Boundary behaviour of $\lambda$-polyharmonic functions on regular trees

TL;DR

This paper investigates the boundary behavior of complex $\lambda$-polyharmonic functions on regular trees, solving boundary value problems and establishing a Fatou theorem for these functions.

Contribution

It provides new results on boundary limits and solves classical boundary value problems for $\lambda$-polyharmonic functions on regular trees.

Findings

Solved Dirichlet and Riquier problems at infinity.

Established a non-tangential Fatou theorem.

Analyzed boundary behavior for $|\lambda|> ho$.

Abstract

This paper studies the boundary behaviour of $\lambda$-polyharmonic functions for the simple random walk operator on a regular tree, where $\lambda$ is complex and $|\lambda|> \rho$, the $\ell^2$-spectral radius of the random walk. In particular, subject to normalisation by spherical, resp. polyspherical functions, Dirichlet and Riquier problems at infinity are solved and a non-tangential Fatou theorem is proved.

Equations

$$f(z)=\sum _{k=0}^{n-1}|z{|}^{2k}{h}_k(z),$$

$$\mathbb{D}=\{z=x+\mathfrak{i}y\in \mathbb{C}:|z|=\sqrt{x^2+y^2}<1\},$$

$$f(z)=\sum _{k=0}^{n-1}{\int }_{\partial \mathbb{D}}|z{|}^{2k}P(z,\xi )d{\nu }_k(\xi ),$$

$$\theta (z,w)=\{\begin{array}{ll}{q}^{-|z\wedge w|},& \text{if }z\ne w,\\ 0,& \text{if }z=w.\end{array}$$

$${\hat{T}}_{x,y}=\{z\in \hat{T}:y\in \pi (x,z)\}.$$

$$g=\sum _{j=1}^n{c}_j{1}_{\partial T_{x_j}}$$

$$\nu :\{\partial T_x:x\in T\}\to \mathbb{C}\text{with}\nu (\partial T_x)=\sum _{y:y^{-}=x}\nu (\partial T_y)\text{for all }x.$$

$${\int }_{\partial T}gd\nu =\sum _{j=1}^n{c}_j\nu (\partial T_{x_j}).$$

$$Pf(x)=\frac{1}{q+1}\sum _{y:y\sim x}f(y),$$

$$F(x,y|\lambda )=\sum _{n=1}^{\infty }{f}^{(n)}(x,y)/{\lambda }^n,$$

$$\begin{gathered}F(x,y|\lambda)=F(\lambda)^{d(x,y)},\quad\text{where}\\ F(\lambda)=\frac{q+1}{2q}\bigl{(}\lambda-s(\lambda)\bigr{)}\quad\text{with}\quad s(\lambda)=\lambda\,\sqrt{1-\rho^{2}/\lambda^{2}}\,,\end{gathered}$$

$$K(x,\xi |\lambda )=\frac{F(x,x\wedge \xi |\lambda )}{F(o,x\wedge \xi |\lambda )}=F(\lambda {)}^{\mathfrak{h}(x,\xi )},x\in T,\xi \in \partial T,$$

$$\mathfrak{h}(x,\xi )=d(x,x\wedge \xi )-d(o,x\wedge \xi )$$

$$h(x)={\int }_{\partial T}K(x,\xi |\lambda )d\nu (\xi ),$$

$$f(x)=\sum _{k=0}^{n-1}{\int }_{\partial T}K(x,\xi |\lambda ){\mathfrak{h}}_k(x,\xi |\lambda )d{\nu }_k(\xi )\text{with}{\mathfrak{h}}_k(x,\xi |\lambda )=\frac{\mathfrak{h}(x,\xi {)}^k}{k!s(\lambda {)}^k},$$

$${\phi }_0(\lambda )=1,{\phi }_1(\lambda )=\lambda ,\text{and}\lambda {\phi }_k(\lambda )=\frac{1}{q+1}{\phi }_{k-1}(\lambda )+\frac{q}{q+1}{\phi }_{k+1}(\lambda )\text{for }k\ge 1.$$

$$\widetilde{F}(\lambda)=\frac{q+1}{2q}\bigl{(}\lambda+s(\lambda)\bigr{)}$$

$$\lambda F(\lambda )=\frac{1}{q+1}+\frac{q}{q+1}F(\lambda {)}^2.$$

$$\begin{array}{c}\Phi (x|\lambda )=a(\lambda )F(\lambda {)}^{|x|}+\tilde{a}(\lambda )\tilde{F}(\lambda {)}^{|x|},\text{where}\\ a(\lambda )=\frac{s(\lambda )-\frac{q-1}{q+1}\lambda }{2s(\lambda )}\text{and}\tilde{a}(\lambda )=\frac{s(\lambda )+\frac{q-1}{q+1}\lambda }{2s(\lambda )}.\end{array}$$

$$0<|F(\lambda)|<1\big{/}\sqrt{q}<|\widetilde{F}(\lambda)|\,,\quad F(1)=1/q\,,\quad\text{and}\quad\widetilde{F}(1)=1.$$

$$\Phi (x|1)=1\text{and}\Phi (x|\lambda )\ne 0\text{for all }x\in T.$$

$$\frac{q+1}{2q}\bigl{(}\lambda\pm\mathfrak{i}\,\sqrt{\rho^{2}-\lambda^{2}}\bigr{)},$$

