Poisson kernels on nilpotent, 3-meta-abelian groups

Abstract

Let $S$ be a semi-direct product $S=N\rtimes A$ where $N$ is a connected and simply connected, nilpotent, non-meta-abelian, 3-abelian Lie group and $A$ is isomorphic with $\mathbb {R}^k,$ $k>1$. On $S$, we consider a class of second-order left-invariant differential operators of the form $\mathcal L_\alpha =L^a+\Delta _\alpha ,$ where $\alpha \in \mathbb {R}^k,$ and for each $a\in \mathbb {R}^k,$ $L^a$ is left-invariant second-order differential operator on $N$ and $\Delta _\alpha =\Delta -\langle \alpha ,\nabla \rangle ,$ where $\Delta$ is the usual Laplacian on $\mathbb {R}^k$. We prove an upper bound for the Poisson kernel for the operator $\mathcal L_\alpha$.

1 Introduction

1.1 Statement of the main result

Let $S$ be a semi-direct product, $S=N\rtimes A$ where $N$ is a connected, simply connected, 3-meta-abelian, nilpotent Lie group and $A$ is isomorphic with $\mathbb {R}^k.$ We identify $A$ with its Lie algebra $\mathfrak a.$ The dimension $k$ of $A$ is called the rank of $S$.

Specifically, we assume that

$$\begin{aligned} N&= N_1\rtimes ( N_2\rtimes N_3)\nonumber \\&\equiv N_1\rtimes N_2\rtimes N_3, \end{aligned}$$

(1.1)

where $N_i,$ $i=1,2,3$, are abelian Lie groups with the corresponding Lie algebras $\mathfrak n_i.$ To avoid trivialities, we assume that $N_3$ is non-central in $N$ (Otherwise, $N$ is meta-abelian.).

Then, there are bases $\{X_1,\ldots ,X_{d_1}\},$ $\{Y_1,\ldots ,Y_{d_2}\},$ and $\{Z_1,\ldots ,Z_{d_3}\}$ for $\mathfrak n_i,$ $i=1,2,3,$ respectively, such that

$$\begin{aligned} \{X_i,\ldots ,X_{d_1},Y_1,\ldots ,Y_{d_2},Z_1,\ldots ,Z_{d_3}\} \end{aligned}$$

forms a Jordan–Hölder basis for the Lie algebra $\mathfrak n$ of $N$. We assume that these bases are ordered so that the matrix of ${{\mathrm{ad}}}_Z$ is strictly lower triangular for all $Z\in \mathfrak n$. We assume in addition that this basis diagonalizes the ${{\mathrm{ad}}}_{\mathfrak {a}}$ action on $\mathfrak n.$ We use these bases to identify $\mathfrak n_1,$ $\mathfrak n_2,$ and $\mathfrak n_3$ with $\mathbb {R}^{d_1},$ $\mathbb {R}^{d_2}$, and $\mathbb {R}^{d_3}$, respectively, and we use the exponential map to identify $N_i$ with the corresponding Lie algebras $\mathfrak n_i=\mathbb {R}^{d_i}.$

For $g\in S$, we write $g=x(g)a(g)=xa=(x,a),$ where $x(g)=x\in N$ and $a(g)=a\in A$ denote the components of $g$ in $N\rtimes A.$ Similarly, for $x\in N$, we write $x=m(x)v(x)w(x)=mvw=(m,v,w)$, where $m(x)=m\in N_1,$ $v(x)=v\in N_2,$ and $w(x)=w\in N_3$ denote the components of $x$ in $N_1\rtimes N_2\rtimes N_3.$

Let

$$\begin{aligned} \Lambda _1&= \{\xi _1,\ldots ,\xi _{d_1}\},\\ \Lambda _2&= \{\vartheta _1,\ldots ,\vartheta _{d_2}\},\\ \Lambda _3&= \{\psi _1,\ldots ,\psi _{d_3}\}, \end{aligned}$$

be the roots of the ${{\mathrm{ad}}}_{\mathfrak a}$ action on $\mathfrak n_1,$ $\mathfrak n_2$ and $\mathfrak n_3$, respectively, corresponding to the given bases. Let

$$\begin{aligned} \Lambda =\Lambda _1\cup \Lambda _2\cup \Lambda _3. \end{aligned}$$

Hence, for all $H\in \mathfrak a,$

$$\begin{aligned} \begin{array}{ll} {{\mathrm{ad}}}_HX_i=[H,X_i]=\xi _i(H)X_i,&{}\quad 1\le i\le d_1,\\ {{\mathrm{ad}}}_HY_j=[H,Y_j]=\vartheta _j(H)Y_j,&{}\quad 1\le j\le d_2,\\ {{\mathrm{ad}}}_HZ_k=[H,Z_k]=\psi _k(H)Z_k,&{}\quad 1\le k\le d_3. \end{array} \end{aligned}$$

(1.2)

Let $d=d_1+d_2+d_3$. For $1\le i\le d$, we set

$$\begin{aligned} \lambda _i= {\left\{ \begin{array}{ll} \xi _i,&{}\quad 1\le i\le d_1,\\ \vartheta _{i-d_1},&{}\quad d_1+1\le i\le d_1+d_2,\\ \psi _{i-d_1-d_2},&{}\quad d_1+d_2+1\le i\le d. \end{array}\right. } \end{aligned}$$

We refer to the class of $NA$ groups defined above as $3$-meta-abelian $NA$ groups. Similarly, we can define a class of $k$-meta-abelian $NA$ groups. This is a quite large class of $NA$ groups. It is not difficult to see that the $AN$ parts of classical semisimple Lie groups of type $A_\ell ,$ $B_\ell ,$, and $D_\ell$ are all $k$-meta-abelian for some $k.$

A simple example of $3$-meta-abelian $NA$ group is the group of all upper triangular $4\times 4$-matrices with positive diagonal.

The principal object of study in this work is the left-invariant differential operator on $S,$

$$\begin{aligned} \mathcal {L}_\alpha =\Delta _\alpha +\sum _{j=1}^{d_1} e^{2\xi _j(a)}X_j^2+\sum _{j=1}^{d_2}e^{2\vartheta _j(a)} Y_j^2+\sum _{j=1}^{d_3}e^{2\psi _j(a)}Z_j^2, \end{aligned}$$

(1.3)

where, for $\alpha =(\alpha _1,\dots ,\alpha _k)\in \mathbb {R}^k$,

$$\begin{aligned} \Delta _\alpha =\sum _{i=1}^k(\partial _{a_i}^2-2\alpha _i\partial _{a_i}), \end{aligned}$$

and the $X_i,$ $Y_j,$ and $Z_k$ are considered as left-invariant differential operators on $N_1$, $N_2$, and $N_3$, respectively. We are particularly interested in the bounded harmonic functions for this operator, i.e., bounded functions $F$ on $S$ satisfying $\mathcal {L}_\alpha F=0$.

