10
GEORGIOS K. ALEXOPOULOS
By writing every v G
Rq
as
= / tiz{v)dz+ (v- [ dz(v)dz)
JM \ JM J
we have the decomposition
w =
v0
e
v1
with
Vb = {v G
Rq
: #z(u) = i;, £ G M} and V1 =
{veRq
: [ $z(v)dz = 0}.
JM
Note that with the notation of section 1.2, we have if = Vi X M.
By chosing a different basis of Rq is necessary, we can assume that {ei,..., en}
is a basis of Vb and {e
n+
i,..., eq} a basis of V\. Then
1. if either 1 i n or 1 j n, then tf^- = £^, where 5^- = 1 for i = j
and ^ j = 0 for z 7^ j and
2. if n i, j q, then (#2ij) = 0.
1. L written as an operator on Rq x M. Let us fix a basis {Zi,..., Z
m
} of the
Lie algenra m of M.
Let L = —(Ei + ... -\- E%) -\- EQ be & centered left invariant sub-Laplacian on
Rq
X M. Then L can be written as
(1.5.2) L = - ^ ZiCLijZj- YJ
ZibiJ~Q^T.~
Yl
]fo~biiZi
li,jrn lim ^ l^q
where the coefficients are analytic functions defined on M and satisfying:
1. dij =const., 1 i, j ra,
2. Gj =const., 1 i,j n,
3. if 1 j n, then bij =const.
4. for all 1 i ra, a^ =eonst.
5. if 1 i n, then 0 = 0 a n d finally,
6. if n i q, (&) = 0.
#. The correctors and the homogenised operator LJJ. Let us fix a function
/ G
C°°(R9)
and let us extend / to
Rq
xM by setting f(x, z) = /(x), (x, z) eNxM.
Then, by (1.5.2),
!*.? 9 l2m J niq
njq
The definition of the correctors t/^, j = l,...,g is motivated by this remark.
Let us first denote by LM the projection of L on M:
LM ~ /
J
Zi&ijZj -f- y aiZ{.
l2,jfm lira
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