Note on the integrability of superharmonic functions

Abstract

Let D D be a domain in R n {{\mathbf {R}}^n} and let S + ( D ) {S^ + }(D) be the set of all nonnegative superharmonic functions on D D . It is shown that if S + ( D ) ⊂ L p ( D ) {S^ + }(D) \subset {L^p}(D) with some p > 0 p > 0 , then for each x 0 ∈ D {x_0} \in D there is a constant C = C ( D , p , x 0 ) > 0 C = C(D,p,{x_0}) > 0 such that the inequality \[ ∫ D u ( x ) p d x ⩽ C u ( x 0 ) p \int _D {u{{(x)}^p}dx \leqslant Cu{{({x_0})}^p}} \] holds for all u ∈ S + ( D ) u \in {S^ + }(D) .