\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 145, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/145\hfil Estimates for the mixed derivatives]
{Estimates for the mixed derivatives of the Green functions on
homogeneous manifolds of negative curvature}

\author[R. Urban\hfil EJDE-2004/145\hfilneg]
{Roman Urban}

\address{Roman Urban \hfill\break
Institute of Mathematics,
University of Wroclaw, Pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland}
\email{urban@math.uni.wroc.pl}

\date{}
\thanks{Submitted October 17, 2003.  Published December 7, 2004.}
\thanks{Partly supported by grant 1PO3A01826 from KBN, and by contract
HPRN-CT-2001-\hfill\break\indent
00273-HARP from RTN Harmonic Analysis and Related Problems}
\subjclass[2000]{22E25, 43A85, 53C30}
\keywords{Green function; second-order differential operators;
$NA$ groups; \hfill\break\indent
Bessel process; evolutions on nilpotent Lie groups}

\begin{abstract}
  We consider the Green functions for second-order left-invariant
  differential operators on homogeneous manifolds of negative
  curvature, being a semi-direct product of a nilpotent Lie group
  $N$ and $A=\mathbb{R}^+$. We obtain estimates for mixed derivatives of the
  Green functions both in the coercive and non-coercive case.
  The current paper completes the previous results obtained by the
  author in a series of papers \cite{amuc,pota,ejde,CM}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}

\section{Introduction}

Let $M$ be a connected and simply connected homogeneous manifold
of negative curvature. Such a manifold is a solvable Lie group
$S=NA$, a semi-direct product of a nilpotent Lie group $N$ and an
Abelian group $A=\mathbb{R}^+$. Moreover, for an $H$ belonging to
the Lie algebra $\mathfrak{a}$ of $A$, the real parts of the
eigenvalues of $\mathop{\rm Ad}_{\exp H}|_{\mathfrak{n}}$, where
$\mathfrak{n}$ is the Lie algebra of $N$, are all greater than 0.
Conversely, every such a group equipped with a suitable
left-invariant metric becomes a homogeneous Riemannian manifold
with negative curvature (see \cite{H}).

On $S$ we consider a second-order left-invariant operator
\begin{equation*}
\mathcal{L} =\sum_{j=0}^mY_j^2+Y.
\end{equation*}
We assume that $Y_0,Y_1,\dots,Y_m$ generate the Lie algebra
$\mathfrak{s}$ of $S$ and $Y\in\mathfrak{s}$. We can always make
$Y_0,\dots, Y_m$ linearly independent and moreover, we can choose
$Y_0,Y_1,\dots,Y_m$ so that $Y_1(e),\dots, Y_m(e)$ belong to
$\mathfrak{n}$ (write $\mathcal{L}$ as $\sum_{i,j=0}^{\mathop{\rm
dim}\mathfrak{n}}\alpha_{i,j}E_iE_j +\sum_{j=0}^{\mathop{\rm
dim}\mathfrak{n}}\beta_jE_j$, $E_0\in\mathfrak{a}$, $\{E_j\}$ is a
basis of $\mathfrak{n}$, $\alpha_{i,j},\beta_j\in\mathbb{R}$ and
then rewrite $\mathcal{L}$ as a sum of squares). Let $\pi:S\to
A=S\slash N$ be the canonical homomorphism. Then the image of
$\mathcal{L}$ under $\pi$ is a second-order left-invariant
operator on $\mathbb{R}^+$,
\begin{equation*}
\pi(\mathcal{L})=(a\partial_a)^2-\gamma a\partial_a,
\end{equation*}
where $\gamma=\gamma_\mathcal{L}\in\mathbb{R}$. We say that a
second order differential operator $\mathcal{L}$ on a Riemannian
manifold is \textit{non-coercive} (\textit{coercive} resp.) if
there is no $\varepsilon>0$ such that
$\mathcal{L}+\varepsilon\text{Id}$ admits the Green function (if
such an $\varepsilon$ exists resp.). It is worth noting that our
definition of coercivity is slightly different than that used e.g.
in \cite{A}. Namely, for us, $\mathcal{L}$ is coercive if it is
weakly coercive in Ancona's terminology. There is a relation
between the notion of coercivity property in the sense used in the
theory of partial differential equations (i.e., that an
appropriate bilinear form is coercive, \cite{Stamp}) and weak
coercivity. For this  the reader is referred to \cite{A}.

In this paper we shall study both coercive and non-coercive
operators. In this case $\mathcal{L}$ can be written as
\begin{equation}\label{operator}
\mathcal{L}=\mathcal{L}_\gamma=\sum_j\Phi_a(X_j)^2+\Phi_a(X)+a^2\partial_a^2
+(1-\gamma)\partial_a,
\end{equation}
where $\gamma=\gamma_{\mathcal{L}}\in\mathbb{R}$, $X,X_1,\dots,X_m$ are
left-invariant vector fields on $N$, moreover, $X_1,\dots,X_m$
are linearly independent and generate $\mathfrak{n}$,
$$\Phi_a={\mathop{\rm Ad}}_{\exp(\log a)Y_0}=e^{(\log a) \mathop{\rm ad}_{Y_0}}=e^{(\log a)
D},$$ where $D=\mathop{\rm ad}_{Y_0}$ is a derivation of the Lie algebra
$\mathfrak{n}$ of the Lie group $N$ such that the real parts $d_j$
of the eigenvalues $\lambda_j$ of $D$ are positive. By multiplying
$\mathcal{L}_\gamma$ by a constant, i.e., changing $Y_0$, we can
make $d_j$ arbitrarily large (see \cite{DHU}).