$$\bigl{|}a(\lambda)\,F(\lambda)^{|x|}\bigr{|}<\bigl{|}\tilde{a}(\lambda)\,\widetilde{F}(\lambda)^{|x|}\bigr{|}$$

$$\mathsf{m}(\partial T_{x})=\begin{cases}1\,,&\text{if }\;x=o\,,\\ 1\big{/}\bigl{(}(q+1)q^{|x|-1}\bigr{)}\,,&\text{if }\;x\neq o\,.\end{cases}$$

$$\Phi (x|\lambda )={\int }_{\partial T}K(x,\xi |\lambda )d\xi .$$

$${\Phi }_n(x|\lambda )={\int }_{\partial T}K(x,\xi |\lambda ){\mathfrak{h}}_n(x,\xi |\lambda )d\xi .$$

$$\frac{(-1{)}^n}{n!}(\lambda \cdot I-P{)}^n{K}^{(n)}(\cdot ,\xi |\lambda )=K(\cdot ,\xi |\lambda ).$$

$$K^{(n)}(x,\xi |\lambda )=K(x,\xi |\lambda )\sum _{k=1}^n\mathfrak{h}(x,\xi {)}^k{g}_{k,n}(\lambda ),$$

$$g_{n,n}(\lambda )=(-1{)}^{n}s(\lambda {)}^{-n}.$$

$$(\lambda \cdot I-P{)}^n{\Phi }_n(\cdot |\lambda )=\Phi (\cdot |\lambda ).$$

Full text

Supported by Austrian Science Fund projects FWF P31237 and W1230. The second author acknowledges the hospitality of Marc Peigné and Kilian Raschel at Institut Denis-Poisson, Université de Tours, France and support in the framework of K.R.’s starting grant from the European Research Council (ERC) under the Grant Agreement No759702.

1. Introduction

A complex-valued function $f$ on a Euclidean domain $D$ is called polyharmonic of order $n$, if it satisfies $\Delta^{n}f\equiv 0$, where $\Delta$ is the classical Euclidean Laplacian. The study of polyharmonic functions originates in work of the $19^{\text{th}}$ century, and is pursued very actively. Basic references are the books by Aronszajn, Creese and Lipkin [2] and by Gazzola, Grunau and Sweers [8] .

A classical theorem of Almansi [1] says that if the domain $D$ is star-like with respect to the origin, then every polyharmonic function of order $n$ has a unique decomposition

[TABLE]

where each $h_{k}$ is harmonic on $D$, and $|z|$ is the Euclidean length of $z\in D$. In particular, if the domain is the unit disk

[TABLE]

then thanks to a Theorem of Helgason [9], Almansi’s decomposition can be written as an integral representation over the boundary $\partial\mathbb{D}$ of the disk, that is, the unit circle, with respect to the Poisson kernel $P(z,\xi)=(1-|z|^{2})/|\xi-z|^{2}\;(z\in\mathbb{D}\,,\;\xi\in\partial\mathbb{D})$. Namely,

[TABLE]

where $\nu_{0},\dots,\nu_{n-1}$ are certain distributions, namely analytic functionals on the unit circle. For details on those functionals, see e.g. the nice exposition by Eymard [7].

A smaller body of work is available on the discrete counterpart, where the Laplacian is a difference operator arising from a reversible Markov chain transition matrix on a graph. Regarding boundary integral representations comparable to (1), Cohen et al. [5] have provided such a result concerning polyharmonic functions for the simple random walk operator on a homogeneous tree. This has recently been generalised by Picardello and Woess [12] to arbitrary nearest neighbour transition operators on arbitrary trees which do not need to be locally finite: [12] provides a boundary integral representation for $\lambda$-polyharmonic functions for suitable complex $\lambda$.

Here we come back to the specific situation of simple random walk on the homogeneous tree $T$ with degree $q+1$, where $q\geq 2$. The necessary preliminaries are outlined in §2. For the transition operator $P$ of the simple random walk on $T$, we study in more detail the boundary behaviour of $\lambda$-polyharmonic functions, that is, $f:T\to\mathbb{C}$ such that $(\lambda\cdot I-P)^{n}f=0$. We assume that $\lambda\in\mathbb{C}\setminus[-\rho\,,\,\rho]$, where $\rho$ is the $\ell^{2}$-spectral radius of $P$ and $[-\rho\,,\,\rho]$ is its $\ell^{2}$-spectrum. Close to the spirit of Korányi and Picardello [11], we extend their results from $\lambda$-harmonic to $\lambda$-polyharmonic functions, and results of the abovementioned work [5] from ordinary polyharmonic functions, i.e. $\lambda=1$, to general complex $\lambda$ in the $\ell^{2}$-resolvent set of $P$.

First, we consider higher order analogues of the Dirichlet problem at infinity: in the classical case $\lambda=1$, one takes any continuous function $g$ on the boundary at infinity $\partial T$ of $T$ and provides a harmonic function on $T$ which provides a continuous extension of $g$ to the compactification $\widehat{T}=T\cup\partial T$. It is given by the (analogue of the) Poisson transform of $g$ with respect to the Martin kernel.

