Λ-separately subharmonic functions.

One of the important problems of potential theory is the study of (sub) harmonicity of separately (sub) harmonic functions. This problem has been studied by many authors and fairly complete results have been obtained. In this paper we will give a survey of results in this area and study a → -separately subharmonic functions. In this paper, we assume that all coefficients of differential forms belong to the class С1, unless additional smoothness conditions are required. We give a definition of Λ-separately subharmonic function, where Λ=(α′,α″) and we show that under additional conditions, these functions belongs to the class α-subharmonic functions, where a = a ′ (z) ∧ a ′ ′ (w) ∧ β , β = d d c ( | z | 2 + | w | 2). If the function u(z, w)∈С2 (D × G) is ∧−separately subharmonic, then by the set of variables ddcu(z, w) ∧α(z, w) ≥0, i.e. the function u(z, w) is α-subharmonic, where a (z , w) = a ′ (z) ∧ a ′ ′ (w) ∧ β , β = d d c ( | z | 2 + | w | 2). If a function u (z, w), (z, w) ∈ D×G is Λ-separately harmonic and coefficients of differential forms α′(z) and α″(w) are real analytic in the domains D and G respectively, then u(z, w) is real analytic α−harmonic function (where α=α′(z) ∧α″(w)∧β) in the domain D×G by the set of variables. [ABSTRACT FROM AUTHOR]