SUB-LAPLACIANS WITH DRIFT
9
Since C is compact we can and shall assume that dc-nieasure(G) = 1.
Also since G/C = 5i X Mi there are Haar measures dsxg and dMxg on S\ and
Mi respectively such that
/ f(g)dg= / f(x,z)dSlxdMl;
JG/C JSx JMX
Since Mi is compact we shall also assume that dM1 -measure(Mi) = 1.
We shall drop the indices G/C, G, S\ and Mi when there is no risk of confusion.
Motivated by the specific nature of the correctors we shall give the following
definition:
We shall say that a function / on Rk x Mi is of type P if there is a G Rfc and
a function 0 G C°°(Mi) such that either
f{x,z) = sm((a,x)))l)(z), (x,z) G
Rfc
x Mu
or
f{x,z) = cos((a,x)))(f)(z), (x,z) G
Rk
x Mi.
Let us denote by TTQ the canonical projection TTQ : G
Rfc
x Mi.
We shall say that / is a function of type P on G if there is another function / '
of type P on
Mfc
x Mi such that f = f on0.
We shall say that / is a function of type QP on G if it is a finite sum of functions
of type P.
If / is a function of type QP then we shall denote by (/) its mean value
(/ = lim - ) - / f(x)dx
where U is a compact neighborhood of the identity element e of G.
Note that if / is a function of type QP then Lf is also a function of type QP.
If we also have (/) = 0, then there is a unique function u of type QP satisfying
Lu = f, (/) = 0.
The correctors ^ will be functions of type QP satisfying (/0J) = 0.
1.5. Calculation of the homogenised sub-Laplacian in the case of the
group R9 X M. In order to illustrate what was said in the previous section, we
shall calculate the homogenised sub-laplacian in the case of the semidirect product
W1
X M of
Rq
by a connected compact Lie group M.
Let us denote hy dz, z £ M the action of M on
Rq.
Note that if (x, z), (y, w) G
1
9
X M then
(x, z)(y, w) = (x + tiz(y),zw).
Let us denote by dZij the entries of the matrix of #z wih respect to the canonical
basis {ei, ...,eg} of
Rq.
If X is a left invariant vector field on
R9
X M satisfying
X(e) = afj, then
(1.5.1) X(x,z)= Y, #*H-fa.
1J9
l
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