However, for $\lambda$-polyharmonic functions of higher order, as well as for $\lambda$-harmonic functions with $\lambda\neq 1$, this needs an additional normalisation, in order to control the Poisson-Martin transforms with respect to the $\lambda$-Martin kernel (and its higher order versions) at infinity. The normalisation is by spherical functions and their higher order analogues, the polyspherical functions. They are introduced in §3, where we also study their asymptotic behaviour at infinity, see Proposition 3.5.

The first two main results are given by the “twin” theorems 4.1 and 4.6 in §4. The (analogue of the) Poisson integral of $g$ with respect to the $n^{\text{th}}$ extension of the $\lambda$-Martin kernel (i.e., the kernel multiplied by the – suitably normalised – $n^{\text{th}}$ power of the Busemann function) is polyharmonic of order $n+1$, and normalised (= divided) by the $n^{\text{th}}$ polyspherical function, it converges to $g$ at the boundary. Next, Theorem 4.6 concerns Fatou type non-tangential convergence of polyharmonic extensions of complex Borel measures on the boundary.

In general, the polyharmonic extension of a continuous boundary function cannot be unique because one may add lower order polyharmonic functions that do not change the limit. However, uniqueness is proved in the case of $\lambda$-harmonic functions ($n=1$), see Theorem 4.7. That is, normalising by the associated spherical function, the solution of the $\lambda$-Dirichlet problem at infinity is unique. Note that since $\lambda$ is in general complex, typical tools from Potential Theory such as the maximum principle cannot be applied here, and are replaced by a new idea, using spherical averages.

As a corollary of these results, a tree-counterpart of the Riquier problem at infinity is provided. In the case of a bounded Euclidean domain $D$ as above, this consists in providing continuous boundary functions $g_{0}\,,\dots,g_{n-1}$ and looking for a polyharmonic function $f$ of order $n$ on $D$ such that $\Delta^{k}f$ is a continuous extension of $g_{k}$ for each $k$. For finite graphs, the analogous problem has been studied in a note by Hirschler and Woess [10], where one can find further references concerning the discrete setting. In the case of $\lambda$-harmonic functions on $T$, the formulation of the analogous problem requires again suitable normalisation, see Definition 4.9 and Corollary 4.10.

2. Homogeneous trees and boundary integral representations

Let $T=T_{q}$ be the homogeneous tree where each vertex has $q+1\geq 3$ neighbours. We need some features of its structure and first recall the well known boundary $\partial T$ of the tree. For $x,y\in T$, there is a unique geodesic path $\pi(x,y)=[x=x_{0}\,,\,x_{1}\,,\dots,x_{n}=y]$ of minimal length $n$, such that $x_{k-1}\sim x_{k}$ for $x=1,\dots,n$, and $d(x,y)=n$ is the graph distance between $x$ and $y$. A geodesic ray is a sequence $[x_{0}\,,\,x_{1}\,,x_{2}\,,...]$ of distinct vertices with $x_{n-1}\sim x_{n}$. Two rays are equivalent if they share all but finitely many among their vertices. An end of $T$ is an equivalence class of geodesic rays, and $\partial T$ is the set of all ends. For any $\xi\in\partial T$ and $x\in T$, there is a unique geodesic $\pi(x,\xi)$ which starts at $x$ and represents $\xi$. Next, we choose a root vertex $o\in T$. We set $\widehat{T}=T\cup\partial T$. For any pair of points $z,w\in\widehat{T}$, their confluent $z\wedge w$ is the last common vertex on the finite or infinite geodesics $\pi(o,w)$ an $\pi(o,z)$, unless $z=w$ is an end, in which case $z\wedge z=z$. Furthermore, for a vertex $x\neq o$, we define its predecessor $x^{-}$ as the neighbour of $x$ on the arc $\pi(o,x)$.

We now equip $\widehat{T}$ with a new metric: we set $|x|=d(x,o)$ for $x\in T$, and let

[TABLE]

This is an ultra-metric which turns $\widehat{T}$ into a compact space with $T$ as an open, discrete and dense subset. A basis of the topology is given by all branches $\widehat{T}_{x,y}\,$, where $x,y\in T$ with $x\neq y$. Here,

[TABLE]

This is a compact-open set, and its boundary $\partial T_{x,y}=\widehat{T}_{x,y}\cap\partial T$ is called a boundary arc. As a matter of fact, a basis of the topology of $\partial T$ is given by the collection of all $\partial T_{x}:=\partial T_{o,x}\,$, including $\partial T_{o}:=\partial T$. A locally constant function on $\partial T$ is a finite linear combination

[TABLE]

of indicator functions of boundary arcs. It can equivalently be written in terms of boundary arcs $\partial T_{x,y_{k}}$ for any fixed vertex $x$. A distribution on $\partial T$ is an element of the dual of the linear space of locally constant functions. Equivalently, it can be written as a finitely additive measure $\nu$ on the collection of all boundary arcs. For this it suffices to consider only the boundary arcs with respect to $o$, so that $\nu$ is characterised as a set function

[TABLE]

For $g$ as above, we write $\nu(g)$ as an integral

[TABLE]

When $\nu$ is non-negative real, compactness yields immediately that it extends to a $\sigma$-additive measure on the Borel $\sigma$-algebra of $\partial T$. In general, $\nu$ does not necessarily extend to a $\sigma$-additive complex measure; see Cohen, Colonna and Singman [6].

