20 GEORGIOS K. ALEXOPOULOS
1.14.9. For all Y £ g, there is a constant c 0 such that
(1.14.9) \Ypt(x,y)-Yp?(x,y)- £ (Y^(x)) XjP»(x,y)\ ct^D+2)^
for all x,y e G and t 1.
Combining (1.8.1), (1.13.2) and (1.14.9) we have the following:
1.14.10. For allY e g and all e £ (0,1) there is a constant c 0
(1.14.10) \Ypt(x,y)-Yp»(x,y)- £ (Yp(x))XjP?(x,y)\
l j m
for all x,y £ G and t 1.
The above results show that the role of the correctors is indeed essential. They
also indicate how we can handle questions about the large time behavior of the
spacial derivatives of the heat kernel.
1.15. Riesz transforms. Let L = —
+ ... -f
+ E$ be a left invariant
sub-Laplacian on a connected Lie group G of polynomial volume growth and let
_ ± _ /
n £ K
Let ei be the linear suspace of g spanned by the vector fields Ei,...,Ep and by
e2 the linear sub-space of g generated by the Lie brackets [Ei, Ej], 1 i, j p.
1.15.1. Let us assume that L is centered and that E0 e t\ + t2-
Then, the Riesz transform operators EiL~1/2 and L~x/2Ei, 1 i p are bounded
on U, for 1 r oo and from
The Riesz transforms are singular integral operators and so, to prove the above
results, we apply the Calderon-Zygmund theory (cf. [SI, S3, CW]). Their kernel
is, in general, singular on the diagonal, as well as at infinity. The singularity near the
diagonal can be treated by using results from the local theory of L. The geometry
and the behavior of G and L at infinity is only used for the singularity at infinity.
In order to isolate these two singularities and treat them separately we consider
- n / 2 , 0
t(n/2)-le-tLdt ^ L-n/2,oo
_ l f°° ^ 2 ) - ! ^ ^
W 2 ) J0 T(n/2) J,
1. Local Riesz transforms. We have the following local result:
1.15.2. Let us assume that E0 e t\-\-t2- Then for allYi,...,Yn £ t\,
the Riesz transform operators
Rn,o = Y1...YnL-n^° and R^0 = L^^Y^Yn
are bounded on LP, for 1 p oo and from L1 to weak-L1.
2. Riesz transforms at infinity. We have the following result: