We consider the iterates of the heat operator
formula here
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
formula here
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
formula here
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
formula here
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.