We now turn to harmonic functions. For a function $f:T\to\mathbb{C}$, we define

[TABLE]

where $y\sim x$ means that the vertices $x,y\in T$ are neighbours. $P$ is the transition operator of the simple random walk on $T$. We recall the very well known fact that as a self-adjoint operator on the space $\ell^{2}(T)$, its spectrum is the interval $[-\rho\,,\,\rho]$, where $\rho=2\sqrt{q}/(q+1)$. In this setting, the discrete counterpart of the Laplacian is $P-I$, where $I$ is the identity operator.

Definition 2.1**.**

For $\lambda\in\mathbb{C}$, a $\lambda$-polyharmonic function of order $n$ is a function $f:T\to\mathbb{C}$ such that $(\lambda\cdot I-P)^{n}f=0$.

For $n=1$, it is called $\lambda$-harmonic, and when $\lambda=1$, we speak of a polyharmonic, resp. harmonic function.

Following [12], for a suitable boundary integral representation, the “eigenvalue” $\lambda$ should belong to the resolvent set $\mathsf{res}(P)=\mathbb{C}\setminus[-\rho\,,\,\rho]$ of $P$ on $\ell^{2}(T)$. In this case, let $G(x,y|\lambda)=(\lambda\cdot I-P)^{-1}\delta_{y}(x)$ be the Green function, that is, the $(x,y)$-matrix element of the resolvent, where $x,y\in T$. By [12, Thm. 4.2], or by direct computation, $G(x,x|\lambda)\neq 0$, and we can define $F(x,y|\lambda)=G(x,y|\lambda)/G(x,x|\lambda)$. These functions depend only on the graph distance $d(x,y)$ between $x$ and $y$.

For $|\lambda|\geq\rho$, one has a combinatorial-probabilistic interpretation:

[TABLE]

where $f^{(n)}(x,y)$ is the probability that the simple random walk starting at $x$ hits $y$ at the $n^{\text{th}}$ step for the first time. Simple and well-known computations yield

[TABLE]

see e.g. [13, Lemma 1.24] (with $z=1/\lambda$). The complex square root is $\sqrt{re^{i\phi}}=\sqrt{r}e^{i\phi/2}$ for $\phi\in(-\pi\,,\,\pi)$.

The $\lambda$-Martin kernel on $T\times\partial T$ is

[TABLE]

where

[TABLE]

is the Busemann function or horocycle index of $x$ with respect to the end $\xi$. Note that for fixed $x$, the function $\xi\mapsto K(x,\xi|\lambda)$ is locally constant.

Now a basic result in the seminal paper of Cartier [4], valid for real $\lambda\geq\rho$, and its extension to complex $\lambda\in\mathsf{res}(P)$ [12] says the following for simple random walk on $T$.

For $\lambda\in\mathbb{C}\setminus[-\rho\,,\,\rho]$, every $\lambda$-harmonic function $h$ on $T$ has a unique integral representation

[TABLE]

where $\nu$ is a distribution on $\partial T$ as in (3). If $\lambda>\rho$ and $h>0$ then $\nu$ is a positive Borel measure. Indeed, this holds for arbitrary nearest neighbour random walks on arbitrary countable trees, and [12] has a method to extend this to a boundary integral representation of $\lambda$-polyharmonic functions. Specialised to simple random walk on $T=T_{q}\,$, this yields the following extension of a result of [5], where the basic case $\lambda=1$ is considered.

Theorem 2.2**.**

[12]** For $\lambda\in\mathbb{C}\setminus[-\rho\,,\,\rho]$, every $\lambda$-polyharmonic harmonic function $f$ of order $n$ on $T$ has a unique integral representation

[TABLE]

where $\nu_{0}\,,\dots,\nu_{n-1}$ are distributions on $\partial T$.

The normalisation by $k!\,s(\lambda)^{k}$, where $s(\lambda)$ is as in (5), is not present in [12, Cor. 5.4]. We shall see below in Lemma 3.4 why it is useful.

3. Polyspherical functions

Definition 3.1**.**

For any $\lambda\in\mathbb{C}$, the spherical function $\Phi(x|\lambda)$ is the unique function on $T$ with $\Phi(o|\lambda)=1$ which is $\lambda$-harmonic and radial, i.e., it depends only on $|x|=d(o,x)$.

Namely, if we set $\varphi_{k}(\lambda)=\Phi(x|\lambda)$ for $|x|=k$, then we have the recursion

[TABLE]

We shall consider the case when $\lambda$ is in the $\ell^{2}$-resolvent set of $P$, that is, $\lambda\in\mathbb{C}\setminus[\rho\,,\,\rho]$. Let $F(\lambda)$ be as in (5), and let

[TABLE]

be the second solution, besides $F(\lambda)$, of the equation

[TABLE]

Then one can solve the above recursion, and

[TABLE]

We collect a few elementary properties.

Lemma 3.2**.**

We have for $\lambda\in\mathbb{C}\setminus[\rho\,,\,\rho]$

[TABLE]

Furthermore,

[TABLE]

Proof.