A fundamental result of Damek [3] implies that bounded $\mathcal {L}_\alpha$-harmonic functions exist provided the following positivity assumption (which we also assume) holds:

$$\begin{aligned} \lambda _i(\alpha )>0, \quad \forall i. \end{aligned}$$

(1.4)

In particular, none of the $\lambda _i$ are identically 0 and the $\lambda _i$ span $\mathfrak a^*$ (Their joint nullspace consists of vectors annihilated by ${{\mathrm{ad}}}_\mathfrak a.$). We set

$$\begin{aligned} A^+=\{a\in \mathbb {R}^k:\lambda _i(a)> 0\text { for }1\le i\le d\}. \end{aligned}$$

It also follows from [3] that under our assumptions, the bounded harmonic functions are precisely the “Poisson integrals” of $L^\infty (N)$. To describe this concept, let $\chi$ be the modular function for left-invariant Haar measure on $S$. Thus, for all $g\in S,$

$$\begin{aligned} \int _Sf(sg)\hbox {d}s=\chi (g)^{-1}\int _Sf(s)\hbox {d}s, \end{aligned}$$

where d$s$ is left-invariant Haar measure on $S.$ Then,

$$\begin{aligned} \chi (g)=\det ({{\mathrm{Ad}}}(g))=e^{\rho _0(a)}, \end{aligned}$$

(1.5)

where

$$\begin{aligned} \rho _0=\sum _{j=1}^d\lambda _j. \end{aligned}$$

(1.6)

Assumption (1.4) together with [3] implies that there exists a Poisson kernel $\nu$ for $\mathcal L_\alpha .$ That is, there is a $C^\infty$ function $\nu$ on $N$ such that every bounded $\mathcal L_\alpha$-harmonic function $F$ on $S$ may be written as a Poisson integral against a bounded function $f$ on the quotient space $A\backslash S=N,$

$$\begin{aligned} F(g)=\int _{A\backslash S}f(gz)\nu (z)\hbox {d}z=\int _Nf(z)\check{\nu }^a(z^{-1}z_o)\hbox {d}z,\;\;g=(z_o,a_o), \end{aligned}$$

where

$$\begin{aligned} \check{\nu }^a(z)=\nu (a^{-1}z^{-1}a)\chi (a)^{-1},\quad \text {where}\quad \check{\nu }(z)=\nu (z^{-1}). \end{aligned}$$

(1.7)

Conversely, the Poisson integral of any $f\in L^\infty (N)$ is a bounded $\mathcal L_\alpha$-harmonic function.

Our goal in this work is to obtain explicit estimates on the rate of decay of $\nu$ on $N$. To describe our results, we require some additional notation.

For $t\in \mathbb {R}^+$ and $\alpha \in A^+,$ let

$$\begin{aligned} \delta _t^\alpha ={{\mathrm{Ad}}}((\log t)\alpha )\big |_N. \end{aligned}$$

Then, $t\mapsto \delta _t^\alpha$ is a one parameter group of automorphisms of $N$ for which the corresponding eigenvalues on $\mathfrak {n}$ are all positive. It is known [10, 12] that then $N$ has $\delta _t^\alpha$-homogeneous norm: A non-negative and subadditive continuous function $|\cdot |_\alpha$ on $N$ which is homogeneous with respect to $\delta _t^\alpha ,$ i.e.,

$$\begin{aligned} |\delta _t^\alpha x|_\alpha =t|x|_\alpha , \end{aligned}$$

and $|n|_\alpha =0$ if and only if $n=e.$

For a subset $\Lambda _o\subseteq \Lambda ,$ and $a\in A,$ define

$$\begin{aligned} \overline{\gamma }_{\Lambda _o}(a)&= \min _{\lambda \in \Lambda _o} \lambda (a)\slash \Vert \lambda \Vert ^2,\\ \gamma _{\Lambda _o}(a)&= \min _{\lambda \in \Lambda _o}\lambda (a). \end{aligned}$$

We set

$$\begin{aligned} K(1)=\{x\in N:|x|_\alpha =1\}. \end{aligned}$$

For every $x\in N$, there is precisely one $x_o\in K(1)$ such that $\delta _{|x|_\alpha }^\alpha x_o=x.$ Thus, we have a map

$$\begin{aligned} N\ni x=(m,v,w)\mapsto x_o=(m_o,v_o,w_o)\in K(1). \end{aligned}$$

The following is our main result.

Theorem 1.1

Let $\nu$ be the Poisson kernel for the operator $\mathcal L_\alpha ,$ defined in (1.3), with $\alpha \in A^+.$ Under the above assumptions, for every $\varepsilon >0$ there exists a constant $c=c_{\varepsilon ,\alpha ,\Lambda }>0$ such that for all points $x=(m,v,w)\in N_1\rtimes N_2\rtimes N_3,$

$$\begin{aligned} \nu (x)\le {\left\{ \begin{array}{ll} c(1+|x|_\alpha )^{-\beta _1},&{}\text { if }\;\;|(m_o,0,0)|_\alpha >\varepsilon ,\\ c(1+|x|_\alpha )^{-\beta _2},&{}\text { if }\;\;|(0,v_o,0)|_\alpha >\varepsilon ,\\ c(1+|x|_\alpha )^{-\beta _3},&{}\text { if }\;\;|(0,0,w_o)|_\alpha >\varepsilon , \end{array}\right. } \end{aligned}$$

(1.8)

where

$$\begin{aligned} \beta _1&= \overline{\gamma }_\Lambda (\alpha )\gamma _\Lambda (\alpha ),\\ \beta _2&= \overline{\gamma }_{\Lambda _2\cup \Lambda _3}(\alpha ) \gamma _{\Lambda _2\cup \Lambda _3}(\alpha ),\\ \beta _3&= \overline{\gamma }_{\Lambda _3}(\alpha )\gamma _{\Lambda _3}(\alpha ). \end{aligned}$$

In particular, there is a constant $c=c_{\alpha ,\Lambda }>0$ such that

$$\begin{aligned} \nu (m,0,0)&\le c(1+|(m,0,0)|_\alpha )^{-\beta _1},&\qquad \quad \text { for all }m&\in N_1,\end{aligned}$$

(1.9)

$$\begin{aligned} \nu (0,v,0)&\le c(1+|(0,v,0)|_\alpha )^{-\beta _2},&\qquad \quad \text { for all }v&\in N_2,\end{aligned}$$

(1.10)

$$\begin{aligned} \nu (0,0,w)&\le c(1+|(0,0,w)|_\alpha )^{-\beta _3},&\qquad \quad \text { for all }w&\in N_3, \end{aligned}$$

(1.11)

Remark 1.2

Notice that $\beta _1\le \beta _2\le \beta _3.$

Remark 1.3

We note that [18, Theorem 1.1] says that on a large class of nilpotent Lie groups $N$ containing the 3-meta-abelian groups one has that for every $q>1$, there is $c=c_{q,\alpha ,\Lambda }>0$ such that for all $x\in N,$

$$\begin{aligned} \nu (x)\le c(1+|x|_\alpha )^{-\frac{2}{q} \overline{\gamma }_\Lambda (\alpha )\gamma _\Lambda (\alpha )}. \end{aligned}$$

(1.12)

In many cases, the estimate in Theorem 1.1 is strictly sharper than (1.12) since clearly the inequalities

$$\begin{aligned} \frac{2}{q}\overline{\gamma }_\Lambda (\alpha ) \gamma _\Lambda (\alpha )<\beta _i \quad \text {for}\,\, i=1,2,3, \end{aligned}$$

can hold for many choices of $\Lambda$ and $\alpha .$ Also, unlike (1.12), the estimates in Theorem 1.1 differentiate between the various directions of approach to infinity.

