A Mean Value Property of Poly‐Temperatures on a Strip Domain

We consider the iterates of the heat operator H = Δ X ‐ ∂ ∂ ton R n +1 ={( X , t ); X =( x 1 , x 2 , …, x n )∈ R n , t ∈ R }. Let Ω⊂ R n +1 be a domain, and let m ⩾1 be an integer. A lower semi‐continuous and locally integrable function u on Ω is called a poly‐supertemperature of degree m if (− H ) m u ⩾0 on Ω (in the sense of distribution). If u and − u are both poly‐supertemperatures of degree m , then u is called a poly‐temperature of degree m . Since H is hypoelliptic, every poly‐temperature belongs to C ∞ (Ω), and hence (− H ) m u ( X , t )=0 ∀( X , t )∈Ω. For the case m =1, we simply call the functions the supertemperature and the temperature. In this paper, we characterise a poly‐temperature and a poly‐supertemperature on a strip D ={( X , t );X∈ R n , 0< t < T } by an integral mean on a hyperplane. To state our result precisely, we define a mean A [·, ·]. This plays an essential role in our argument.