First of all, by (8), $F(\lambda)\neq 0$ and $F(\lambda)\widetilde{F}(\lambda)=1/q$. Next, by (4), |F(\lambda)|\leq F(|\lambda|)<F(\rho)=1\big{/}\sqrt{q}\, for $|\lambda|>\rho$. Also when $|\lambda|=\rho$ and $\lambda\neq\pm\rho$, we have |F(\lambda)|<F(\rho)=1\big{/}\sqrt{q}\,. At last, for $\lambda$ in the real interval $(-\rho\,,\rho)$, the limits of $F(\cdot)$ are

[TABLE]

according to whether $\lambda$ is approached within the upper or lower half plane. Thus, in the upper open semidisk $\{z\in\mathbb{C}:|z|<\rho\,,\;\Re z>0\}$, as well as in the corresponding lower open semidisk, $F(\lambda)$ is analytic, and its absolute values at the boundary are \leq 1\big{/}\sqrt{q}\,. By the Maximum Modulus Principle, $|F(\lambda)|<1/\sqrt{q}\,$ within each of those two semidisks. We see that the last inequality holds in all of $\mathbb{C}\setminus[\rho\,,\,\rho]$.

Consequently, $|q\widetilde{F}(\lambda)|=1/|F(\lambda)|>\sqrt{q}$. The values for $\lambda=1$ are obvious.

Finally, we claim that for the coefficient functions in (9) one has $|a(\lambda)|<|\tilde{a}(\lambda)|$. For $|\lambda|>\rho$, as well as for $|\lambda|=\rho$ and $\lambda\neq\pm\rho$, one can see this from the fact that $1-\rho^{2}/\lambda^{2}$ belongs to the complex half-plane with positive real part. For $\lambda$ in one of the above two semidisks, one can proceed as above: one checks that $\tilde{a}(\lambda)\neq 0$. Then the function $a(\lambda)/\tilde{a}(\lambda)$ is analytic in each semidisk, with boundary values whose absolute values are $\leq 1$, and the desired inequality follows. Therefore

[TABLE]

for every $x\in T$, and $\Phi(x|\lambda)\neq 0$. ∎

We can describe the spherical functions via their integral representation (6). Let $\mathsf{m}$ stand for the uniform distribution on $\partial T$. This is the Borel probability measure which for each $k\in\mathbb{N}_{0}$ assigns equal mass to all boundary arcs $\partial T_{x}\,$, where $x\in T$ with $|x|=k$. That is,

[TABLE]

We shall often write $d\mathsf{m}(\xi)=d\xi\,$. Then

[TABLE]

Indeed, the right hand side satisfies all requirements of Definition 3.1, which determine the spherical function. A comparison with Theorem 2.2 leads us to the following.

Definition 3.3**.**

For $n\geq 0$, the $n^{\text{th}}$ polyspherical function is

[TABLE]

It is $\lambda$-polyharmonic of order $n+1$, and it is radial. With respect to those two properties, it is uniquely determined by its values for $|x|=0,1,\dots,n$. For $n\geq 1$, its value at $x=o$ is [math]. For $n=0$ it is of course the spherical function (10).

In particular, $(\lambda\cdot I-P)^{n}\,\Phi_{n}(\cdot|\lambda)$ is $\lambda$-harmonic and radial, so that it must be a multiple of $\Phi(\cdot|\lambda)$. In order to determine the factor, we need to recall part of how Theorem 2.2 was obtained in [12]. Let $K^{(n)}(x,\xi|\lambda)$ be the $n^{\textrm{th}}$ derivative of $K(x,\xi|\lambda)$ with respect to $\lambda$. Then

[TABLE]

In [12, equation (5.2)], it is shown that

[TABLE]

where the functions $g_{k,n}(\lambda)$ are given recursively; in particular, with $s(\lambda)$ as in (5),

[TABLE]

Combining (11) and (12), we get

Lemma 3.4**.**

(\lambda\cdot I-P)^{n}\bigl{[}K(\cdot,\xi|\lambda)\,\mathfrak{h}_{n}(x,\xi|\lambda)\bigr{]}=K(\cdot,\xi|\lambda)\,.**

Integrating with respect to $d\xi$, we also obtain the following.

[TABLE]

We shall need the asymptotic behaviour of $\Phi_{n}(x|\lambda)$ as $|x|\to\infty$.

Proposition 3.5**.**

Let $\lambda\in\mathbb{C}$ with $|\lambda|>\rho$. Then, as $|x|\to\infty$,

[TABLE]

with $\tilde{a}(\lambda)$ given by (9). In particular, in the standard case $\lambda=1$, we have $\widetilde{F}(1)=\tilde{a}(1)=1$.

Therefore there is $R=R_{n,\lambda}>0$ such that

[TABLE]

Furthermore,

[TABLE]

Proof.

By Lemma 3.2,

[TABLE]

Now let $x\in T\setminus\{o\}$. For $\ell\in\{0,1,\dots,|x|\}$, let $A_{\ell}=\{\xi\in\partial T:|x\wedge\xi|=\ell\}$, and set $\mathsf{m}_{\ell}=m(A_{\ell})$. Then

[TABLE]

We use F(\lambda)=\bigl{(}q\widetilde{F}(\lambda)\bigr{)}^{-1} and set $k=|x|-\ell$. Then the integral formula of Definition 3.3 translates into

[TABLE]

The last term within the big parentheses tends to [math] as $|x|\to\infty\,$. Decompose the sum into the two pieces where in the first one, summation is over $1\leq k\leq\sqrt{|x|}$ and in the second one, summation is over $k>\sqrt{|x|}$. Then the second part is a remainder of a convergent series, so that it also tends to [math] as $|x|\to\infty$. Now, in the range $1\leq k\leq\sqrt{|x|}$, the quotients $(|x|-2k)/|x|$ tend to $1$ uniformly as $|x|\to\infty$. Therefore the first part of the sum converges to

