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Some characterizations of the class Em(Ω) and applications

In this paper, we give some characterizations of the class Em(Ω) and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class. | ANNALES POLONICI MATHEMATICI 115.2 2015 Some characterizations of the class Em Ω and applications by Hai Mau Le Hanoi Hong Xuan Nguyen Hanoi and Hung Viet Vu Son La Abstract. We give some characterizations of the class Em Ω and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class. 1. Introduction. The complex Monge Ampère operator has a central role in pluripotential theory and has been extensively studied for many years. This operator was used to obtain many important results of pluripoten- tial theory in Cn n gt 1. An example of such application is the proof of quasi-continuity of plurisubharmonic functions yielding the pluripolarity of negligible sets. In BT1 Bedford and Taylor have shown that this operator is well defined on the class of locally bounded plurisubharmonic functions with range in the class of nonnegative measures. Recently to extend the domain of definition of this operator to plurisubharmonic functions which may or not be locally bounded Cegrell C1 C2 has introduced and investigated the classes E0 Ω F Ω and E Ω on which the complex Monge Ampère oper- ator is well defined. He has developed pluripotential theory on these classes. To extend the class of plurisubharmonic functions and to study a class of complex differential operators more general than the Monge Ampère oper- ator in B1 and DK2 the authors introduced m-subharmonic functions and studied the complex Hessian operator. They were also interested in the complex Hessian equations in Cn and on compact Kähler manifolds. In order to continue the study of the complex Hessian operator for m-subharmonic functions which are not locally bounded in a recent preprint Lu Chinh Hoang Lu introduced the Cegrell classes Em 0 Ω F Ω and E Ω asso- m m ciated to m-subharmonic functions and proved that the complex Hessian operator is well defined on these classes. Thus it is of interest to obtain a characterization of these classes analytically and