The Generalised Laplace Operator and the Topological Characteristic of Removable S¯ - Singular Sets of Subharmonic Functions

The class of harmonic and subharmonic functions have been studied by many authors, defined in different ways, by the Laplace differential operator, averaging, generalised Laplace operators, etc. The well-known theorem of Blaschke-Privalov gives an excellent criterion for subharmonicity in terms of the generalised Laplace operators: an upper semi-continuous in the domain D ⊂ R n function u ( x ), u ( x ) ≢ - ∞ , is subharmonic if and only if △ ¯ u ( x ) ≥ 0 ∀ x 0 ∈ D \ u - ∞ . One of the notable results is Privalov’s theorem, where he got more deeper result with an exceptional set E : if the function u ( x ), u ( x ) ≢ - ∞ , is upper semi-continuous in the domain D ⊂ R n and the following two conditions hold: △ ¯ B u ( x 0 ) ≥ 0 ∀ x 0 ∈ D \ [ E ∪ u - ∞ ] , where E ⊂ D is a closed in D set, m e s E = 0 ; △ ¯ B u ( x 0 ) > - ∞ ∀ x 0 ∈ E \ P , where P ⊂ E is some polar set. Then the function u ( x ) is subharmonic in D . The purpose of this paper is to characterise completely this type of exceptional sets. For this, we introduce the so-called S ̲ and S ¯ singular-sets, which are directly related to the exceptional set of I. Privalov. We prove: E ∈ S ̲ if and only if m e s E = 0 ; E ∈ S ¯ if and only if E ∘ = ∅ .