Remark 1.4

We should also remark that in [20], we proved the analog of Theorem 1.1 in the 2-meta-abelian case. We found that there are essential difficulties if one wants to prove a similar estimate for the Poisson kernel on $k$-meta-abelian group with $k\ge 3.$ In Sect. 1.2 below, we describe these difficulties in more details (See in particular Remarks 1.5 and 1.6.).

1.2 Strategy of the proof

Let $\mu _t$ (resp., $T_t$) be the semigroups of measures (resp., operators) generated by $\mathcal L_\alpha .$ It is known (see Sect. 6.1) that the Poisson kernel $\nu$ is equal to $\lim _{t\rightarrow \infty }\pi _N(\mu _t),$ where $\pi _N(g)=x(g)$ is a projection from $S$ onto $N.$ To get some information on $\mu _t$, we use a well-known formula which express $T_t$ as a skew-product of the diffusions on $N$ and $A.$ (The idea of such a decomposition goes back to [13, 14, 22]. In the context of $NA$ groups with $\dim A=1$, this decomposition was used in [4–7], and later was generalized by the authors and applied for $\dim A>1,$ see, e.g., [17, 19].)

Specifically, for $\sigma \in C([0,\infty ),A),$ $A=\mathbb {R}^k,$ let

$$\begin{aligned} \mathcal L_{N}^{\sigma }=\sum _{j=1}^{d_1}e^{2\xi _j(\sigma (t))} X_j^2+\sum _{j=1}^{d_2}e^{2\vartheta _j(\sigma (t))}Y_j^2+ \sum _{j=1}^{d_3}e^{2\psi _j(\sigma (t))}Z_j^2, \end{aligned}$$

(1.13)

considered as a time-dependent left-invariant differential operator on $N$. It is known that then $\mathcal L_{N}^{\sigma }$ generates a time-inhomogeneous diffusion on $N$ with transition kernel $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s},$ $t\ge s\ge 0,$ and the corresponding evolution operators $U^{N_1\rtimes N_2\rtimes N_3,\sigma }_{s,t}$ (see [2, 21]). The skew-product formula says that

$$\begin{aligned} T_tf(x,a)=\mathbf{E}_a^\sigma U^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}f(\cdot ,\sigma (t))\big |_x=\mathbf{E}_a^\sigma (f*_{N}P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0})(x,\sigma (t)), \end{aligned}$$

(1.14)

for $f\in C_c(N\times \mathbb {R}^k)$ and $t\ge 0,$ where the expectation is taken with respect to the distribution of the Brownian motion $\sigma (t)\in \mathbb {R}^k$ with drift $-2\alpha ,$ starting from $a,$ i.e., $\sigma (0)=a,$ and generated by $\Delta _\alpha .$ The subscript $N$ in the convolution $*_N$ means that $f$ is convolved with the kernel $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}$ with respect to the first variable in $N.$

The next step is to disintegrate the kernel $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}$ using [20, Theorem 1.2] (see also section 3 in [19]) which is a skew-product formula similar to formula (1.14). Specifically, from the decomposition (1.1), the time-dependent family of operators

$$\begin{aligned} \mathcal L_{N_2\rtimes N_3}^{\sigma }=\sum _{j=1}^{d_2} e^{2\vartheta _j(\sigma (t))}Y_j^2+\sum _{j=1}^{d_3}e^{2\psi _j (\sigma (t))}Z_j^2, \end{aligned}$$

(1.15)

gives rise to an evolution on $N_2\rtimes N_3=\mathbb {R}^{d_2}\rtimes \mathbb {R}^{d_3}$ that is described by a kernel $P^{N_2\rtimes N_3,\sigma }_{t,s}.$ The semi-direct product $N_2\rtimes N_3$ is a 2-meta-abelian group. Thus, we have a relatively good knowledge about the kernel $P^{N_2\rtimes N_3,\sigma }_{t,s}$ (see [20]). Let $\eta (t)=(\eta ^{N_2}(t),\eta ^{N_3}(t))$ be the process generated by $\mathcal L_{N_2\rtimes N_3}^\sigma .$ The skew-product formula from [20] gives that

$$\begin{aligned} U^{N_1\rtimes N_2\rtimes N_3,\sigma }_{s,t}f(m,v,w)= \mathbf{E}_{s,(v,w)}^{\eta }U_{s,t}^{N_1,\sigma ,\eta }f(\cdot ,\eta ^{N_2}(t), \eta ^{N_3}(t))\big |_{m}, \end{aligned}$$

(1.16)

where the subscript in the expectation means that $\eta (s)=(v,w),$ and $U_{s,t}^{N_1,\sigma ,\eta }$ is the family of evolution operators generated by the operator

$$\begin{aligned} \mathcal L_{N_1}^{\sigma ,\eta }=\sum _{j=1}^{d_1}e^{2\xi _j(\sigma (t))} \left( {{\mathrm{Ad}}}(\eta (t))\big |_{{\mathfrak n}_1}X_j\right) ^2. \end{aligned}$$

Formula (1.16) allows us to compute and estimate in Sects. 5 and 6 the kernel $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}$ on certain subsets of $N$ which, it turns out, are sufficient for our purposes (See formulas (5.1), (6.3), and (6.5).).

Remark 1.5

In [20], we proved a result analogous to Theorem 1.1 for meta-abelian groups, i.e., in the 2-meta-abelian case. The generalization to the 3-meta-abelian case is complicated by the lack of an appropriate estimate for $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}$ valid on all of $N$. The fact that it is possible to obtain a good estimate for the Poisson kernel on all of $N$ by piecing together estimates of $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}$ on subsets is somewhat surprising.

Remark 1.6

In order to consider $k$-meta-abelian $NA$ groups with $k\ge 4$, we need a better understanding of the situation for $k=3.$ In particular, we need to invent some methods which allow us to produce a good and global Gaussian estimate for $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}.$ If we try to apply methods from [20], the problem of estimating $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}$ amounts to the good knowledge of the properties of the time-inhomogeneous process $\eta (t)$ generated by the time-dependent operator (1.15). We hope that if this estimate is obtained, then we will be able to apply induction argument and eventually get estimates for the Poisson kernel for all $k.$

1.3 Structure of the paper

The outline of the rest of the paper is as follows. In Sect. 2, we recall some basic facts about exponential functionals of Brownian motion. In Sect. 3, we consider the evolution process $\eta (t)$ on $N_2\rtimes N_3$ generated by the operator (1.15) and state the estimate for the corresponding transition kernels. Next, in Sect. 4, we study the evolution kernel on $N_1$ which is the second ingredient of the skew-product formula (1.16). In Sect. 5, we prove an estimate for the kernel $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}.$ Finally in Sect. 6, we construct the Poisson kernel and prove our main theorem.