[TABLE]

as $|x|\to\infty$. This yields the proposed asymptotic formula, with some elementary computations for getting the factor $\tilde{a}(\lambda)$. ∎

4. Dirichlet, Riquier and Fatou type convergence

In the classical case of harmonic functions, that is, when $\lambda=1$, the Dirichlet problem asks whether for any real or complex valued function $g\in\mathcal{C}(\partial T)$, there is a continuous extension to $\widehat{T}$ which is harmonic in $T$. That is, we look for a function $h=h_{g}$ on $T$ such that

[TABLE]

If a solution exists then it is necessarily unique by the minimum (maximum) principle. For our simple random walk on $T$, it is folklore that the Dirichlet problem is solvable, and that the solution is given as the Poisson integral of $g\,$:

[TABLE]

We are now interested in the general case when $\lambda\in\mathbb{C}\setminus[-\rho\,,\,\rho]$, which will remain fixed throughout this section. First of all, the above question is not well-posed. Indeed, if for example $\lambda>1$ is real, then the “Poisson integral” of the constant function $\mathbf{1}$ on $\partial T$ is $\Phi(x|\lambda)$. By Proposition 3.5, it tends to $\infty$ as $|x|\to\infty$, since $\widetilde{F}(\lambda)>1$. Thus, we need to normalise, compare with [11]. The same is necessary for the polyharmonic versions of higher order.

Theorem 4.1**.**

Let $\lambda\in\mathbb{C}$ with $|\lambda|>\rho$. For $g\in\mathcal{C}(\partial T)$ and $n\geq 0$, set

[TABLE]

Then $f$ is $\lambda$-polyharmonic of order $n+1$ and

[TABLE]

Before the proof of this result, we introduce the normalized kernel

[TABLE]

We only need it for large $|x|$, and then $\Phi_{n}(x|\lambda)\neq 0$ by Proposition 3.5, so that the division in (15) and the definition of $\mathcal{K}_{n}$ are legitimate. If we fix such an $x\in T$ with $|x|\geq R$, the function $\xi\mapsto\mathcal{K}_{n}(x,\xi|\lambda)$ is locally constant, since it depends only on $x\wedge\xi$ which ranges within the finite geodesic $\pi(o,x)$. Therefore it is continuous.

Lemma 4.2**.**

Let $y\in T$. Then

[TABLE]

uniformly for $\xi\in\partial T\setminus\partial T_{y}\,$.

Proof.

If $x\in T_{y}$ and $\xi\in\partial T\setminus\partial T_{y}$ then $x\wedge\xi=y\wedge\xi\in\pi(o,y)$. We have

[TABLE]

Therefore, using Lemma 3.2 and Proposition 3.5,

[TABLE]

which tends to [math] as proposed. ∎

Proof of Theorem 4.1..

For $x\in T$ with $|x|\geq R$,

[TABLE]

defines a complex Borel measure on $\partial T$. (It also depends on $\lambda$ and $n$, which we omit in the present notation.) We have $\mu_{x}(\partial T)=1\,.$ We write $|\mu|_{x}$ for its total variation measure. Its density with respect to $d\xi$ is \bigl{|}K(x,\xi|\lambda)\,\mathfrak{h}_{n}(x,\xi|\lambda)\bigr{|}\big{/}\bigl{|}\Phi_{n}(x|\lambda)\bigr{|}. Let us write

[TABLE]

A computation completely analogous to the one in the proof of Proposition 3.5 shows that

[TABLE]

Therefore

[TABLE]

We can now prove (15) along classical lines. Let $\xi_{0}\in\partial T$ and $\varepsilon>0$. Then, given $g\in C(\partial T)$, there is a neighbourhood of $\xi_{0}$ on which $|g(\xi)-g(x_{0})|<\varepsilon$. We may assume that this neighbourhood is of the form $\partial T_{y}\,$, where $y\in\pi(o,\xi_{0})$. If $x\to\xi_{0}$ then $x\in T_{y}$ when $|x|$ is sufficiently large. Then

[TABLE]

Now Lemma (4.2) implies that for $x\to\xi$ we have $|\mu|_{x}(\partial T\setminus\partial T_{y})\to 0\,,$ while $|\mu|_{x}(\partial T_{y})$ remains bounded by Lemma (17). ∎

Next, we consider a Fatou-type theorem for polyharmonic functions. That is, in the integral of Theorem 4.1 we replace $g(\xi)\,d\xi$ by a complex Borel measure $\nu$ on $\partial T$. We need to consider a restricted type of convergence to the boundary.

Definition 4.3**.**

Let $\xi\in\partial T$ and $a\geq 0$. The cone at $\xi$ of width $a$ is

[TABLE]

The motivation for this definition is well-known: in the open unit disk, consider a cone $C_{\alpha}(z)$ whose vertex is a point $z$ on the unit circle, whose axes connects the origin with $z$, and whose opening angle is $\alpha<\pi$. Then, passing to the hyperbolic metric on the disk, all elements of the cone are at bounded distance (depending on $\alpha$) from the axes. The standard graph metric of $T$ should be seen as an analogue of the hyperbolic metric on the disk, while a tree-analogue of the Euclidean metric is the ultrametric $\theta$ of (2). Compare with Boiko and Woess [3] for a “dictionary” concerning the many of the other analogies between the potential theory on the unit disk and $T$. Thus, $a$ is a substitute for the angle $\alpha$, and of course, if $|x|\to\infty$ within $\Gamma_{a}(\xi)$ then $x\to\xi$ in the topology of $\widehat{T}$. We shall use the following tools.

