16 GEORGIOS K. ALEXOPOULOS
THEOREM
1.10.1. Let L be a centered left invariant sub-Laplacian on a con-
nected nilpotent Lie group N. Then every L-harmonic function u on N (i.e. satis-
fying Lu = 0 on N), which grows polynomially, is equal to a polynomial.
We shall give the generalisation of the above result to connected Lie groups of
polynomial volume growth in section 1.12 below, where we shall also make some
further remarks and give some related references.
1.11. A Taylor formula for the heat functions on Lie groups of poly-
nomial volume growth.
1. The corrected monomials. We shall use the notation of sections 1.4 and 1.9.
Let G be a connected Lie group of polynomial volume growth. We define monomials
on G by extending the monomials P(x) on (Si) A T
m
the following way:
We extend first the monomials P(x) on (SI)N to monomials
Pf(x,
z) on S\ \M\ =
G/C by setting
P'(x,z) = P(x), (x,z) eS1\M1
and then to monomials P"{g) on G by setting
P"(g) = P'(n(g)), g€G.
If the original monomial P on
(SI)JV
satisfies (1.9.1) then its extention P" to G
also satisfies (1.9.1) i.e. P and its extention P" to G have the same homogeneous
degree.
As in the nilpotent case, we shall say that P(t,g) is a monomial on R x G if
P{t,g) = trnQ{g), with Q(g) a monomial on G. We shall define the homogeneous de-
gree
&egHP{t,g)oiP{t,g)by
degH P(t, g) = 2m + deg# Q(g).
As in the nilpotent case, we shall denote by
Po(t,g),P2(t,g),...,P1/k(t,g)
the monomials with homogeneous degree k.
Let L be a centered left invariant sub-Laplacian on G and let LH be the asso-
ciated homogenised operator.
Let us also associate to LH and to the monomials Pi(t,x), i = 0,1,2,... poly-
nomials Qpi(t,x), i 0,1,2,..., as in section 1.9.1.
PROPOSITION
1.11.1. To every monomial Pi (t, g) with degH Pi = h, as above,
we can associate another "corrected monomial" Qp (£, g) which can be written as
(l.ll.l) QtM9) = Pifrg) + Yl W9)Pj(t,g),
0jisk-i
where the functions ij)1- are functions of type QP and which satisfies
(l-H-2) (§t+LH) QpSt'9) ={.Ft+L) QP^9)-
Note that the polynomials Qp in (1.11.1) and (1.11.2) above, are not neces-
sarily unique.
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