2 Preliminaries

2.1 Exponential functionals of Brownian motion

Let $b(s),$ $s\ge 0,$ be the Brownian motion on $\mathbb {R}$ staring from $a\in \mathbb {R}$ and normalized so that

$$\begin{aligned} \mathbf{E}_af(b(s))=\frac{1}{\sqrt{4\pi s}}\int _{\mathbb {R}}f(x+a)e^{-x^2\slash 4s}\hbox {d}x. \end{aligned}$$

(2.1)

Hence, $b(s)$ has a normal distribution with mean $\mathbf{E}b(s)=a$ and variance ${{\mathrm{Var}}}b(s)=2s.$

Remark

Our normalization of the Brownian motion $b(s)$ is different than that typically used by probabilists who tend to assume that ${{\mathrm{Var}}}b(s)=s.$

For $d>0$ and $\mu >0$, we define the following exponential functional

$$\begin{aligned} I_{d,\mu }=\int _0^\infty e^{d(b(s)-\mu s)}\hbox {d}s. \end{aligned}$$

(2.2)

Theorem 2.1

(Dufresne, [9]) Let $b(0)=0.$ Then, the functional $I_{2,\mu }$ is distributed as $(4\gamma _{\mu \slash 2})^{-1},$ where $\gamma _{\mu \slash 2}$ denotes a gamma random variable with parameter $\mu \slash 2,$ i.e., $\gamma _{\mu \slash 2}$ has a density $(1\slash \Gamma (\mu \slash 2))x^{\frac{\mu }{2}-1}e^{-x}\mathbf 1_{[0,+\infty )}(x).$

As a corollary of Theorem 2.1, by scaling the Brownian motion and changing the variable, we get the following theorem (See [17, Lemma 5.4] for details.).

Theorem 2.2

Let $\sigma (u)=b(u)-2\alpha u$ be the $k$-dimensional Brownian motion with a drift $-2\alpha ,$ $d>0,$ and let $\ell \in (\mathbb {R}^k)^*$ be such that $\ell (\alpha )>0.$ Then,

$$\begin{aligned} \mathbf{E}_af\left( \int _0^\infty e^{d\ell (\sigma (u))}\hbox {d}u\right) =c_{d,\ell ,\alpha }e^{\ell (a)}\int _0^\infty f(u)u^{-\varrho \slash d}\exp \left( -\frac{e^{d\ell (a)}}{2d^2\ell ^2u}\right) \frac{\hbox {d}u}{u}, \end{aligned}$$

where $\varrho =2\ell (\alpha )\slash \Vert \ell \Vert ^2.$

Remark

Exponential functionals of type (2.2) are called perpetual functionals in financial mathematics, and they play an important role there (see, e.g., [15, 16, 23]). In particular, the distribution of the integral over finite interval $(0,t)$ in (2.2) has many applications in Asian options (see, e.g., [1, 8, 11]).

2.2 Notation for exponential functionals

For a continuous function $\sigma :[0,\infty )\rightarrow A=\mathbb {R}^k,$

$$\begin{aligned} A_{N_1,i}^\sigma (s,t)&= \int _s^te^{2\xi _{i}(\sigma (u))} \hbox {d}u,\quad i=1,\ldots ,d_1,\\ A_{N_2,j}^\sigma (s,t)&= \int _s^te^{2\vartheta _{j}(\sigma (u))} \hbox {d}u,\quad \;j=1,\ldots ,d_2,\\ A_{N_3,k}^\sigma (s,t)&= \int _s^te^{2\psi _{k}(\sigma (u))} \hbox {d}u, \quad k=1,\ldots ,d_3, \end{aligned}$$

and

$$\begin{aligned} A_{N,j}^\sigma (s,t)=\int _s^te^{2\lambda _{j}(\sigma (u))}\hbox {d}u,\quad j=1,\ldots ,d. \end{aligned}$$

We also define, for $i=1,2,3,$

$$\begin{aligned} A_{N_i,\Sigma }^\sigma (s,t)=\sum _{j=1}^{d_i}A_{N_i,j}^\sigma (s,t),\qquad A_{N_i,\Pi }^\sigma (s,t)=\prod _{j=1}^{d_i}A_{N_i,j}^\sigma (s,t). \end{aligned}$$

Finally, we put

$$\begin{aligned} A_{N,\Sigma }^\sigma (s,t)=\sum _{i=1}^3A_{N_i,\Sigma }^\sigma (s,t),\qquad A_{N,\Pi }^\sigma (s,t)=\prod _{i=1}^3A_{N_i,\Pi }^\sigma (s,t). \end{aligned}$$

2.3 Moments of exponential functionals

The following lemma follows from Theorem 2.2.

Lemma 2.3

The functional $A_{N,j}^\sigma (0,\infty ),$ $j=1,\ldots ,d,$ has a finite $s$-th moment (for every $a\in \mathbb {R}^k$)

$$\begin{aligned} \mathbf{E}^\sigma A_{N,j}^\sigma (0,\infty )^s<+\infty \end{aligned}$$

if and only if $\lambda _j(\alpha )\slash \Vert \lambda \Vert ^2>s$, where $\Vert \cdot \Vert$ is $\ell ^2$ norm on $\mathbb {R}^k$. In particular, $A_{N,j}^\sigma (0,\infty )$ has all negative moments.

2.4 Action of $A$ on $N$

For $a\in \mathbb {R}^k,$ $\Phi (a)$ denotes the action of $a$ on $\mathfrak n,$ see (1.2). By (1.2), the automorphisms $\{\Phi (a)\}_{a\in \mathbb {R}^k}$ leave $\mathfrak {n}_i,$ $i=1,2,3$ invariant. We identify linear transformation $\Phi (a)$ on $\mathfrak n$ with $d\times d$ matrix,

$$\begin{aligned} \Phi (a)=\begin{bmatrix} \Phi (a)\big |_{\mathfrak {n}_1}&0&0\\ 0&\Phi (a)\big |_{\mathfrak {n}_2}&0\\ 0&0&\Phi (a)\big |_{\mathfrak {n}_3} \end{bmatrix}, \end{aligned}$$

where

$$\begin{aligned} \Phi (a)\big |_{\mathfrak {n}_1}&= {{\mathrm{diag}}}\left[ e^{\xi _1(a)},\ldots ,e^{\xi _{d_1}(a)}\right] ,\\ \Phi (a)\big |_{\mathfrak {n}_2}&= {{\mathrm{diag}}}\left[ e^{\vartheta _1(a)},\ldots ,e^{\vartheta _{d_2}(a)}\right] ,\\ \Phi (a)\big |_{\mathfrak {n}_3}&= {{\mathrm{diag}}}\left[ e^{\psi _1(a)},\ldots ,e^{\psi _{d_3}(a)}\right] . \end{aligned}$$

Let $\sigma$ be a continuous function from $[0,+\infty )$ to $A=\mathbb {R}^k.$ We define

$$\begin{aligned} \Phi ^\sigma (t)=\Phi (\sigma (t)). \end{aligned}$$

(2.3)

3 Evolution kernel on $N_2\rtimes N_3$

In this section, we consider time-dependent operator on $N_2\rtimes N_3,$

$$\begin{aligned} \mathcal L_{N_2\rtimes N_3}^{\sigma }=\sum _{j=1}^{d_2} e^{2\vartheta _j(\sigma (t))}Y_j^2+\sum _{j=1}^{d_3} e^{2\psi _j(\sigma (t))}Z_j^2. \end{aligned}$$

The operator $\mathcal L_{N_2\rtimes N_3}^{\sigma }$ gives rise to an evolution on $N_2\rtimes N_3=\mathbb {R}^{d_2}\rtimes \mathbb {R}^{d_3}$ that is described by a kernel $P^{N_2\rtimes N_3,\sigma }_{t,s}$ and the corresponding operator $U^{N_2\rtimes N_3,\sigma }_{t,s}.$ The semi-direct product $N_2\rtimes N_3$ is a 2-meta-abelian group.

