We consider the iterates of the heat operator

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on Rn+1={(X, t); X =(x1, x2, …, xn) ∈Rn, t∈R}. Let Ω⊂Rn+1 be a domain, and let m[ges ]1 be an integer. A lower semi-continuous and locally integrable function u on Ω is called a poly-supertemperature of degree m if

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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

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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

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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.