SUB-LAPLACIANS WIT H DRIF T
19
Combining (1.14.1) and (1.14.2) we have the following:
COROLLARY
1.14.2. There are constants c,
CL
0 such that
(1.14.3) \Pt(e, e) -
CLrD'2\ ct^D+^2,
t 1.
Also by interpolating (1.8.1) and (1.14.1) we have the following:
COROLLARY
1.14.3. For all e G (0,1) there is a constant c 0 such that
(1.14.4) \pt(x,e)-p°t(x,e)\
ct^D+e^2exp
(-^\ ,
x
G N, t 1.
2. Lie groups of polynomial volume growth. Let L be a centered left invariant
sub-Laplacian on a connected Lie group of polynomial volume growth G and let
LH be the associated homogenised operator.
Let also pt(x,y) and p^(x,y) be the heat kernels of L and LH respectively.
Using the notation of section 1.4.7, we extend pf(x,y) to S\ X M\ = G/C
by setting pf((x,z), (y,w)) pj*(x,y), (x,z), (y,w) G S\ X Mi and then to G by
setting p?{g,h) = p f ((#), ?r(/i)), g,heG.
If X, y, ...,Z are left invariant vector fields on (Si)AT, then we also extend, in
the same way, the kernel Xy...Zpf (z,2/) to G.
We have the following Berry-Esseen type of estimate (cf. [Fe, Pe, JKO, Kol,
Ko2, Z]):
THEOREM
1.14.5. Tftere are constants c 0 srzcft £fta£
(1.14.5) |p*(x,2/) - p f (x,y)| cr
D+1
/
2
, x , i / 6 G , t l .
Combining the above result with (1.14.3) we have the following:
COROLLARY
1.14.6. There are constants c,
CL
0 such that
(1.14.6) |pt(e, e) - C
L
t~
D
/
2
| c£-^
+ 1
)/
2
, t 1.
Also by interpolating (1.14.5) and (1.8.1) we have the following:
COROLLARY
1.14.7. For all e G (0,1) there is a constant c 0 such that
(1.14.7) \pt(x,y)-p?(x,y)\ ci-( D + £ )/ 2 exp ( - ^ ^ ) , x,y€G,tl.
Concerning the time derivative of pt(x,y) we have the following result:
THEOREM
1.14.8. There is a constant c 0 such that
(1.14.8) \§iPt(x,y) - ^P"(^V)\
ct~{D+3)/\
x,y€G,tl.
3. A Berry-Esseen estimate for the spacial derivatives. For the space deriva-
tives of pt(x,y), the situation is quite different. More precisely, if X\,...,Xq is a
convenient basis of (S\)N and if
i[)1,...,
^
n i
are the associated first order correctors,
then we have the following result:
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