The Euclidean space $\mathbb {R}^n$ is endowed with the usual scalar product $\langle x,y\rangle =x\cdot y=\sum _{i=1}^nx_iy_i$ and the corresponding $\ell ^2$ norm $\Vert x\Vert =\langle x,x\rangle ^{1\slash 2}.$

The following estimate is proved in [20, Theorem 4.1].

Theorem 3.1

There are positive constants $C,D$ and $k_o$ such that for all $t>s\ge 0$ and all $(v,w)\in N_2\rtimes N_3,$

$$\begin{aligned}&A_{N_2,\Pi }^{\sigma }(s,t)^{1\slash 2}A_{N_3,\Pi }^{\sigma }(s,t)^{1\slash 2}P^{N_2\rtimes N_3,\sigma }_{t,s}(v,w)\\&\quad \le C(\Vert v\Vert ^{\frac{1}{2k_o}}+1) \exp \left( -\frac{D\Vert w\Vert ^ 2}{A_{N_3,\Sigma }^\sigma (s,t)}-\frac{D\Vert v\Vert ^2}{(\Vert v\Vert ^{\frac{1}{2k_o}}+\Vert w\Vert +2)^{2k_o}A_{N_2,\Sigma }^\sigma (s,t)}\right) \\&\qquad +\,CA_{N_3,\Sigma }^\sigma (s,t)^{1\slash 2}\exp \left( -D\frac{\Vert v\Vert ^{\frac{1}{k_o}}+\Vert w\Vert ^ 2}{A_{N_3,\Sigma }^\sigma (s,t)}\right) . \end{aligned}$$

Remark

Here, $k_{o}$ is the smallest non-negative integer such that

$$\begin{aligned} ({{\mathrm{ad}}}_X)^{k_o+1}\big |_{\mathfrak {n}_2}=0, \quad \forall X\in \mathfrak {n}_3. \end{aligned}$$

Note that $k_o>0$ since by hypothesis $N_3$ is non-central.

4 Evolution kernel on $N_1$

Let

$$\begin{aligned} \eta (t)=\left( \eta ^{N_2}(t),\eta ^{N_3}(t)\right) = \left( \eta ^1(t),\ldots ,\eta ^{d_2+d_3}(t)\right) \end{aligned}$$

be the time-inhomogeneous Markov process generated by the operator

$$\begin{aligned} \mathcal L_{N_2\rtimes N_3}^{\sigma }=\sum _{j=1}^{d_2} e^{2\vartheta _j(\sigma (t))}Y_j^2+\sum _{j=1}^{d_3} e^{2\psi _j(\sigma (t))}Z_j^2 \end{aligned}$$

considered in Sect. 3.

Now on $N_1$, we consider time-dependent operator

$$\begin{aligned} \mathcal L_{N_1}^{\sigma ,\eta }=\sum _{j=1}^{d_1} e^{2\xi _j(\sigma (t))}({{\mathrm{Ad}}}(\eta (t))\big |_{\mathfrak {n}_1}X_j)^2. \end{aligned}$$

The following notation will be useful. For a $n\times n$ invertible matrix $A$, we set

$$\begin{aligned} \mathcal B(A)(x)=2^{-1}A^{-1}x\cdot x\quad \text { and }\quad \mathcal D(A)=(2\pi )^{-n\slash 2}(\det A)^{-1\slash 2}. \end{aligned}$$

(4.1)

Since $N_1$ is abelian, the transition kernels $P_{t,s}^{N_1,\sigma ,\eta }$ of the time-inhomogeneous process $\omega (t)$ generated by $\mathcal L_{N_1}^{\sigma ,\eta }$ are given (see Proposition 2.10 in [19]) by

$$\begin{aligned} P_{t,s}^{N_1,\sigma ,\eta }(m;m^\prime )=\mathcal {D} \left( A_{N_1}^{\sigma ,\eta }(s,t)\right) e^{-\mathcal {B}\left( A_{N_1}^{\sigma ,\eta }(s,t)\right) \left( m-m^\prime \right) }, \end{aligned}$$

(4.2)

where

$$\begin{aligned} A_{N_1}^{\sigma ,\eta }(s,t) =2\int _0^t\left[ {{\mathrm{Ad}}}\left( \eta (u)\right) \big |_{\mathfrak n_1}\Phi ^\sigma (u)\big |_{\mathfrak n_1}\right] \left[ {{\mathrm{Ad}}}\left( \eta (u)\right) \big |_{\mathfrak n_1}\Phi ^\sigma (u)\big |_{\mathfrak n_1}\right] ^*\hbox {d}u, \end{aligned}$$

where $\Phi ^\sigma (u)$ is defined in (2.3).

Exactly in the same way as Lemma 3.3 in [20], one can prove the following lemma.

Lemma 4.1

There is a constant $C>0$ such that

$$\begin{aligned} \mathcal {D}(A^{\sigma ,\eta }_{N_1}(s,t))\le CA_{N_1,\Pi }^\sigma (s,t)^{-1\slash 2}. \end{aligned}$$

5 Evolution kernel on $N_1\rtimes N_2\rtimes N_3$

To get the kernel $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}$, we apply the skew-product formula (1.16) (for the proof, see [20, Theorem 1.2]),

$$\begin{aligned} P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}(0,0,0;m,v,w)= \lim _{\varepsilon \rightarrow 0}\mathbf{E}_{0,(0,0)}^{\eta }P^{N_1,\sigma ,\eta }_{t,s} (0;m)\psi _\varepsilon (\eta (t)), \end{aligned}$$

(5.1)

where for $(v,w)\in N_2\rtimes N_3=\mathbb {R}^{d_2}\rtimes \mathbb {R}^{d_3}$ given and $\varepsilon >0$,

$$\begin{aligned} \psi _\varepsilon (v^\prime ,w^\prime )=\varepsilon ^{-d_2-d_3} \mathbf 1_{B_\varepsilon (v)}(v^\prime )\mathbf 1_{B_\varepsilon (w)}(w^\prime ) \end{aligned}$$

is the (normalized in $L^1$) indicator function of the product of two $\varepsilon$-balls around $v$ and $w,$

$$\begin{aligned} B_\varepsilon (v)=\prod _{j=1}^{d_2}B_\varepsilon ^1(v_j),\qquad B_\varepsilon (w)=\prod _{j=1}^{d_3}B_\varepsilon ^1(w_j) \end{aligned}$$

and

$$\begin{aligned} B_\varepsilon ^1(x)=[x-\varepsilon \slash 2,x+\varepsilon \slash 2]. \end{aligned}$$

The following theorem will be used in order to get the estimates for the evolution kernel $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}$ on those sets which are necessary for the upper bound for the Poisson kernel.