Lemma 4.4**.**

[11]** For $g\in L^{1}(\partial T,\mathsf{m})$, let

[TABLE]

be the associated Hardy-Littlewood maximal function on $\partial T$. Then the operator $g\mapsto\mathcal{M}g$ is weak type (1,1), that is, there is $C>0$ such that for every $t>0$,

[TABLE]

With $R$ as in Proposition 3.5, we now define for $a\geq 0$ and $g\in L^{1}(\partial T,\mathsf{m})$,

[TABLE]

Proposition 4.5**.**

For every $a\geq 0$ there is a constant $C_{a}$ such that

[TABLE]

Proof.

Let $\pi(o,\xi)=[o=x_{0}\,,x_{1}\,,x_{2}\,,\dots].$ First, let $a=0$. Fix $x=x_{r}$ with $r\geq R$. Then, with $A_{\ell}=\{\eta\in\partial T:|x\wedge\eta|=\ell\}$ as above, we use the properties listed in Lemma 3.2 and compute

[TABLE]

For general $a\in\mathbb{N}$, let $y\in T$ with $|y|\geq R$ and d\bigl{(}y,\pi(o,\xi)\bigr{)}\leq a. Then $d(x,y)\leq 2a$, where $x$ is the element on $\pi(o,\xi)$ with $|x|=|y|$. Recall that $|F(\lambda)|<1$. Since $|\mathfrak{h}(x,\eta)-\mathfrak{h}(y,\eta)|\leq d(x,y)$, we have

[TABLE]

for every $\eta\in\partial T$. Therefore

[TABLE]

Setting $C_{a}=(1+2a)^{n}\,|F(\lambda)|^{-2a}\,C_{0}\,$, the proposition follows. ∎

After Lemma 4.4 and Proposition 4.5, also the proof of the following theorem now follows the strategy of [11]. For the sake of providing a complete picture in the situation of trees, we also include some of the “standard” details in its proof.

Theorem 4.6**.**

Let $\lambda\in\mathbb{C}$ with $|\lambda|>\rho$, and let $\nu$ be a complex Borel measure on $\partial T$. For $n\geq 0$, set

[TABLE]

Then $f$ is $\lambda$-polyharmonic of order $n+1$ and

[TABLE]

where $g$ is the Radon-Nikodym derivative of the absolutely continuous part of $\nu$ with respect to the uniform distribution $\mathsf{m}$ on $\partial T$.

Proof.

We first give an outline of the standard fact that the limit in (19) is [math] when $\nu$ is singular with respect to equidistribution. The latter means that there is a Borel set $E\subset\partial T$ with uniform measure [math] such that $\partial T\setminus E$ is a $\nu$-null-set. For every $\varepsilon>0$ there are disjoint boundary arcs $\partial T_{y_{1}}\,,\dots,\partial T_{y_{k}}$ depending on $\varepsilon$, whose union $E_{\varepsilon}$ contains $E$ and has uniform measure $<\varepsilon$. Let $|\nu|$ be the total variation measure of $\nu$. If $x\to\xi_{0}\in\partial T\setminus E_{\varepsilon}\,$, then by Lemma 4.2,

[TABLE]

Since this holds for every $\varepsilon>0$, we get that $f(x)/\Phi_{n}(x|\lambda)\to 0$ almost everywhere on $\partial T$.

Now we may assume without loss of generality that we have $g=d\nu/d\mathsf{m}\in L^{1}(\partial T,\mathsf{m})$. Then there is a sequence $(g_{k})_{k\in\mathbb{N}}$ of continuous functions on $\partial T$ such that

[TABLE]

Set

[TABLE]

By Lemma 4.4 and Proposition 4.5,

[TABLE]

for every $\varepsilon>0$. By the Borel-Cantelli Lemma, this yields that

[TABLE]

For each $k$, the function on $\widehat{T}$ with values $g_{k}(\xi)$ for $\xi\in\partial T$ and $f_{k}(x)/\Phi_{n}(x|\lambda)$ for $x\in T$ is continuous on $\widehat{T}$ by Theorem 4.1. This readily implies that for $\xi\in A$, we have convergence as proposed in (19). ∎

We now come back to continuous boundary functions and Theorem 4.1. For $n\geq 1$, we cannot expect uniqueness of $f$ as a polyharmonic function of order $n+1$ which has the asymptotic behaviour of (15). Indeed, (14) shows that we can add polyharmonic functions of lower order such that the limit in Theorem 4.1 remains the same. However, for the case $n=0$, i.e., for $\lambda$-harmonic functions, we can investigate uniqueness: this case corresponds to the classical Dirichlet problem at infinity. Indeed, for real $\lambda>\rho$ one can use the typical argument, namely the maximum principle, to prove uniqueness. However, for complex $\lambda$, this is not available, and we have to introduce another method.