Let $\mathcal{G}_\gamma(xa,yb)$ be the {\it Green function} for
$\mathcal{L}_\gamma$. $\mathcal{G}_\gamma$ is (uniquely) defined
by two conditions:
\begin{itemize}
\item [(i)] $\mathcal{L}_\gamma\mathcal{G}_\gamma(\cdot,yb)=-\delta_{yb}$ as distributions
(functions are identified with distributions via the right Haar
measure),
\item [(ii)] for every $yb\in S$, $\mathcal{G}_\gamma(\cdot,yb)$ is a potential for $\mathcal{L}_\gamma$,
i.e, is a positive superharmonic function such that its largest
harmonic minorant is equal to zero (cf. \cite{CC}).
\end{itemize}
Let
\begin{equation}\label{gf}
\mathcal G_\gamma(x,a):=\mathcal G_\gamma(e,xa),
\end{equation}
where $e$ is the identity element of the group $S$. Since
$\mathcal{L}_\gamma$ is left-invariant it is easily seen
that
\begin{equation*}
\mathcal G_\gamma(xa,yb)=\mathcal G_\gamma(e,yb(xa)^{-1})=\mathcal
G_\gamma(yb(xa)^{-1}).
\end{equation*}
In this article we call $\mathcal G_\gamma(x,a)$ defined in
\eqref{gf} the Green function for $\mathcal L_\gamma$.

The main goal of this paper is to give estimates for derivatives
of the Green function \eqref{gf} for $\mathcal{L}_\gamma$.

To illustrate the general set up, before we proceed further, we
would like to give a simple and explicit example of the operator
$\mathcal L$ in coordinates. Consider $S=\mathbb{R}^n\times\mathbb{R}^+$. Let
$d_1,\dots,d_n$ be positive constants. For every $a>0$, define
$\Phi_a(\partial_{x_j})=a^{d_j}\partial_{x_j}$. Then $\Phi_a$ on
$\mathbb{R}^n$ becomes
$\Phi_a(x)=\Phi_a(x_1,\dots,x_n)=(a^{d_1}x_1,\dots,a^{d_n}x_n)$
and we get on $\mathbb{R}^n$ a structure of the homogeneous group with the
homogeneous dimension $Q=\sum d_j$ (see \cite{FS}). The
multiplication law in $S$ is given by the formula
$(x,a)\cdot(y,b)=(x+\Phi_a(y),ab)$. In this example the operator
\eqref{operator} is $\mathcal{L}=\sum_j
a^{2d_j}\partial_{x_j}^2+a^2\partial_a^2+(1-\gamma)a\partial_a$.
The Green function for $\mathcal{L}$ is
$\mathcal{G}((x,a),(y,b))=\int_0^\infty p_t(x,a;y,b)dt$, where
$p_t(x,a;y,b)$ is the heat diffusion kernel on $S$, such that
$u(t,y,b):=p_t(x,a;y,b)$ is the minimal solution of
$\mathcal{L}u=\partial_t u$, $u(0,y,b)=\delta_{(x,a)}(y,b)$ and
$\delta_{(x,a)}(\cdot)$ stands for Dirac's delta.

Let us go back to the general setting. We are going to prove (or
at least to sketch the proof of) the following estimates. Let
$\gamma\geq 0$. For every neighborhood $\mathcal U$ of the
identity $e$ of $NA$ there is a constant $C=C(\gamma)$ such that
we have
\begin{equation}\label{ge}
|\partial_a^k\mathcal{X}^I\mathcal{G}_{-\gamma}(x,a)|\leq
\begin{cases}
C(|x|+a)^{-\|I\|-Q-\gamma}a^{-k}&\\
\times (1+|\log(|x|+a)^{-1}|)^{\|I\|_0}
&\text{ for $(x,a)\in(\mathcal Q\cup\mathcal U)^c$,}\\[3pt]
Ca^{-k}&\text{ for $(x,a)\in \mathcal Q\setminus\mathcal U$}
\end{cases}
\end{equation}
and
\begin{equation}\label{ge1}
|\partial_a^k\mathcal{X}^I\mathcal{G}_{\gamma}(x,a)|\leq
\begin{cases}
C(|x|+a)^{-\|I\|-Q-\gamma}a^{\gamma-k}&\\
\times (1+|\log(|x|+a)^{-1}|)^{\|I\|_0}
&\text{ for $(x,a)\in(\mathcal Q\cup \mathcal U)^c$,}\\[3pt]
Ca^{\gamma-k}&\text{ for $(x,a)\in \mathcal Q\setminus \mathcal
U,$}
\end{cases}
\end{equation}
where $|\cdot|$ stands for a ``homogeneous norm" on $N$, $\mathcal
Q=\{|x|\leq 1, a\leq 1\}$, $\|I\|$ is a suitably defined length of
the multi-index $I$ and $\|I\|_0$ is a certain number depending on
$I$ and the nilpotent part of the derivation $D$. In particular,
$\|I\|_0$ is equal to 0 if the action of $A=\mathbb{R}^+$ on $N$, given by
$\Phi_a$, is diagonal or, if $I=0$.
$\mathcal{X}_1,\dots,\mathcal{X}_n$ is an appropriately chosen
basis of $\mathfrak{n}$. For the precise definitions of all the
notions that have appeared here see Sect. \ref{preliminaries}.


