Conditions for subharmonicity and subharmonic extensions of functions

It is shown that the well-known local Blaschke-Privalov condition, which distinguishes the subharmonic functions in the set of real upper semicontinuous functions in a fixed Euclidean domain $G$ in terms of integral mean values over balls, can be replaced by other, a priori weaker, local conditions of this type on certain subsets of $G$. Both classical and new results on removable singularities of harmonic and subharmonic functions are obtained as consequences of the central theorem.
Bibliography: 28 titles.