Theorem 5.1

There is a constant $C>0$ such that for every $t>s\ge 0,$ and for every $(m,v,w)\in N_1\rtimes N_2\rtimes N_3,$

$$\begin{aligned} P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}(0,0,0;m,v,w)\le CA_{N_1,\Pi }^\sigma (s,t)^{-1\slash 2}P^{N_2\rtimes N_3,\sigma }_{t,s}(0,0;v,w). \end{aligned}$$

Proof

It follows from (4.2) and Lemma 4.1 that

$$\begin{aligned} P^{N_1,\sigma ,\eta }_{t,s}(0;m)\le P^{N_1,\sigma ,\eta }_{t,s} (0;0)=\mathcal {D}\left( A_{N_1}^{\sigma ,\eta }(s,t)\right) \le CA_{N_1,\Pi }^\sigma (s,t)^{-1\slash 2}. \end{aligned}$$

Then, from (5.1)

$$\begin{aligned} P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,s}(0,0,0;m,v,w)\le CA_{N_1,\Pi }^\sigma (s,t)^{-1\slash 2}\lim _{\varepsilon \rightarrow 0}\mathbf{E}_{0,(0,0)}^{\eta }\psi _\varepsilon (\eta (t)) \end{aligned}$$

Clearly,

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\mathbf{E}_{0,(0,0)}^{\eta }\psi _\varepsilon (\eta (t))=P^{N_2\rtimes N_3,\sigma }_{t,s}(0,0;v,w) \end{aligned}$$

and the lemma follows. $\square$

6 Poisson kernel on $N_1\rtimes N_2\rtimes N_3$

In this section, we will give the proof of Theorem 1.1.

6.1 Construction of the Poisson kernel

Here, we recall the construction of the Poisson kernel $\nu$—the principle object of our study. At the same time, this construction provides convenient formula for the Poisson kernel. Let $\mu _t$ be the semigroup of probability measures on $S=N\rtimes \mathbb {R}^k$ generated by $\mathcal {L}_\alpha .$ It is known [5] that

$$\begin{aligned} \lim _{t\rightarrow \infty }(\pi _N(\check{\mu }_t),f)=(\nu ,f), \end{aligned}$$

where $\pi _N$ denotes the projection from $S$ onto $N$ and $(\check{\mu },f)=(\mu ,\check{f}), \check{f}(x)=f(x^{-1}).$ Let $a\in \mathbb {R}^k$ and let $\mu$ be a measure on $N.$ We define

$$\begin{aligned} (\mu ^a,f)=(\mu ,f\circ {{\mathrm{Ad}}}(a)). \end{aligned}$$

For $a\in \mathbb {R}^k$, we have

$$\begin{aligned} \nu ^a(x)=\nu (a^{-1}xa)\chi (a)^{-1},\;\;x\in N, \end{aligned}$$

(6.1)

where $\chi$ is as in (1.5).

We will need the following fact (see [17, Lemma 4.1] for a proof).

Lemma 6.1

We have

$$\begin{aligned} (\nu ^a,f)=\lim _{t\rightarrow \infty }(\pi _N(\check{\mu }_t)^a,f) =\lim _{t\rightarrow \infty }(\mathbf{E}_a^\sigma \check{P}^{N_3\rtimes N_2\rtimes N_1,\sigma }_{t,0}(0,0,0;\cdot ,\cdot ,\cdot ),f), \end{aligned}$$

where $P^{N_3\rtimes N_2\rtimes N_1,\sigma }_{t,0}$ is the evolution kernel for the operator $\mathcal L_N^{\sigma }$ defined in (1.13).

6.2 Upper bound for $\nu ^{s\alpha }$

Our main aim in this subsection is to obtain an upper bound for

$$\begin{aligned} \nu ^{s\alpha }(x)=\nu ((s\alpha )^{-1}x(s\alpha ))\chi (s\alpha )^{-1} \end{aligned}$$

for all $s<0,$ where $\alpha \in A^+$ is a drift vector of the operator (1.3). Then, in Sect. 6.4, in order to get an upper bound for $\nu$ we will apply a simple homogeneity argument together with some comparison results about evolution kernels.

By Lemma 6.1,

$$\begin{aligned} \nu ^{s\alpha }(m,v,w)=\lim _{t\rightarrow \infty }\mathbf{E}_{s\alpha }^\sigma \check{P}^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}(0,0,0;m,v,w). \end{aligned}$$

(6.2)

Hence, in order to estimate $\nu ^{s\alpha }$ we need to estimate $P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}.$ The main results of this section are the following upper bounds for $\nu ^{s\alpha }.$

Theorem 6.2

Let $\alpha \in A^+,$ and let $K_3$ be a compact subset of $N_3$ such that $0\not \in N_3.$ Then, there is a constant $c=c_{\Lambda ,K_3,\alpha }>0$ such that for every $s<0,$ and all $w\in K_3\subset N_3,$

$$\begin{aligned} \nu ^{s\alpha }(0,0,w)\le c e^{-\rho _0(s\alpha )} e^{s\overline{\gamma }_{\Lambda _3}(\alpha )\gamma _{\Lambda _3}(\alpha )}. \end{aligned}$$

Theorem 6.3

Let $\alpha \in A^+,$ and let $K_2$ be a compact subset of $N_2$ such that $0\not \in N_2.$ Then, there is a constant $c=c_{\Lambda ,K_2,\alpha }>0$ such that for every $s<0,$ and all $v\in K_2\subset N_2,$

$$\begin{aligned} \nu ^{s\alpha }(0,v,0)\le c e^{-\rho _0(s\alpha )} e^{s\overline{\gamma }_{\Lambda _2\cup \Lambda _3}(\alpha ) \gamma _{\Lambda _2\cup \Lambda _3}(\alpha )}. \end{aligned}$$

6.3 Proofs of Theorems 6.2 and 6.3

We start with three lemmas. In all of them, the exponential functionals are on the interval $(0,\infty ),$ i.e., $A_{\star ,\star }^\sigma$ denotes $A_{\star ,\star }^\sigma (0,\infty ).$

Lemma 6.4

There is a constant $C>0$ such that for all $s<0$ and $\alpha \in A^+,$

$$\begin{aligned} \mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1}\le Ce^{-2\rho _{0}(s\alpha )} \end{aligned}$$

and

$$\begin{aligned} \mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1}A_{N_3,\Sigma }^\sigma \le Ce^{-2\rho _{0}(s\alpha )}. \end{aligned}$$

Proof

We have

$$\begin{aligned} \mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1}=e^{-2\rho _{0}(s\alpha )} \mathbf{E}_{0}^\sigma (A_{N,\Pi }^\sigma )^{-1}. \end{aligned}$$

The expected value $\mathbf{E}_{0}^\sigma (A_{N,\Pi }^\sigma )^{-1}$ is finite. This follows by applying Cauchy–Schwarz inequality succesively and the fact that exponential functionals $A_{N,j}^\sigma$ have negative moments (Lemma 2.3).