It should be said that the estimate for the Green function itself
(i.e., $I$=0) with $\gamma>0$, also from below, was proved by E.
Damek in \cite{D} and then by the author for $\gamma=0$ in
\cite{CM} but at that time it was impossible to prove analogous
estimate for derivatives. The reason was that we did not have
sufficient estimates for the derivatives of the transition
probabilities of the evolution on $N$ generated by an appropriate
operator which appears as the ``horizontal" component of the
diffusion on $N\times\mathbb{R}^+$ generated by $a^{-2}\mathcal
L_{-\gamma}$ (cf. \cite{DHU}). These estimates have been obtained
by the author in \cite{MathZ} and eventually led up to the
estimates for derivatives of the Green functions in the
non-coercive case, i.e., $\gamma=\gamma_{\mathcal{L}}=0$ (see
\cite{pota} for derivatives with respect to $N$ and $A$-variables
separately and \cite {ejde} for the mixed derivatives which
required a slightly different approach). Next, in \cite{amuc} the
results from \cite{pota} have been used to get estimates for
derivatives in the coercive case. This note completes the previous
works of the author in that we provide a proof of the estimates
which is valid for both the coercive and non-coercive cases.

The proofs of \eqref{ge} and \eqref{ge1} require both analytic and
probabilistic techniques. Some of them have been introduced in
\cite{DHZ,DHU} and \cite{CM}.

The structure of the paper is as follows. In Sect.
\ref{preliminaries} we set the notation and give all necessary
definitions. In particular, we recall a definition of the Bessel
process which appears as the``vertical" component of the diffusion
generated by $a^{-2}\mathcal{L}_{-\gamma}$ on
$N\times\mathbb{R}^+$  as well as the notion of the evolution on
$N$ generated by an appropriate operator which appears as
the``horizontal" component of the diffusion on
$N\times\mathbb{R}^+$ mentioned in the above (cf. \cite{DHU,
pota}). Moreover, we state Theorem \ref{r} which is the main tool
in the proof of Theorem \ref{main}.

Finally, in Sect. \ref{proof of theorem} we state the
estimates \eqref{ge}, \eqref{ge1} precisely (see Theorem
\ref{main}) and we give a sketch of its proofs.

\section{Preliminaries.}\label{preliminaries}
\subsection*{$NA$ groups} Good reference for this topic are
\cite{DHZ, DHU} and \cite{FS}. Let $N$ be a connected and simply
connected nilpotent Lie group. Let $D$ be a derivation of the Lie
algebra $\mathfrak{n}$ of $N$. For every $a\in\mathbb{R}^+$ we define an
automorphism $\Phi_a$ of $\mathfrak{n}$ by the formula
\begin{equation*}
\Phi_a=e^{(\log a)D}.
\end{equation*}
Writing $x=\exp X$ we put
\begin{equation*}
\Phi_a(x):=\exp\Phi_a(X).
\end{equation*}
Let $\mathfrak{n}^\mathbb{C}$ be the complexification of $\mathfrak{n}$.
Define
\begin{equation*}
\mathfrak{n}_\lambda^\mathbb{C}=\{X\in\mathfrak{n}^\mathbb{C}:\exists k>0\text{
such that }(D-\lambda I)^k=0\}.
\end{equation*}
Then
\begin{equation}\label{gradation}
\mathfrak{n}=\bigoplus_{\text{Im}\lambda\geq 0}V_\lambda,
\end{equation}
where
\begin{equation*}
V_\lambda=
\begin{cases}
V_{\bar\lambda}=(\mathfrak{n}_\lambda^\mathbb{C}\oplus\mathfrak{n}_{\bar\lambda}^\mathbb{C})\cap\mathfrak{n}&\text{if Im$\lambda\not=0$,}\\
\mathfrak{n}_\lambda^\mathbb{C}\cap\mathfrak{n}&\text{if Im$\lambda=0.$}
\end{cases}
\end{equation*}
We assume that the real parts $d_j$ of the eigenvalues $\lambda_j$
of the matrix $D$ are strictly greater than 0. We define the
number
\begin{equation}\label{qdef}
Q=\sum_j\text{Re }\lambda_j=\sum_j d_j
\end{equation}
and we refer to this as a \textit{``homogeneous dimension"} of
$N$. In this paper $D=\mathop{\rm ad}_{Y_0}$ (see Introduction). Under the
assumption on positivity of $d_j$, \eqref{gradation} is a
gradation of $\mathfrak{n}$.

