Some Remarks on the Dirichlet Problem on Infinite Trees

We consider the Dirichlet problem on infinite and locally finite rooted trees, and we prove that the set of irregular points for continuous data has zero capacity. We also give some uniqueness results for solutions in Sobolev $ W^{1,p} $ of the tree.


Introduction
A tree T is a connected graph without loops. In this note we further assume that a tree has a distinguished edge ω called the root, that is locally finite (meaning that no vertex is connected with infinitely many other vertices) and has no leaves. By leaf we mean a vertex x which is the endpoint of a unique edge α, α = ω. We call V (T ) the vertex set end E(T ) the edge set of the tree. Trees have become a subject of interest because, due to their relatively simple structure, they can be used as a toy model for complicated situations arising in the study of problems of real as well as complex analysis. This point of view has been adopted in a variety of ways and it has been proven fruitful, for example in the study of the Dirichlet space of analytic function in the unit disc. (See for example [2] [6]). Here we are interested in the tree as an object per se. In particular, in analogy with the Potential Theory in the Euclidean space, we investigate the interplay between the capacity of the boundary of a tree and the Dirichlet problem. The paper is organized as follows. In Section 1 we introduce a Nonlinear Potential Theory on T which allows us to define equilibrium measures, equilibrium potentials and the capacity c(E) of a subset E of the boundary ∂T . As in the continuous case, irregular points are points where the equilibrium potential fails to attain the value 1 [Definition 1.3]. In Section 3 show that in fact such points can be identified by a Wiener's like series of capacities [Theorem 3.1]. Furthermore, we show that the classical probabilistic solution of the Dirichlet problem with continuous boundary data actually converges to the given data, except at irregular points [Theorem 3 .5]. As a corollary we get the corresponding Kellogg's Theorem, i.e. the set of irregular points for the Dirichlet problem on a tree has capacity zero. Section 4 is devoted to the study of uniqueness of the solution of the Dirichlet problem. Here the results are more rudimentary for general trees [Example 1]. We introduce Sobolev spaces on a tree and discuss some results about their boundary values.

Preliminaries
Our approach on tree capacities is in the framework of an abstract Potential Theory that can be found for example in [1]. We give a brief exposition of the theory in the particular case of trees here. First let us introduce a piece of notation. A geodesic {α i } is a (finite or infinite) sequence of edges such that for every α j ∈ {α i }, {α 0 , . . . , α j } is the shortest path between α 0 and α j . Notice that for every edge α there exists a unique geodesic {α 0 = ω, α 1 , . . . , α N = α} =: [ω, α] which starts at the root and ends at α. We call |α| := N the level of α. A rooted tree has also a natural partial order attached. For two edges α, β, α ≥ β if [ω, α] ⊇ [ω, β]. Hence it makes sense to define successor sets S(α) = {β ∈ E(T ) : β ≥ α} and predecessor sets P (α) = {β ∈ E(T ) : β ≤ α}, as well as the set of sons of α, s(α) := {β ∈ E(T ), β ≥ α, |β| = α + 1} and the (unique) parent of α denoted p(α) ∈ E(T ) which satisfies α ∈ s(p(α)). We call T α the subtree of T rooted at α which has as vertex the set S(α). The boundary of a tree T is defined as the set of infinite geodesics with starting point ω, and has a topology generated by the basis {∂T α } α∈E(T ) where ∂T α is the set of infinite geodesics passing through α. It turns out that this space is metrizable and T ∪ ∂T is a compactification of T with the edge counting metric. Note that the order relation extends naturally to an order on the set E(T ) ∪ V (T ) ∪ ∂T . Let g be a real valued function defined on the vertices if T . Given a point ζ = {x j } ∞ j=1 ∈ ∂T , we define the radial limit of g at ζ as The Fatou's set of g is The boundary value of g is the map g * : ∂T → R which on Fatou's points ζ ∈ F (g) is defined by From now on we simply write g in place of g * for the extension of g to the boundary since no confusion can arise. In order to define a capacity of a set E ⊆ ∂T we need the notion of the potential of a function.
The potential extends to a function from V (T ) ∪ ∂T to R ∪ {±∞}, It is clear that if f is a nonnegative function then If is defined on the all boundary, possibly taking value +∞, while in general its Fatou's set is non trivial. Let p ∈ (1, +∞) be a fixed exponent and p ′ its Hölder conjugate, 1/p + 1/p ′ = 1.

