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