We consider a group $S$ which is a \textit{semi-direct} product of
$N$ and the multiplicative group $A=\mathbb{R}^+= \{\exp
tY_0:t\in\mathbb{R}\}:$
\begin{equation*}
S=NA=\{xa:x\in N,a\in A\}
\end{equation*}
with multiplication given by the formula
\begin{equation*}
(xa)(yb)=(x\Phi_a(y)\ ab).
\end{equation*}
On $N$ we define a \textit{``homogeneous norm"}, $|\cdot|$ (cf.
\cite{DHZ,DHU}) as follows. Let $(\cdot,\cdot)$ be a fixed inner
product in $\mathfrak{n}$. We define a new inner product
\begin{equation}\label{product}
\langle X,Y\rangle =\int_0^1\Big(\Phi_a(X),\Phi_a(Y)\Big)\frac{da}{a}
\end{equation}
and the corresponding norm
\begin{equation*}
\|X\|=\langle X,X\rangle^{1\slash 2}.
\end{equation*}
We put
\begin{equation*}
|X|=\left(\inf\{a>0:\|\Phi_a(X)\|\geq 1\}\right)^{-1}.
\end{equation*}
One can easily show that for every $Y\ne 0$ there exists precisely
one $a>0$ such that $Y=\Phi_a(X)$ with $|X|=1$. Then we have
$|Y|=a$.

Finally, we define the homogeneous norm on $N$. For $x=\exp X$ we
put
\begin{equation*}
|x|:=|X|.
\end{equation*}
Notice that if the action of $A=\mathbb{R}^+$  on $N$ (given by
$\Phi_a$) is diagonal the norm we have just defined is the usual
homogeneous norm on $N$ and the number $Q$ in \eqref{qdef} is
simply the homogeneous dimension of $N$ (see \cite{FS}).

Having all that in mind we define appropriate derivatives (see
also \cite{DHZ}). We fix an inner product \eqref{product} in
$\mathfrak{n}$ so that $V_{\lambda_j}$, $j=1,\dots,k$ are
mutually orthogonal and an orthonormal basis
$\mathcal{X}_1,\dots,\mathcal{X}_n$ of $\mathfrak{n}$. The
enveloping algebra $\mathfrak{U}(\mathfrak{n})$ of $\mathfrak{n}$
is identified with the polynomials in
$\mathcal{X}_1,\dots,\mathcal{X}_n$. In
$\mathfrak{U}(\mathfrak{n})$ we define $\langle
\mathcal{X}_1\otimes\dots\otimes\mathcal{X}_r,\mathcal{Y}_1\otimes\dots\otimes\mathcal{Y}_r\rangle
= \prod_{j=1}^r\langle \mathcal{X}_j,\mathcal{Y}_j\rangle$. Let
$V_j^r$ be the symmetric tensor product of $r$ copies of
$V_{\lambda_j}$. For $I=(i_1,\dots,i_k)\in(\mathbb{N}\cup\{0\})^k$ let
\begin{equation*}
\mathcal{X}^I=\mathcal{X}_1^{(i_1)}\dots\mathcal{X}_k^{(i_k)},\text{
where }\mathcal{X}_j^{(i_j)}\in V_j^{i_j}.
\end{equation*}
Then for $\mathcal{X}\in V_{\lambda_j},$
\begin{equation*}
\|\Phi_a(\mathcal{X})\|\leq c\exp(d_j\log a+D_j\log(1+|\log a|)),
\end{equation*}
where $d_j=\text{Re}\lambda_j$ and $D_j=\dim V_{\lambda_j}-1$, and
so
\begin{equation}\label{den}
\|\Phi_a(\mathcal{X}^I)\|\leq\exp\Big(\sum_{j=1}^ki_j(d_j\log
a+D_j\log(1+|\log
a|))\Big)\prod_{j=1}^k\|\mathcal{X}_j^{(i_j)}\|
\end{equation}

\subsection*{Bessel process}
Let $\sigma(t)$ denote the Bessel process with a parameter
$\alpha\geq 0$ (cf. \cite{RY}), i.e., a continuous Markov process
with the state space $[0,+\infty)$ generated by
$\partial_a^2+\frac{2\alpha+1}{a}\partial_a$. The transition
function with respect to the measure $y^{2\alpha+1}dy$ is given
(cf. \cite{Borodin, RY}) by:
\begin{equation*}
p_t(x,y)=\begin{cases}
\frac{1}{2t}\exp\big(\frac{-x^2-y^2}{4t}\big) I_\alpha\left(
\frac{xy}{2t}\right) \frac{1}{(xy)^\alpha}
&\text{for $x,y>0,$}\\
\frac{1}{2^\alpha (2t)^{\alpha+1}\Gamma(\alpha+1)}
\exp\big(\frac{-y^2}{4t}\big) &\text{for $x=0$, $y>0,$}
\end{cases}
\end{equation*}
where
\begin{equation*}
I_\alpha(x)= \sum_{k=0}^\infty\frac{(x\slash
2)^{2k+\alpha}}{k!\Gamma(k+\alpha+1)}
\end{equation*}
is \textit{the Bessel function} (see \cite{L}). Therefore for
$x\geq 0$ and a measurable set $B\subset(0,\infty)$:
\begin{equation*}
\mathbf{P}_x(\sigma(t)\in B)=\int_Bp_t(x,y)y^{2\alpha+1}dy.
\end{equation*}
If $\sigma(t)$ is the Bessel process with a parameter $\alpha$
starting from $x$, i.e. $\sigma(0)=x$, then we will write that
$\sigma(t)\in\mathop{\rm BESS}_x(\alpha)$ or simply
$\sigma(t)\in\mathop{\rm BESS}(\alpha)$ if the starting point is not important
or is clear from the context.