Definition 1.2.
Suppose that E ⊆ ∂T is a Borel set. We define the p−capacity of E, Some remarks are in order. It is customary to say that a property holds p−capacity almost everywhere or c p almost everywhere if the set on which it does not hold has p−capacity zero. With this terminology one can prove [1, Theorem 2.3.10] that given a Borel set E ⊆ ∂T there exists a unique function f E : E(T ) → [0, +∞), such that If = 1, c p −a.e. on E and f E p ℓ p = c p (E). This function is called the p−equilibrium function for the set E.
There exists a quite useful equivalent definition of capacities in terms of measures. We call charge a signed finite Borel measure on ∂T . The co-potential of a charge µ is defined by For brevity we shall write M instead of (I * µ) when the implied charge is clear from the context. The p-energy of a charge is just We define also the mutual energy of a charge µ and a function f on edges admitting boundary values µ−almost everywhere to be If the mutual energy is finite, we can switch sums and integrals and write E(µ, f ) = β∈E(T ) f (β)M (β).
For functions on edges we use the footnote notation f p : α → f (α) p ′ −1 , where powers of negative quantities has to be intended as follows: a s := sgn(a)|a| s , for each a ∈ R, s > 0. Hence, if E p (µ) < ∞, we can switch sums and integrals and get The following is what is usually called the dual definition of capacity.
Moreover, there exists a unique positive charge µ E supported in E, called the p−equilibrium measure of E, such that and

On p-harmonic functions on trees
If g : V (T ) → R is a function of the vertices, we define its gradient on the edges to be the difference operator, where b(α), e(α) denote the beginning and the end vertex of α, with respect to the order relation.
It is immediate to see that the following fundamental theorem of calculus holds.
We say that a function f : It is immediate that the potential of a charge defines a forward additive function. Next proposition characterizes forward additive functions that can be obtained as potentials of charges.
if and only if there exists a (unique) charge µ on ∂T such that f = I * µ.
Proof. Note that the limit in condition (3) is in fact a supremum. For f forward additive we have For each α write ζ(α) for an arbitrary point in ∂T α . For each k ∈ N, define a Borel measure, The total variation of the measure µ k is given by from (3) it follows that the family of measures µ k is uniformly bounded, so that it has a weak * -limit µ which is positive. For each edge α we have For the uniqueness part, if f = I * ν for some other charge ν, then ν(∂T α ) = µ(∂T α ) for each α ∈ E(T ) and hence µ ≡ ν.
Viceversa, let µ be a charge on ∂T and consider the forward additive function f = I * µ. Then Observe that if f ≥ 0 then condition 3 is automatically satisfied. Given g : V (T ) → R, its p-Laplacian at the vertex x is given by As usual, we simply call Laplacian the linear operator ∆ := ∆ 2 and we say that g is harmonic if ∆g ≡ 0. Observe that harmonicity coincide with the mean value property For more details on p−harmonic functions on trees and their boundary behaviour see for example [7].
It follows that ∆ p If p ≡ 0 if and only if (2) holds.
Putting together the last two Propositions we get the following.

The Dirichlet problem
There is an extensive literature on the discrete Dirichlet problem and its variations on graphs (See for example [11], [10] and [9]). In the particular case of trees we derive more precise results about the exceptional set.
In the proof we shall need the following rescaling property of equilibrium measures on trees (see [3]).
Lemma 3.2. Let µ, µ α be the p-equilibrium measures for the sets E ⊆ ∂T and E α , respectively. Then it holds the relation Moreover, for every α ∈ E(T ), µ solves the following equation: Proof of Theorem 3.1. Let µ be the equilibrium measure for E, and M its co-potential. Set ε := 1 − IM p (ζ) ≥ 0 to be the deficit of regularity of the point ζ ∈ E. Let {α j } = P (ζ), and set t n = j≥n M p (α j ). Clearly t n is monotonically decreasing to zero, being the tail of the converging sum IM p (ζ). By Lemma 3.2, Now, the sum n c n converges if and only if n (1−c n ) > 0. The partial product can be explicitly calculated thanks to its telescopic structure, Since t 0 = µ(∂T ) p/p ′ > 0, it follows that Observe that the Wiener condition (5) can be re-written purely in terms of capacities on the all boundary, in the following sense.
Proof. Let T be any rooted tree, E ⊆ ∂T and consider a tent T α , with |α| = n. If µ is the equilibrium measure for E α = E ∩ ∂T α ⊆ ∂T , then the associated co-potential M is supported on E(T α ) ∪ P (b(α)), since the equilibrium function M p must minimize the p-norm. By forward additivity, M must be constant on P (α), i.e. M (β) = M (ω) = c p (E α ) for β ≤ α. By the rescaling properties we know that from which follows Substituting this expression in the Wiener condition (5) we get the result.

