On the mean value property of superharmonic and subharmonic functions

Recall that a function u is harmonic (superharmonic, subharmonic) in an open set U ⊂ Rn (n ≥ 1) if u ∈ C2(U) and Δu = 0 (Δu ≤ 0,Δu ≥ 0) on U . Denote by H(U) the space of harmonic functions in U and SH(U) (sH(U)) the subset of C2(U) consisting of superharmonic (subharmonic) functions in U . If A ⊂ Rn is Lebesgue measurable, L1(A) denotes the space of Lebesgue integrable functions on A and |A| denotes the Lebesgue measure of A when A is bounded. We also recall the mean value property of harmonic, superharmonic, and subharmonic functions in U ([2]): if x ∈ U and B(x,r) = {y ∈ Rn; ‖y− x‖ < r}, r > 0, is such that B(x,r)⊂U , then for all u∈H(U) (SH(U),sH(U)), u(x)= (≥,≤) 1 ∣ B(x,r) ∣ ∣ ∫