Properties of the Bessel process are very well known and their
proofs are rather standard. They can be found e.g. in \cite{RY,
DHU, pams, Phd}. However, in our paper we will not explicitly make
use of any particular property of the Bessel process.

\subsection*{Evolutions}
Let $X,X_1,\dots, X_m$ be as in \eqref{operator}. Let $\sigma:
[0,\infty)\longrightarrow [0,\infty)$ be a continuous function
such that $\sigma(t)>0$ for every $t>0$. We consider the family of
evolutions operators $L_{\sigma(t)}-\partial_t$, where
\begin{equation}\label{operatorsigma}
L_{\sigma(t)}=\sigma(t)^{-2}\Big(\sum_j\Phi_{\sigma(t)}(X_j)^2+
\Phi_{\sigma(t)}(X)\Big).
\end{equation}
For the main result of the paper we are mainly interested in the
operator \eqref{operatorsigma} with $\sigma(t)$ being a trajectory
of an appropriate Bessel process.

Since we assume $X_1,\dots,X_m$ being linearly independent,
we select $X_{m+1},\dots, X_n$ so that $X_1,\dots ,X_n$ form a
basis of $\mathfrak{n}$. For a multi-index $I=(i_1,\dots,i_n)$,
$i_j\in\mathbb{Z}^+$ and the basis $X_1,\dots,X_n$ of the Lie algebra
$\mathfrak{n}$ of $N$ we write: $X^I=X_1^{i_1}\dots X_n^{i_n}$
and $|I|=i_1+\dots+i_n$. For $k=0,1,\dots,\infty$ we define:
\begin{equation*}
C^k=\{f:X^If\in C(N),\text{ for }|I|<k+1\}
\end{equation*}
and
\begin{equation*}
C_\infty^k=\{f\in C^k:\lim_{x\to\infty}X^If(x)\text{ exists for
}|I|<k+1\}.
\end{equation*}
For $k<\infty$ the space $C^k_\infty$ is a Banach space with the
norm
\begin{equation*}
\|f\|_{C^k_\infty}=\sum_{|I|\leq k}\|X^If\|_{C(N)}.
\end{equation*}

Let $\{U^\sigma(s,t):0\leq s\leq t\}$ be the unique family of
bounded operators on $C_\infty=C_\infty^0$ which satisfy
\begin{itemize}
\item [(i)] $U^\sigma(s,s)=I$,
\item [(ii)] $U^\sigma(s,r)U^\sigma(r,t)=U^\sigma(s,t)$, $s<r<t$,
\item [(iii)] $\partial_sU^\sigma(s,t)f=-L_{\sigma(s)}U^\sigma(s,t)f$
for every $f\in C_\infty$,
\item [(iv)]
$\partial_tU^\sigma(s,t)f=U^\sigma(s,t)L_{\sigma(t)}f$ for every
$f\in C_\infty$,
\item [(v)] $U^\sigma(s,t):C^2_\infty\longrightarrow C^2_\infty$.
\end{itemize}
Note that $U^\sigma(s,t)$ is a convolution operator. Namely,
$U^\sigma(s,t)f=f*p^\sigma(t,s)$, where $p^\sigma(t,s)$ is a
smooth density of a probability measure. By ii) we have
$p^\sigma(t,r)*p^\sigma(r,s)=p^\sigma(t,s)$ for $t>r>s$. The
existence of the family $U^\sigma(s,t)$ follows from \cite{T}.

For $\alpha\geq 0$, on a direct product $G=N\times\mathbb{R}^+$ we
consider the following operator
$$
\mathbf{L}_\alpha=a^{-2}\sum_j
\Phi_a(X_j)^2+a^{-2}\Phi_a(X)+\partial_a^2+\frac{2\alpha+1}{a}\partial_a.$$
For $f\in C^\infty_c(G)$, we define on $G$ the following function
\begin{equation}\label{rozw}
u(t,x,a):=\mathbf{E}_a U^\sigma (0,t)f(x,\sigma(t)),
\end{equation}
where the expectation is taken with respect to the distribution of
the Bessel process $\sigma(t)$ starting from $a$. The following
theorem, which gives the formula for the solution of the heat
equation for $L_\alpha$ in terms of the evolution on $N$ driven by
the Bessel process, is one of the main tool in the proof of our
results.

\begin{theorem}\label{r} Let $u=u(t,x,a)$ be a function on $G$ defined by \eqref{rozw}. Then
\begin{equation}\label{3.2}
\mathbf{L}_\alpha u=\partial_tu,\ \ u(0,x,a)=f(x,a).
\end{equation} Moreover, there
exists the only one bounded from below solution $u$ of
\eqref{3.2}.
\end{theorem}
\begin{proof} For the proof of the first part of Theorem
\ref{r} see \cite{DHU}. The uniqueness of the bounded from below
solution $u$ follows by some kind of the maximum principle which
is a modification of Theorem. 3.1.1 in \cite{SV}. The proof for a
diagonal action given in \cite{Phd} can be easily generalized.
\end{proof}

\section{The main result and its proof.}\label{proof of theorem}
In this section we obtain pointwise estimates for derivatives of
the Green function \eqref{gf}.