Similarly, since $s<0$ and $\alpha \in A^+,$

$$\begin{aligned} \mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1}A_{N_3,\Sigma }^\sigma&= \sum _{j=1}^{d_3}\mathbf{E}_{s\alpha }^\sigma \prod \limits _{\begin{array}{c} k=1\\ k\not =j \end{array}}^d(A_{N,k}^\sigma )^{-1}\\&= \sum _{j=1}^{d_3}e^{-2\rho _{0}(s\alpha )+\psi _j(s\alpha )} \mathbf{E}_{0}^\sigma \prod \limits _{\begin{array}{c} k=1\\ k\not =j \end{array}}^d(A_{N,k}^\sigma )^{-1}\\&\le e^{-2\rho _{0}(s\alpha )}\sum _{j=1}^{d_3} \mathbf{E}_{0}^\sigma \prod \limits _{\begin{array}{c} k=1\\ k\not =j \end{array}}^d(A_{N,k}^\sigma )^{-1}. \end{aligned}$$

Again, by Lemma 2.3, the expectations $\mathbf{E}_{0}^\sigma \prod \limits _{\begin{array}{c} k=1\\ k\not =j \end{array}}^d(A_{N,k}^\sigma )^{-1}$ for $j=1,\ldots ,d_3$ are finite. $\square$

The next two lemmas follows immediately from the proof of [17, Lemma 6.2] and the inequality (6.3) on p. 269 in [17].

Lemma 6.5

Let $\alpha \in A^+.$ For every $\beta >0$, there is a constant $c=c_{\Lambda _3,\alpha ,\beta }>~0$ such that for every $s<0,$

$$\begin{aligned} \mathbf{E}_{s\alpha }^\sigma e^{-\beta \slash A_{N_3,\Sigma }^\sigma }\le ce^{2s\overline{\gamma }_{\Lambda _3}(\alpha )\gamma _{\Lambda _3}(\alpha )}. \end{aligned}$$

Lemma 6.6

Let $\alpha \in A^+.$ For every $\beta >0$, there is a constant $c=c_{\Lambda _2,\Lambda _3,\alpha ,\beta }>0$ such that for every $s<0,$

$$\begin{aligned} \mathbf{E}_{s\alpha }^\sigma e^{-\beta \slash (A_{N_2,\Sigma }^\sigma +A_{N_3,\Sigma }^\sigma )}\le ce^{2s\overline{\gamma }_{\Lambda _2\cup \Lambda _3} (\alpha )\gamma _{\Lambda _2\cup \Lambda _3}(\alpha )}. \end{aligned}$$

Proof of Theorem 6.2

By Theorem 5.1,

$$\begin{aligned} P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}(0,0,0;0,0,w) \le CA_{N_1,\Pi }^\sigma (0,t)^{-1\slash 2}P^{N_2\rtimes N_3,\sigma }_{t,0}(0,w). \end{aligned}$$

From Theorem 3.1, since $0\not \in K_3$ there is a constant $c>0$ such that for all $w\in K_3,$

$$\begin{aligned}&P^{N_2\rtimes N_3,\sigma }_{t,0}(0,w)\le \\&CA_{N_2,\Pi }^{\sigma }(0,t)^{-1\slash 2}A_{N_3,\Pi }^{\sigma }(0,t)^{-1\slash 2}\left( 1+A_{N_3,\Sigma }^\sigma (0,t)^{1\slash 2}\right) e^{-c\slash A_{N_3,\Sigma }^\sigma (0,t)}. \end{aligned}$$

Consequently, for $w\in K_3,$

$$\begin{aligned}&P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}(0,0,0;0,0,w)\nonumber \\&\quad \le CA_{N_3,\Pi }^{\sigma }(0,t)^{-1\slash 2}\left( 1+A_{N_3,\Sigma }^\sigma (0,t)^{1\slash 2}\right) e^{-c\slash A_{N_3,\Sigma }^\sigma (0,t)}. \end{aligned}$$

(6.3)

Using (6.2), we get that for all $w\in K_3,$ (note that below the range of integration in all functionals is $(0,+\infty )$)

$$\begin{aligned} \nu ^{s\alpha }(0,0,w)&\le C\mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1\slash 2}e^{-c\slash A_{N_3,\Sigma }^\sigma }\nonumber \\&+\,\,C\mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1\slash 2}(A_{N_3,\Sigma }^\sigma )^{1\slash 2}e^{-c\slash A_{N_3,\Sigma }^\sigma }. \end{aligned}$$

(6.4)

By the Cauchy–Schwarz inequality

$$\begin{aligned} \nu ^{s\alpha }(0,0,w)&\le C\left( \mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1 }\right) ^{1\slash 2} \left( \mathbf{E}_{s\alpha }^\sigma e^{-2c\slash A_{N_3,\Sigma }^\sigma }\right) ^{1\slash 2}\\&\quad +C\left( \mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1 }A_{N_3,\Sigma }^\sigma \right) ^{1\slash 2}\left( \mathbf{E}_{s\alpha }^\sigma e^{-2c\slash A_{N_3,\Sigma }^\sigma }\right) ^{1\slash 2}. \end{aligned}$$

Now Theorem 6.2 follows from Lemma 6.4 and Lemma 6.5. $\square$

Proof of Theorem 6.3

By Theorem 5.1,

$$\begin{aligned} P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}(0,0,0;0,v,0) \le CA_{N_1,\Pi }^\sigma (0,t)^{-1\slash 2}P^{N_2\rtimes N_3,\sigma }_{t,0}(v,0). \end{aligned}$$

(6.5)

From Theorem 3.1 with $w=0,$ since $0\not \in K_2$ there is a constant $c>0$ such that for all $v\in K_2,$

$$\begin{aligned}&A_{N_2,\Pi }^{\sigma }(0,t)^{1\slash 2}A_{N_3,\Pi }^{\sigma }(0,t)^{1\slash 2}P^{N_2\rtimes N_3,\sigma }_{t,0}(v,0)\\&\qquad \le C e^{-c_1\slash A_{N_2,\Sigma }^\sigma (0,t)} +CA_{N_3,\Sigma }^\sigma (0,t)^{1\slash 2}e^{-c_2\slash A_{N_3,\Sigma }^\sigma (0,t)}\\&\qquad \le C(1+A_{N_3,\Sigma }^\sigma (0,t)^{1\slash 2})e^{-c\slash (A_{N_2,\Sigma }^\sigma (0,t)+A_{N_3,\Sigma }^\sigma (0,t))}. \end{aligned}$$

As in the proof of Theorem 6.2, we conclude that on $K_2,$ (the range of integration in all functionals below is $(0,+\infty )$)

$$\begin{aligned} \nu ^{s\alpha }(0,v,0)&\le C\mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1\slash 2}e^{-c\slash (A_{N_2,\Sigma }^\sigma (0,t)+A_{N_3,\Sigma }^\sigma (0,t))}\\&+\,C\mathbf{E}_{s\alpha }^\sigma (A_{N,\Pi }^\sigma )^{-1\slash 2}(A_{N_3,\Sigma }^\sigma )^{1\slash 2}e^{-c\slash (A_{N_2,\Sigma }^\sigma (0,t)+A_{N_3,\Sigma }^\sigma (0,t))}. \end{aligned}$$

This together with Lemma 6.4 and Lemma 6.6 finish the proof. $\square$

6.4 Upper bound for the Poisson kernel $\nu$

Having Theorem 5.1, Theorem 6.2, and Theorem 6.3, we are ready to prove the estimate for the Poisson kernel.

