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, for f and g pluriharmonic functions, SfSg = SgSf on (D-h)(perpendicular to) if and only if f and g satisfy one of the following conditions: (1) both f and g are holomorphic; (2) both (f) over bar and (g) over bar are holomorphic; (3) there are constants alpha and beta, both not being zero, such that alpha f + beta g is constant.