For a positive $\delta<1\slash 2$ define
\begin{align*}
T_\delta=&\{(x,a)\in N\times\mathbb{R}^+:1-\delta<a<1+\delta,|x|<\delta\},\\
\mathcal Q=&\{(x,a)\in N\times\mathbb{R}^+:|x|\leq 1, a\leq 1\}.
\end{align*}
\begin{theorem}\label{main}
For a multi-index $I=(i_1,\dots,i_k)$, $\gamma\geq 0$, $k\in\mathbb{Z}^+$
and all operators $\mathcal{X}^I=\mathcal{X}_1^{(i_1)}\dots
\mathcal{X}_k^{(i_k)}$, where $\mathcal{X}_j^{(i_j)}\in
V_j^{i_j}$, with $\|\mathcal{X}^I\|\leq 1$, there are constants
$C$ such that
\begin{equation*}
|\partial_a^k\mathcal{X}^I\mathcal{G}_{-\gamma}(x,a)|\leq
\begin{cases}
C(|x|+a)^{-\|I\|-Q-\gamma}a^{-k}&\\
\times (1+|\log(|x|+a)^{-1}|)^{\|I\|_0}&\text{ for $(x,a)\in(\mathcal Q\cup T_\delta)^c$,}\\
Ca^{-k}&\text{ for $(x,a)\in \mathcal Q\setminus T_\delta$}
\end{cases}
\end{equation*}
and
\begin{equation*}
|\partial_a^k\mathcal{X}^I\mathcal{G}_{\gamma}(x,a)|\leq
\begin{cases}
C(|x|+a)^{-\|I\|-Q-\gamma}a^{\gamma-k}&\\
\times (1+|\log(|x|+a)^{-1}|)^{\|I\|_0}&\text{ for $(x,a)\in(\mathcal Q\cup T_\delta)^c$,}\\
Ca^{\gamma-k}&\text{ for $(x,a)\in \mathcal Q\setminus T_\delta$}
\end{cases}
\end{equation*}
where $\|I\|=\sum_{j=1}^ki_jd_j$, $d_j=\text{Re}\lambda_j$, and
$\|I\|_0=\sum_{j=1}^ki_jD_j$, $D_j=\mathop{\rm dim}V_{\lambda_j}-1$.
\end{theorem}

Let $\alpha\geq 0$ and $\gamma\geq 0$. Along with the operator
$\mathcal L_{-\gamma}$ defined in \eqref{operator} we consider the
corresponding operator $\mathbf{L}_\alpha$,
\begin{equation*}
\mathbf{L}_\alpha=a^{-2}\sum_j\Phi_a(X_j)^2+a^{-2}\Phi_a(X)+\partial_a^2+\frac{2\alpha+1}{a}
\partial_a=a^{-2}\mathcal L_{-\gamma},
\end{equation*}
where $\alpha=\gamma\slash 2$. The Green function $G_\alpha$ for
$\mathbf{L}_\alpha$ is given by
\begin{equation*}
G_\alpha(x,a;y,b)=\int_0^\infty p_t(x,a;y,b)d t,
\end{equation*}
where
\begin{equation*}
T_tf(x,a)=\int f(y,b)p_t(x,a;y,b)d y b^{2\alpha+1}d b
\end{equation*} is the heat semigroup on
$L^2(N\times\mathbb{R}^+,dyb^{2\alpha+1}db)$ given by Theorem \ref{r} with
the infinitesimal generator $\mathbf{L}_\alpha$ .

On $N\times\mathbb{R}^+$ we define \textit{dilations}:
\begin{equation*}
D_t(x,a)=(\Phi_t(x),ta),\qquad\ t>0.
\end{equation*}
It is not difficult to check that although the operator
$\mathbf{L}_\alpha$ is not left-invariant it has some homogeneity with
respect to the family of dilations introduced above:
\begin{equation*}
\mathbf{L}_\alpha(f\circ D_t)=t^2\mathbf{L}_\alpha f\circ D_t.
\end{equation*}
This implies that
\begin{equation}\label{homogeneity}
G_\alpha(x,a;y,b)=t^{-Q-2\alpha}G_\alpha(D_{t^{-1}}(x,a);D_{t^{-1}}(y,b)).
\end{equation}
It turns out (see (1.17) in \cite{DHU}) that
\begin{equation}\label{g-g}
\mathcal G_{-\gamma}(x,a)=G_{\gamma\slash
2}(e,1;x,a)=G^*_{\gamma\slash 2}(x,a;e,1),
\end{equation}
where $G^*_{\alpha}$ is the Green function for the operator
\begin{equation*}
\mathbf{L}^*_\alpha=a^{-2}\sum_j\Phi_a(X_j)^2-a^{-2}\Phi_a(X)+\partial_a^2+\frac{2\alpha+1}{a}\partial_a
\end{equation*}
formally conjugate to $\mathbf{L}_\alpha$ with respect to the measure
$a^{2\alpha+1}d ad x$. Moreover,
\begin{equation*}
G^*_\alpha(x,a;e,1)=\lim_{\eta\to
0}\int_0^\infty\mathbf{E}_1p^\sigma(t,0)(x)m_\alpha(I_{a,\eta})^{-1}1_{I_{a,\eta}}(\sigma_t)d
t,
\end{equation*}
where
\begin{equation*}
m_\alpha(I)=\int_Ia^{2\alpha+1}da
\end{equation*}
and the
expectation is taken with respect to the distribution of the
Bessel process with the parameter $\alpha$ starting from 1, i.e.,
$\mathop{\rm BESS}_1(\alpha)$ on the space $C((0,\infty),(0,\infty))$,
$p^\sigma(t,0)$ is the transition function of the evolution
generated by the operator \eqref{operatorsigma} and
$I_{a,\eta}=[a-\eta,a+\eta]$.

