Some Properties of Plurisubharmonic Measures

Abstract

Publisher Summary This chapter presents a study on some properties of plurisubharmonic measures. Let U be an open subset of C n and denoted by M(U) the positive measures μ on U which can be written Δφ = μ where φ is plurisubharmonic on U and Δ is the Laplace operator. The plurisubharmonic functions on U is denoted by PSH(U). The elements in M(U) are called plurisubharmonic measures. For n = 1 every positive measure is a (pluri) subharmonic measure, but for n > 1 the situation is different. A positive measure μ on U is called a local plurisubharmonic measure. The chapter also introduces a class between M and M 1oc . A subharmonic function φ is said to be n-subharmonic if it is subharmonic in each z p , 1 ≤ p ≤ n, the other variables fixed. In particular, every plurisubharmonic function is n-subharmonic. The chapter discusses determinative sets, let U be an open and connected subset of C n and let P be closed in U. The complement of P in U is said to be determinative for PSH(U) if plurisubharmonic functions on U which are equal on U\P have to be equal to U . In the same way one can speak about determinative sets for M (U), m(U) and so on.