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We discuss an inverse mean value property of solutions of the heat equation. We show that, under certain conditions, a volume mean value identity characterizes heat balls.

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) .

In this article we prove the following: If u u is a nonzero superharmonic function on a proper subdomain D D in R n {R^n} , then \[ ∫ D | u ( x ) | p δ D ( x ) n p − n − 2 p d x = ∞ , {\int _D {\left | {u(x)} \right |} ^p}{\delta _D}{(x)^{np - n - 2p}}dx = \infty , \] where 0 > p ≤ 1 0 > p \leq 1 and δ D ( x ) {\delta _D}(x) denotes the distance between x x and the boundary of D D .