Filomat 2024 Volume 38, Issue 5, Pages: 1613-1622
https://doi.org/10.2298/FIL2405613H
Full text ( 232 KB)
Convergence for m−positive currents and m−subharmonic functions
Hbil Jawhar (Department of Mathematics, Jouf University, Sakaka, Saudi Arabia), jmhbil@ju.edu.sa
Zaway Mohamed (Department of Mathematics, Faculty of Sciences of Sfax, Sfax University, Sfax, Tunisia + Irescomath Laboratory, Gabes University, Zrig Gabes, Tunisia), mohamed_zaway@yahoo.fr
In this paper we give sufficient conditions on m−subharmonic functions fk and
m−positive currents Rk of bidegree (p,p) to ensure the convergence of
fk.Rk∧γm−p in the sense of currents. As application of this result, we treat
the special case Rk=(ddcfk)p and we show that one of the condition of the
main result is necessary in this case.
Keywords: Monge-Ampère operator, m−subharmonic function, Capacity, m−positive closed current
Show references
E. Bedford and B. A. Taylor, Fine Topology, Silov Boundary, and (ddc)n, Journal Of Functionnal Analysis.
Z. Blocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier(Grenoble). 55 (5) (2005), 1735-1756.
L. H. Chinh, Equations Hessiennes complexes, (PhD Thesis), Toulouse III: UT3 Paul Sabatier University. (2012).
L. H. Chinh, A variational approach to complex Hessian equations in Cn, J. Math. Anal. Appl. 431 (2015), 228-259.
J. P. Demailly, Monge-Amp‘ere operators, Lelong numbers and intersection theory, Complex Analysis and Geometry, Univ. Ser. Math. Plenum, New York. (1993), 115-193.
A. Dhouib and F. Elkhadhra, m−Potential theory associated to a positive closed current in the class of m-sh functions, Complex Variables and Elliptic Equations. 61 (7) (2016), 1-28.
J. E. Fornaess and N. Sibony, Oka’s inequality for currents and applications. Math.Ann. 301 (1995), 399-419.
C. O. Kiselman, Sur la definition de l’opérateur de Monge-Ampère complexe. Analyse Complexe: Proceedings, Toulouse Springer- Verleg LNM 1094(1983), 139-150.
P. Lelong, Discontinuitee et annulation de l’operateur de Monge-Ampère complexe, Lecture Notes in Math. Springer-Verlag, Berlin. 1028 (1983), 219-224 (French).
A. S. Sadullaev and B. I. Abdullaev, Potential theory in the class of msubharmonic functions, Tr. Mat. Inst. Steklova. 279 (2012), 166-192.
Y. Xing, Weak convergence of currents, Mathematische Zeitschrift. 260 (2008), 253-264.
A. Zygmund, Trigonometric Series-I. Cambridge University Press (1959).