Proof of Theorem 1.1

By continuity of $\nu$, there is a constant $C_\alpha >0$ such that for all $x\in N$ with the norm $|x|_\alpha \le 1$, we have $\nu (x)\le C_{\alpha }.$

Consider $x\in N$ with $|x|_\alpha >1.$ Let $\delta _t^\alpha ={{\mathrm{Ad}}}((\log t)\alpha ).$ Then, $|\delta _t^\alpha x|_\alpha =t|x|_\alpha .$ We write $x$ as $x=\delta ^\alpha _{\exp (-s)}x_o$ with $|x_o|_\alpha =1$ and $s<0.$ Then, $|x|_\alpha =e^{-s}>1.$ Let $K(1)=\{x_o:\;|x_o|_\alpha =1\}.$ By definition (6.1) of $\nu ^{s\alpha },$ we get

$$\begin{aligned} \nu (x)=\nu (\delta ^\alpha _{\exp (-s)}x_o)=\nu ((s\alpha )^{-1}x_o(s\alpha )) =e^{\rho _0(s\alpha )}\nu ^{s\alpha }(x_o), \end{aligned}$$

(6.6)

where $\rho _0=\sum _{j=1}^d\lambda _j.$ Now, estimates (1.10) and (1.11) follow from Theorems 6.3 and 6.2, respectively, if we apply (6.6) to $x=(0,v,0)$ and $x=(0,0,w),$ respectively. Then, $x_o=(0,v_o,0)$ and $x_o=(0,0,w_o).$ Estimate (1.9) is a consequence of (1.12). Finally, in order to prove (1.8), we proceed as follows.

Let $\mathcal U_t^\sigma (v,w),$ $(v,w)\in N_2\rtimes N_3,$ be the estimate for the evolution kernel $P^{N_2\rtimes N_3,\sigma }_{t,0}$ $(0,0;v,w)$ given by Theorem 3.1.

By Theorem 5.1,

$$\begin{aligned} P^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}(0,0,0;m,v,w)&\le CA_{N_1,\Pi }^\sigma (0,t)^{-1\slash 2}P^{N_2\rtimes N_3,\sigma }_{t,0}(v,w)\nonumber \\&\le CA_{N_1,\Pi }^\sigma (0,t)^{-1\slash 2}\mathcal U_t^\sigma (v,w). \end{aligned}$$

(6.7)

Note that $\check{P}^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}(0,0,0;m,v,w)$ has the same estimate. Now we consider $(m,v,w)$ in a compact sets $K(1)$ (clearly, $0\not \in K(1)$). By Lemma 6.1 and (6.7),

$$\begin{aligned} \nu ^{s\alpha }(m,v,w)&= \lim _{t\rightarrow \infty }\mathbf{E}_{s\alpha }^\sigma \check{P}^{N_1\rtimes N_2\rtimes N_3,\sigma }_{t,0}(0,0,0;m,v,w)\nonumber \\&\le C\mathbf{E}_{s\alpha }^\sigma A_{N_1,\Pi }^\sigma (0,\infty )^{-1\slash 2}\mathcal U^\sigma _\infty (v,w). \end{aligned}$$

(6.8)

Notice that it follows from Theorem 3.1 that there is $c>0$ such that for all $(u,w)\in K(1)\cap (N_2\rtimes N_3),$ and all $t>0,$

$$\begin{aligned} \mathcal U_t^\sigma (v,w)\le c\mathcal U_t^\sigma (0,w)\quad \text {and}\quad \mathcal U_t^\sigma (v,w)\le c\mathcal U_t^\sigma (v,0). \end{aligned}$$

Applying the above inequalities to (6.8), we get

$$\begin{aligned}&\nu ^{s\alpha }(m,v,w)\le \mathbf{E}_{s\alpha }^\sigma A_{N_1,\Pi }^\sigma (0,\infty )^{-1\slash 2}\mathcal U_\infty ^\sigma (0,w)\\&\qquad \text { and }\nu ^{s\alpha }(m,v,w)\le \mathbf{E}_{s\alpha }^\sigma A_{N_1,\Pi }^\sigma (0,\infty )^{-1\slash 2}\mathcal U_\infty ^\sigma (v,0). \end{aligned}$$

In fact, the quantities

$$\begin{aligned} \mathbf{E}_{s\alpha }^\sigma A_{N_1,\Pi }^\sigma (0,\infty )^{-1\slash 2}\mathcal U_\infty ^\sigma (0,w)\quad \text {and}\quad A_{N_1,\Pi }^\sigma (0,\infty )^{-1\slash 2}\mathbf{E}_{s\alpha }^\sigma \mathcal U_\infty ^\sigma (v,0) \end{aligned}$$

are estimated in the proofs of Theorem 6.2 and Theorem 6.3, respectively. Therefore, we have that for every $\varepsilon >0$, there exists a constant $c=c_{\varepsilon ,\Lambda ,\alpha }$ such that for all $(m,v,w)\in K(1),$

$$\begin{aligned} \nu ^{s\alpha }(m,v,w)\le ce^{-\rho _0(s\alpha )} e^{s\overline{\gamma }_{\Lambda _3}(\alpha )\gamma _{\Lambda _3} (\alpha )}\text { if }\Vert w\Vert \ge \varepsilon >0 \end{aligned}$$

and

$$\begin{aligned} \nu ^{s\alpha }(m,v,w)\le c e^{-\rho _0(s\alpha )} e^{s\overline{\gamma }_{\Lambda _2\cup \Lambda _3}(\alpha ) \gamma _{\Lambda _2\cup \Lambda _3}(\alpha )}\text { if }\Vert v\Vert \ge \varepsilon >0. \end{aligned}$$

Thus by the homogeneity (6.6),

$$\begin{aligned} \nu (m,v,w)\le c(1+|(m,v,w)|_\alpha )^{-\overline{\gamma }_{\Lambda _3} (\alpha )\gamma _{\Lambda _3}(\alpha )}\text { if }\Vert w_o\Vert \ge \varepsilon >0 \end{aligned}$$

and

$$\begin{aligned} \nu (m,v,w)\le c(1+|(m,v,w)|_\alpha )^{-\overline{\gamma }_{\Lambda _2 \cup \Lambda _3}(\alpha )\gamma _{\Lambda _2\cup \Lambda _3}(\alpha )}\text { if }\Vert v_o\Vert \ge \varepsilon >0. \end{aligned}$$

The inequality (1.8) follows. $\square$