Continuation of Analytic and Pluriharmonic Functions in the Given Direction by the ChirkaMethod: a Survey

In this paper, we provide a survey of results on analytic and plurisubharmonic continuations of functions that have this set of singularities along a fixed direction. We show the advantages of using the pluripotential theory and the Jacobi-Hartogs series for description of the singular set of such functions.

Properties of Commutativity of Dual Toeplitz Operators on the Orthogonal Complement of Pluriharmonic Dirichlet Space over the Ball

We completely characterize the pluriharmonic symbols for (semi)commuting dual Toeplitz operators on the orthogonal complement of the pluriharmonic Dirichlet space in Sobolev space of the unit ball. We show that, forfandgpluriharmonic functions,SfSg=SgSfon(Dh)⊥if and only iffandgsatisfy one of the following conditions:(1)bothfandgare holomorphic;(2)bothf¯andg¯are holomorphic;(3)there are constantsαandβ, both not being zero, such thatαf+βgis constant.