Harmonic functions representable by integrals of Green type. I

We introduce the concepts of an integral of Green–Lebesgue type and the integral of Green–Lebesgue which generalizes the classical representation of a harmonic function inside a closed surface $S$ by means of Green's formula. We give conditions for representability of a harmonic function by integrals of the above types. In this part of the work we consider the case of the unit sphere $S$.