Large time estimates for non-symmetric heat kernel on the afﬁne group

We consider the heat kernel ~t associated to the left invariant Laplacian with a drift term, on the affine group of the line. We obtain a large time upper estimate for ~t.


Introduction
Let G be the two dimensional Lie group of affine transformations on IR, T : ~ + x (ç E l~) with 0 y = et E I~+ and t, x e R.This group is the only non abelian two dimensional Lie group, and it can be seen as the semidirect product ?X I~+ since .03C3103C32 : 03B6 ~ y1y203BE + x2y1 + x1, where the action of R*+ on R is X ~ xy.
The Lie algebra g of G is spanned by the left-invariant vector fields X = y~ ~x and Y = y~ ~y.
A left invariant distance d on G, called the control distance, is associated to these vector fields (cf.[9]).We denote by |g| = d(e, g), where e = (o,1) is the identity element of G.
The group G is not unimodular, indeed drg = y-1dxdy is the right Haar measure whereas dlg = y-2dxdy is the left Haar measure.The modular func- tion m(g) =  is thus m(g) = y, g = (x, y).
It is a solvable Lie group, and the lack of unimodularity implies that G is of exponential growth.where and v are real numbers.
The operator L generates a diffusion semigroup e-tL (cf.[2], [7]).We denote by 03C6t the kernel of e-tL with respect to the left Haar measure on G, i.e.
Many authors have studied the heat kernel associated to a driftless Laplacian on various Lie groups and Riemannian manifolds (cf.for instance [6], and references there, for an interesting survey).In our setting, the affine group, an explicit formula of the (driftless) kernel is known (see e.g.[4] and (2.4) below).Note that G ~ H2 (the hyperbolic space) as a manifold.Further the control distance coincides with the Riemannian distance and the left invariant mea- sure is the Riemannian measure.Since the action of G on itself is isometric the vectors fields X, Y form on the tangent space a basis at every point of G. Finally ~~ f ~2 = (X f )2 + (Y coincides with the square of the norm of the Riemannian gradient. The first order terms of (1.1) are not invariant under the isometry group of H2 but they are invariant under the action of G.This makes it natural to study L as an invariant operator on G.
The small time behavior of the kernel associated to Laplacians with drift has been studied by Varopoulos in [11] and [12].In our setting we have the upper estimate (cf.[15], [14], [11] and [12]) exp(-~~t ), 0 t 1, g = (x, y) E G. (1.2)   In the setting of Lie groups of polynomial growth, a large time upper estimate for the kernel has been obtained by Alexopoulos (cf.[1]).
In this paper we study the large time behavior of 03C6t on the affine group.Note that the non unimodularity of the affine group and the consequent exponential volume growth lead to additional difficulties.The techniques used in this paper are therefore completely different from those used in ~1~.
Throughout this paper positive constants are denoted either by the letters c or C.These may differ from one line to another.

Preliminaries
The left invariance of the distance d forces us to work with the left Haar measure.The driftless Laplacian is formally self-adjoint with respect to dr g.It follows that the conjugated A with the modular function m gives rise to an operator which is formally self-adjoint with respect to dig.
Let us remark that if we conjugate the vector field X with m we have rrzl~2X(m-1~2~) = X, , and it follows that the left-invariant field X is anti-symmetric with respect to dl 9 too.

Let Lo
the conjugated Laplacian with the modular function m.
It is easy to see that also Lo is a left-invariant operator on G.
Let Tt = be the diffusion semigroup generated by Lo and let be its kernel with respect to dig.By (2.4) we have Z't = and therefore t(g) = m1/2(g)03C6t(g). (2.5) Because of the anti-symmetry of the field X, the adjoint operator Lo* of (2.4) on is given by Lo We denote by Tt = the semigroup generated by Lo* and by $[ its kernel with respect to dlg.We have that ~t~9) _ ~t~9 1).
In the following of this paper we shall prove the upper Gaussian estimate for the kernel ~t~t( 9) Then (1.3) follows from (2.5).
We prove the following lemma, whose proof is inspired by the well-known Nash-Varopoulos method.Lemma 2.1: We have that ~t~~ ~ Ct-3/2, t > O.
Applying the same argument to the adjoint operator Tt we obtain the esti- In this section we shall prove the upper Gaussian estimate for the kernel t ~t ~ > 1, g E G.
The method we shall use to deduce the above Gaussian estimate directly from lemma 2.1 appeared first in the work of Ushakov [10].
By the change of variables ~-1g The left invariance of the control distance implies that 2(~~-1I2-t-Ig~I2).
The idea of the proof (cf.[5]) is to show that if the constant C in the integrals (3.12) and (3.13) is large enough we have ~ %(~) t > 1 (3.14) and ~~)exp(~)~C~, Note that the operator Lo and its adjoint Lo are operators of the same kind, so it suffices to prove (3.14).
We begin by showing that (3.14) is a consequence of the following esti- mate : there exists Co > 0 such that for every R > 0 and t > 1 On one hand, we have that and on the other hand, applying (3.15) Choosing R~ = Ct we deduce (3.14).
We claim that F(t) is a decreasing function.Indeed ~~ t) 2 ~c ~t + ~c ~t e~ at Using the anti-symmetry of the fields X, Y with respect to dr g and (3.16) we estimate F'(t) = -2 G ( X ( 0 3 C 6 t e ) X 0 3 C 6 t and since F is decreasing for every 0 T t, and 0 R' R, we have and therefore H(R, t) ~ H ( R ' , ) + e x p ( ( R ' -R ) 2 4 ( 1 + 2 ( -s ) ) ~~~, 0 T « s, R' R.
4 Final remark In section 3 we adapted to a non-symmetric case the method decribed in ~5~.
We managed to overcome the difficulties imposed by the presence of the drift term X, thanks to its special nature.First, the anti-symmetry of X with respect to the left Haar measure too allowed us to evaluate the two integrals (3.14) in the same way.Second, since X = (X, Y] we are able to prove that F(t) is a deceasing function.
Assuming that an upper estimate of the form ~~ Ct-03B4 holds for some 6 > 0 and every t > 1, this method could be generalized in an natural way to the semi-direct product G = N X R*+, where N is a stratified group and ~8+ acts on N by dilations.Let us remark that the proof of Lemma 2.1 works provided the local dimension of the group is less or equal to 3.
As far as we can see this method does not work in more general contexts.
One can consider on G the Laplacian with drift term L = -( X 2 + Y 2 ) + X + 0 3 B D Y ( 1