Since $\mathcal L_{-\gamma}(\cdot)=a^{-\gamma}\mathcal
L_\gamma(a^\gamma\cdot)$ it follows that
\begin{equation}\label{conj}
\mathcal G_\gamma(xa,yb)=a^{\gamma}\mathcal
G_{-\gamma}(xa,yb)b^{-\gamma}
\end{equation}
and therefore, by \eqref{g-g} and \eqref{conj},
\begin{equation*}
\mathcal G_\gamma(x,a)=G^*_{\gamma\slash 2}(x,a;e,1)a^{\gamma}.
\end{equation*}

Before we go to the proof of our main result we note the following
important proposition which gives estimates on the set $\mathcal
Q\setminus T_\delta$ of some functional of the evolution
$p^\sigma$ which plays the crucial role in the proof of Theorem
\ref{main}.

\begin{proposition}\label{mainproposition}
i) For every $1>\delta>0$ and for every multi-index $I$ such that
$|I|>0$ there exists a constant $C>0$ such that for every
$(x,a)\in(\mathcal{Q}\setminus T_\delta)\cap\{(x,a)\in
N\times\mathbb{R}^+:a\leq 1-\delta\}$ and for every $0\leq l\leq
k-1$,
\begin{equation*}
\sup_{0<\eta<\delta\slash 2}|\int_0^\infty\mathbf{E}_1
X^Ip^\sigma(t,0)(x)\partial_a^lm_\alpha(I_{a,\eta})^{-1}\partial_a^{k-l}1_{I_{a,\eta}}(\sigma_t)dt|\leq
Ca^{-k}.
\end{equation*}

ii) For every $0<\chi_0\leq 1,$ $0<r_0\leq 1$ and for every
multi-index $I$ such that $|I|>0$ there exists a constant $C>0$
such that for every $\chi\leq\chi_0,$ for every $(x,a)\in\{0<a\leq
1,r_0\leq |x|\leq 1\},$ and for every $0\leq l\leq k-1$,
\begin{equation*}
\sup_{0<\eta<1}|\int_0^\infty\mathbf{E}_\chi
X^Ip^\sigma(t,0)(x)\partial_a^lm_\alpha(I_{a,\eta})^{-1}\partial_a^{k-l}1_{I_{a,\eta}}(\sigma_t)dt|\leq
Ca^{-k}.
\end{equation*}

iii) For every $1>\delta>1\slash 2$ and for every multi-index $I$
such that $|I|>0$ there exists a constant $C>0$ such that for
every $\chi\leq 1\slash 2-\delta,$ for every
$(x,a)\in\{(1-\delta)\slash 2\leq a\leq 1\slash 2\}$ and for every
$0\leq l\leq k-1$,
\begin{equation*}
\sup_{0<\eta<\delta\slash 4}|\int_0^\infty\mathbf{E}_\chi
X^Ip^\sigma(t,0)(x)\partial_a^lm_\alpha(I_{a,\eta})^{-1}\partial_a^{k-l}1_{I_{a,\eta}}(\sigma_t)dt|\leq
Ca^{-k}.
\end{equation*}
\end{proposition}
\begin{proof}[Sketch of the proof] The case $\alpha=0$ has been
proved in \cite{ejde}. (See the proof of Proposition 3.1 in
\cite{ejde}. We take the opportunity to say that the formulation
of Proposition 3.1 in \cite{ejde} is wrong and that it should have
been stated exactly as above with $\alpha=0.$) The generalization
from $\alpha=0$ to an arbitrary $\alpha>0$ is (almost)
straightforward. One only needs to imitate the proof of
Proposition 3.1 in \cite{ejde}.
\end{proof}

After this preparatory facts we are ready to give

\begin{proof}[Sketch of the proof of Theorem \ref{main}]
Let $0<\delta<1\slash 2$, $k\in\mathbb{Z}^+$ and a multi-index $I$ be
fixed.

\noindent \textbf{Case 1.} We consider the set
\begin{equation*}
S_1=\mathcal{Q}\setminus T_\delta.
\end{equation*}
By Proposition \ref{mainproposition} and Theorem \ref{r} it
follows that there exists a positive constant $C$ such that
\begin{equation}\label{c1}
|\partial_a^k\mathcal X^IG^*_{\gamma\slash 2}(x,a;e,1)|\leq
Ca^{-k}
\end{equation}
for every $(x,a)\in \tilde S_1:=S_1\cap\{(x,a)\in
N\times\mathbb{R}^+:a\leq 1-\delta\}$. But $S_1\setminus\text{Int}\tilde
S_1$ is a compact set and  $G^*_{\gamma\slash 2}$ is a continuous
function so we get \eqref{c1} on $S_1$. Therefore, on $S_1$ we have
that
\begin{equation*}
|\partial_a^k\mathcal X^I\mathcal
G_{-\gamma}(x,a)|=|\partial_a^k\mathcal X^IG^*_{\gamma\slash
2}(x,a;e,1)|\leq Ca^{-k}
\end{equation*}
and
\begin{equation*}
|\partial_a^k\mathcal X^I\mathcal
G_{\gamma}(x,a)|=|\partial_a^k\mathcal X^IG^*_{\gamma\slash
2}(x,a;e,1)a^{\gamma}|\leq Ca^{\gamma-k}.
\end{equation*}

