Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


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


Last Page



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).

Included in

Analysis Commons



To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.