Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences


In this work some extremal function and its properties are studied for the class of m-subharmonic functions. We study weighted (m,δ)-Green function Vm*(z,K,ψ,δ), defined by the class ℒmδ = {u(z)∈shm(ℂn): u(z)≤δ, z∈ℂn}, δ > 0. We see that the regularity of the points with respect to different numbers δ differ from each other. Nevertheless, we will prove that if the compact K ⊂ ℂn is (m,δ,ψ)-regular, then weighted (m,δ)-Green function is continuous in the whole space ℂn.

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1. Blocki Z. Weak solutions to the complex Hessian equation. Ann. Inst. Fourier Vol. 55, Issue 5, 1735-1756 (2005).

2. Sadullaev A., Abdullaev B. Potential theory in the class of m-subharmonic functions. Proc. Stek. Inst. Math. Vol. 279, No 1, 155-180 (2012).

3. Sadullaev A. Pluripotential theory. Applications. Palmarium Academic Publishing. 307 pp. (2012) (in Russian).

4. Sadullaev A. Further developments of the pluripotential theory (survey). Algebra, complex analysis, and pluripotential theory. Springer Proc. Math. Stat. Vol. 264, Springer, Cham. 167-182 (2018).

5. Sadullaev A. Plurisubharmonic measures and capacities on complex manifolds. Uspekhi Mat. Nauk. Vol. 36, No. 4, 53-105 (1981).

6. Klimek M. Pluripotential theory. Oxford University Press, New York. 274 pp. (1991).

7. Brelot M. On topologies and boundaries in potential theory. Berlin: Springer-Verlag, 176 pp. (1971).

8. Landkof N.S. Foundations of Modern Potential Theory. New York, Berlin, Heidelberg: Springer-Verlag, 517 pp. (1972).

9. Alan M.A. Weighted regularity and two problems of Sadullaev on weighted regularity. Complex Analysis and its Synergies. Springer, Vol. 8, No. 5, 1- 7 (2019). DOI:10.1007/s40627-019-0026-4.

10. Narzillaev N.Kh. Delta-extremal functions in Cn. Journal of Siberian Federal University. Mathematics & Physics. Vol. 14, Issue 3, 389–398 (2021).

11. Narzillaev N.Kh. About ψ-regular points. Tashkent, Acta NUUz, No. 2.2. 173-175 (2017).

12. Whitney H. Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36 pp. 63–89 (1934).

13. Abdullaev B.I., Sharipov R.A. m-subharmonic functions in the whole space Cn. Green's function. Uzbek Mathematical journal. No. 3, 3-8 (2013) (in Russian).

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