\noindent \textbf{Case 2.} Outside the set $S_1$ we imitate the proof of
Theorem 3.1 in \cite{amuc}, where derivative with respect to
$\mathcal{X}$ have been considered. The homogeneity
\eqref{homogeneity} of the Green function plus \eqref{den} play
the crucial role.
\end{proof}


\begin{thebibliography}{10}

\bibitem{A}
A.~Ancona.
\newblock Negatively curved manifolds, elliptic operators, and the {M}artin
  boundary.
\newblock {\em Ann. of Math.}, 125: 495--536, 1987.

\bibitem{CC}
C.~Constantinescu and A.~Cornea.
\newblock {\em Potential theory on harmonic spaces}.
\newblock Springer-Verlag, New York, 1972.
\newblock With a preface by H. Bauer, Die Grundlehren der mathematischen
  Wissenschaften, Band 158.

\bibitem{Borodin}
A.N. Borodin and P.~Salminen.
\newblock {\em Handbook of {B}rownian motion - facts and formulae}.
\newblock Birkh{\"{a}}user Verlag, 1996.

\bibitem{D}
E.~Damek.
\newblock Fundamental solutions of differential operators on homogeneous
  manifolds of negative curvature and related {R}iesz transforms.
\newblock {\em Coll. Math}, 73(2): 229--249, 1997.

\bibitem{DHU}
E.~Damek, A.~Hulanicki, and R.~Urban.
\newblock Martin boundary for homogeneous {R}iemannian manifolds of negative
  curvature at the bottom of the spectrum.
\newblock {\em Rev. Mat. Iberoamericana}, 17(2): 257--293, 2001.

\bibitem{DHZ}
E.~Damek, A.~Hulanicki, and J.~Zienkiewicz.
\newblock Estimates for the {P}oisson kernels and their derivatives on rank one
  {$NA$} groups.
\newblock {\em Studia Math.}, 126(2): 115--148, 1997.

\bibitem{FS}
G. B. Folland and E.~Stein.
\newblock {\em Hardy spaces on homogeneous groups}.
\newblock Princeton Univ. Press, 1982.

\bibitem{H}
E.~Heintze.
\newblock On homogeneous manifolds of negative curvature.
\newblock {\em Math. Ann.}, 211: 23--34, 1974.

\bibitem{Stamp}
D.~Kinderlehrer and G.~Stampacchia.
\newblock {\em An introduction to variational inequalities and their
  applications}.
\newblock Academic Press, 1980.

\bibitem{L}
N. N. Lebedev.
\newblock {\em Special functions and their applications}.
\newblock Dover Publications, Inc., 1972.

\bibitem{RY}
D.~Revuz and M.~Yor.
\newblock {\em Continuous martingales and {B}rownian motion}.
\newblock Springer-{V}erlag, 1991.

\bibitem{T}
H.~Tanabe.
\newblock {\em Equations of evolutions}.
\newblock London, Pitman, 1979.

\bibitem{SV}
D.~Stroock and S.R. Varadhan.
\newblock {\em Multidimensional difussion processes}.
\newblock Springer, 1979.

\bibitem{amuc}
R.~Urban.
\newblock Estimates for the derivatives of the {G}reen functions on homogeneous manifolds of negative curvature.
\newblock {\em Acta Math. Univ. Comenianae (N.S.)}, 73(1): 43--53, 2004.

\bibitem{pota}
R.~Urban.
\newblock Estimates for derivatives of the {G}reen functions for the
  noncoercive differential operators on homogeneous manifolds of negative
  curvature.
\newblock {\em Potential {A}nalysis}, 19(4): 317--339, 2003.

\bibitem{ejde}
R.~Urban.
\newblock Estimates for derivatives of the {G}reen functions for the
  noncoercive differential operators on homogeneous manifolds of negative
  curvature, {II}.
\newblock {\em Electron. J. Differential Equations}, 2003(86): 1--8, 2003.

\bibitem{Phd}
R.~Urban.
\newblock {\em Estimates for the {P}oisson kernels on {$NA$} groups. A
  probabilistic method}.
\newblock PhD thesis, Wroclaw {U}niversity, 1999.

\bibitem{pams}
R.~Urban.
\newblock Estimates for the {P}oisson kernels on homogeneous manifolds of
  negative curvature and the boundary {H}arnack inequality in the noncoercive
  case.
\newblock {\em Probab. Math. Statist.}, 21(1): 213--229, 2001.

\bibitem{CM}
R.~Urban.
\newblock Noncoercive differential operators on homogeneous manifolds of
  negative curvature and their {G}reen functions.
\newblock {\em Colloq. Math.}, 88(1): 121--134, 2001.

\bibitem{MathZ}
R.~Urban.
\newblock Estimates for derivatives of the {P}oisson kernels on homogeneous
  manifolds of negative curvature.
\newblock {\em Math. Z.}, 240(4): 745--765, 2002.

\end{thebibliography}


\end{document}