Theorem 4.7**.**

Let $\lambda\in\mathbb{C}\setminus[-\rho\,,\,\rho]$. For $g\in\mathcal{C}(\partial T)$, the function

[TABLE]

is the unique solution of the $\lambda$-Dirichlet problem with boundary function $g$, i.e., the unique $\lambda$-harmonic function such that

[TABLE]

Proof.

Continuity holds by Theorem 4.1. By linearity, we need to prove uniqueness only in the case when $g\equiv 0$. Thus, we assume that $\lambda\cdot h=Ph$ and that $\lim_{|y|\to\infty}h(y)/\Phi(y|\lambda)=0$, and we have to show that $h\equiv 0$.

We extend the notion of the spherical functions as follows:

[TABLE]

where the functions $\varphi_{k}$ are given by (9). For fixed $x\in T$, this is the unique $\lambda$-harmonic function of $y$ with value $1$ at $x$ which is radial with respect to the point $x$. Now let us define the spherical average of $h$ around $x$, that is, the function defined by

[TABLE]

A short computation shows that $\bar{h}$ is $\lambda$-harmonic, whence $\bar{h}(y)=h(x)\Phi(x,y|\lambda)$. By assumption, the function $\widehat{T}\to\mathbb{C}$ with value [math] on $\partial T$ and value $h(y)/\Phi(y|\lambda)$ at $y\in T$ is continuous. By uniform continuity

[TABLE]

Let $y\in T$ be such that $d(y,x)\geq N+|x|$. Then every $v\in T$ with $d(y,x)=d(y,v)$ satisfies $|v|\geq N$, so that

[TABLE]

Applying Proposition 3.5 once more, to both $\Phi(v|\lambda)$ and $\Phi(x,v|\lambda)\,$,

[TABLE]

Since $\widetilde{F}(\lambda)^{|v|-d(v,x)}$ is bounded in absolute value by $\max\{|\widetilde{F}(\lambda)|^{|x|},|\widetilde{F}(\lambda)|^{-|x|}\}$, we see that there is a finite upper bound, say $M_{x}(\lambda)$, depending only on $x$ and $\lambda$, such that

[TABLE]

Consequently, also the absolute value of the average $\bar{h}(y)$ has the same upper bound. We get

[TABLE]

Letting $N\to\infty\,$, we conclude that $h(x)=0$, and this holds for any $x\in T$, as required. ∎

Theorem 4.1 tells us that for considering the boundary behaviour of a $\lambda$-polyharmonic function $f$ of order $n$, it first should be normalised by dividing by $\Phi^{(n-1)}(\cdot|\lambda)$.

Lemma 4.8**.**

Let $f$ be polyharmonic of order $n$ and such that the $\lambda$-harmonic function $h=(\lambda\cdot I-P)^{n-1}f$ satisfies

[TABLE]

where $g\in\mathcal{C}(\partial T)$. Then

[TABLE]

where $g$ is $\lambda$-polyharmonic of order $n-1$.

Proof.

It follows from Theorems 4.7 that

[TABLE]

Set

[TABLE]

By Lemma 3.4,

[TABLE]

Therefore $g=f-f_{g}$ satisfies $(\lambda\cdot I-P)^{n-1}g=0$. ∎

If in the above lemma, the natural normalisation g\,\big{/}\,\Phi^{(n-2)}(\cdot|\lambda) has continuous boundary values, then g\,\big{/}\,\Phi^{(n-1)}(\cdot|\lambda) tends to [math] at the boundary of the tree by (14). Thus, by Theorem 4.1, f\,\big{/}\,\Phi^{(n-1)}(\cdot|\lambda) has the same boundary limit $g$ as (\lambda\cdot I-P)^{n-1}f\,\big{/}\,\Phi(\cdot|\lambda).

We conclude that for considering an analogue of the classical Riquier problem, with given boundary functions $g_{0}\,,\dots,g_{n-1}\,$, our solution $f$ should be obtained step-wise: first, (\lambda\cdot I-P)^{n-1}f\,\big{/}\,\Phi(\cdot|\lambda) should have boundary limit $g_{n-1}\,$, and we take $f_{n-1}=f_{g_{n-1}}$ according to Lemma 4.8. Next, the function $f-f_{n-1}$ should be polyharmonic of order $n-1$, and (\lambda\cdot I-P)^{n-2}(f-f_{n-1})\,\big{/}\,\Phi(\cdot|\lambda) should have boundary limit $g_{n-2}$. We then proceed recursively. We clarify this by the next definition.

Definition 4.9**.**

Let $\lambda\in\mathbb{C}\setminus[-\rho\,,\,\rho]$ and $g_{0}\,,\dots\,,g_{n-1}\in\mathcal{C}(\partial T)$. Then a solution of the associated Riquier problem at infinity is a polyharmonic function

[TABLE]

of order $n$, where each $f_{k}$ is polyharmonic of order $k+1$ and

[TABLE]

Corollary 4.10**.**

A solution of the Riquier problem as stated in Definition 4.9 is given by the functions

[TABLE]

One also has

[TABLE]

As already outlined further above, the solution is not unique. We can add to $f_{k}$ some suitable $\lambda$-polyharmonic function of lower order: normalised by $\Phi_{k}(x|\lambda)$, by (14) the latter will tend to zero, as $|x|\to\infty$. What is unique is – by Theorem 4.7 – the solution $(\lambda\cdot I-P)^{k}f_{k}=h_{g_{k}}\,$.