On Some Weighted Classes of m-Subharmonic Functions
DOI:
https://doi.org/10.15407/mag20.01.112Анотація
У цiй роботi ми вивчаємо клас $\mathcal{E}_m(\Omega)$ $m$-субгармонiчних функцiй, введений Лю в [18]. Ми доводимо, що збiжнiсть мiр Гессе виводиться зi збiжностi вiдносно $m$-ємностi для функцiй, що належать $\mathcal{E}_m(\Omega)$ та
задовольняють певнi додатковi умови. Далi ми розповсюджуємо цi результати на клас $\mathcal{E}_{m,\chi}(\Omega)$, який залежить вiд заданої дiйсної функцiї $\chi$. Дано повну характеризацiю цих класiв за допомогою мiри Гессе, а
також теорему пiдпродовження вiдносно $\mathcal{E}_{m,\chi}(\Omega)$.
Mathematical Subject Classification 2020: 32W20, 32U05, 32U15, 32U40
Ключові слова:
m-субгармонiчна функцiя, ємнiсть, оператор Гессе, збiжнiсть вiдносно m-ємностiПосилання
E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1--40. https://doi.org/10.1007/BF02392348
E. Bedford and B.A.Taylor, The Dirichlet problem for a complex Monge-Ampère operator, Invent. Math. 37 (1976), 1--44. https://doi.org/10.1007/BF01418826
S. Benelkourchi, Approximation of weakly singular plurisubharmonic functions, Internat. J. Math. 22 (2011) 937--946. https://doi.org/10.1142/S0129167X11007094
S. Benelkourchi, Weighted pluricomplex energy, Potential Anal. 31 (2009), 1--20. https://doi.org/10.1007/s11118-009-9119-7
S. Benelkourchi, V.Guedj, and A.Zeriahi, Plurisubharmonic functions with weak singularities, Complex Analysis and Digital Geometry Ed. M. Passare. Proceedings from the Kiselmanfest, Uppsala Universitet, 2007, 57--73.
Z. Błocki, Weak solutions to the complex Hessian equation, Ann. Inst. Fourier (Grenoble) 55,(2005), 1735--1756. https://doi.org/10.5802/aif.2137
U. Cegrell, Pluricomplex energy, Acta. Math. 180 (1998), 187--217. https://doi.org/10.1007/BF02392899
U. Cegrell, The general definition of the complex Monge-Ampère operator, Ann. Inst.Fourier (Grenoble) 54 (2004), 159--179. https://doi.org/10.5802/aif.2014
U. Cegrell and A. Zeriahi, A Subextension of plurisubharmonic functions with bounded Monge-Ampère operator mass, C.R. Acad. Sci. Paris Ser. 336 (2003), 30--308. https://doi.org/10.1016/S1631-073X(03)00031-1
L.M. Hai and T. V. Dung, Subextension of $m$-subharmonic functions, Vietnam Journal of Mathematics 48 (2020), 47--57. https://doi.org/10.1007/s10013-019-00343-9
L.M. Hai, P.H. Hiep, N.X. Hong, and N.V. Phu, The Monge-Ampère type equation in the weighted pluricomplex energy class, Int. J. Math. 25 (2014), 1450042. https://doi.org/10.1142/S0129167X14500426
L.M. Hai and V. V. Quan, Weak solutions to the complex $m$-Hessian equation on open subsets of $C^n$, Complex Anal. Oper. Theory 279 (2019), 4007--4025. https://doi.org/10.1007/s11785-019-00948-5
P.H. Hiep, Convergence in capacity, Ann. Polon. Math. 93 (2008), 91-99. https://doi.org/10.4064/ap93-1-8
L. Hormander, Notion of Convexity, Progess in Mathematics, Birkhäuser,Boston, 127, 1994.
V.V. Hung, Local property of a class of m-subharmonic functions, Vietnam J. Math. 44 (2016), 621--630. https://doi.org/10.1007/s10013-015-0176-5
V.V. Hung and N.V Phu, Hessian measures on $m$-polar sets and applications to the complex Hessian equations, Complex Var. Elliptic Equ. 8 (2017), 1135--1164. https://doi.org/10.1080/17476933.2016.1273907
A. El Gasmi The Dirichlet problem for the complex Hessian operator in the class $N_m(Ω,f)$, Mathematica scandinavica 127 (2021), 287--316. https://doi.org/10.7146/math.scand.a-125994
H.C. Lu, A variational approach to complex Hessian equations in $C^n$, J. Math. Anal. Appl. 431 (2015), No. 1, 228--259. https://doi.org/10.1016/j.jmaa.2015.05.067
H.C. Lu, Equations Hessiennes Complexes, Ph.D. Thesis, Université Paul Sabatier, Toulouse, France, 2012. https://theses.fr/2012TOU30154
V. T. Nguyen, Maximal $m$-subharmonic functions and the Cegrell class $N_m$, Indagationes Mathematicae 30 (2019), 717--739. https://doi.org/10.1016/j.indag.2019.03.005
A.S. Sadullaev and B.I. Abdullaev, Potential theory in the class of $m$-subharmonic functions, Tr. Mat. Inst. Steklova 279 (2012), 166--192. https://doi.org/10.1134/S0081543812080111
Y. Xing, Continuity of the complex Monge-Ampère operator, Proc. Amer. Math. Soc. 124 (1996), 457--467. https://doi.org/10.1090/S0002-9939-96-03316-3
Y. Xing, Complex Monge-Ampère measures of plurisubharmonic functions with bounded values near the boundary, Can. J. Math. 52 (2000), 1085--1100. https://doi.org/10.4153/CJM-2000-045-x
