). (in Russian).
Recommended Citation
Akramov, Nurali and Egamberganov, Khakimboy
(2023)
"A new capacity in the class of shm functions defined by laplace operator,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 6:
Iss.
3, Article 4.
DOI: https://doi.org/10.56017/2181-1318.1255