3.2.
A probabilistic interpretation of capacity. The connection between random walks on graphs and electrical networks is nothing new (see for example [10]). Here we give an interpretation of the capacity of the boundary of a tree, which will be helpful in the solution of the Dirichlet problem. As usual we work on a general rooted locally finite tree T without leaves. Consider the simple random walk (Z n ) on the vertices of T which stops when it hits the root vertex b(ω). In this context we consider o = b(ω) part of the extended boundary ∂T = ∂T ∪ {o} of T . Then there exists a ∂T − valued random variable Z ∞ such that Z n converges to Z ∞ , P xalmost surely for every x ∈ T , where P x is the probability measure of the random walk starting at x ∈ T . We can now associate to any vertex x ∈ T the harmonic measure λ x (E) : where E is a Borel subset of ∂T . For a general tree T not necessarily finite , let T n be the truncation of T up to level n. Then from the finite case we have that c Tn (∂T n ) = P e(ω) (sup i |Z i | ≥ n). By monotonicity of measures the last quantity converges to P e(ω) (Z ∞ ∈ ∂T ), as n → +∞. It remains to show that c Tn (∂T n ) → c T (∂T ), as n → ∞. By definition of capacity we get that c Tn (∂T n ) ≥ c T (∂T ), since the equilibrium function f ∂Tn is an admissible function for ∂T .
To prove the other inequality we use the dual expression for capacity. Suppose that µ n is a measure on ∂T n such that E(µ n ) = |α|≤n M n (α) 2 ≤ 1 and µ 2 n (∂T n ) = c Tn (∂T n ). Consider now the corresponding charges on ∂T where ζ α any point in ∂T α and δ ζα the corresponding Dirac mass. Since µ n (∂T ) ≤ 1 we can find a weak * -limit point µ of the sequence { µ n }. We have that Therefore, letting m → ∞ we get that E(µ) ≤ 1, and hence by the dual definition of capacity, Given a function ϕ defined on ∂T we define its harmonic extension (or its Poisson integral ) to be the function P(ϕ) : V (T ) → R given by The harmonicity of P(ϕ) follows by the Markov property, since Proof. Pick α ∈ E(T ) and let ζ = {x j } ∞ j=0 ∈ ∂T α . Write {α j } ∞ j=1 = P (ζ), so that x j = e(α j ). For n ≥ |α| we have 0 ≤ 1 − λ xn (∂T α ) ≤ P xn (Z n hits α before hitting ∂T α ) = n j=|α| P xj (Z n hits x j−1 before hitting ∂T αj ) = n j=|α| 1 − c 2,αj (∂T αj ) .
By the Wiener condition (5) we have that the right handside vanishes as n → +∞ if and only if ζ is a regular point for ∂T . Hence, for any regular point ζ ∈ ∂T α we have lim n→∞ λ xn (∂T α ) = 1, from which it follows that for any regular point ζ in the boundary Any open set E ⊆ ∂T can be written as a disjoint union of tents {∂T α k }. Let ζ be a regular point of regular of ∂T . Then, It follows that λ xn w * −→ δ ζ , as n → ∞. Therefore, for any ϕ ∈ C(∂T ),

Uniqueness results
We give a first uniqueness result for the class of spherically symmetric trees, which are trees where the degree is constant on levels. Clearly homogeneous trees belong to this class. We define the Lebesgue measure on ∂T to be the measure λ which is equidistribuded among sons of any edge and is normalized with λ(∂T ) = 1. Namely, for each β ∈ E(T ), if we write p(β) for the edge parent of β, we have λ(∂T β ) = λ(∂T p(β) )/ deg (e(p(β))) = 1/ α<β deg(e(α)).
It is clear that on spherically symmetric trees I * λ is constant on levels. In what follows, we write λ(k) in place of λ(∂T β ) when |β| = k. One can check that the equilibrium measure of a spherically symmetric tree is a scalar multiple of the Lebesgue measure. and define the measures λ αj on ∂T in the following way It is clear that λ αj is absolutely continuous with respect to λ. Integrating on the tent rooted in α, using the fact that λ(∂T β ) depends only on the level of β, for each j we get Note that the last quantity is finite because the capacity of the boundary is positive. Being the same true for each j, M must be constant on s(α). It follows that µ = M (ω)λ, i.e. the measure µ is a scalar multiple of the Lebesgue measure. Hence, which gives M ≡ 0 on E(T ) which is the thesis.
The same is not true for a general tree. In fact, there exists a sub-dyadic tree T , with no irregular boundary points and a charge µ on ∂T such that IM = 0 everywhere except at a point, but µ = 0, as shown in the next example. Example 1. The following diagram represents an infinite subdyadic tree and the copotential M of a charge µ on its boundary, where the number r over an edge indicates how many times the edge is repeated. Also, the label T (a) means that the vertex is the root vertex of a dyadic tree which caries a total measure of a on the boundary, and the measure M is divided equally at each edge.
By applying Wiener's test we see that the ζ 0 is a regular point of the boundary, while all other points are clearly regular by symmetry. 4.1. Energy conditions. The situation is different if we work with measure with finite energy. Proposition 4.2. Let µ, ν be charges on ∂T , with E p (µ), E p (ν) < ∞. Denote by M and V their potentials , respectively. If IM p = IV p both µ-a.e. ad ν-a.e., then µ ≡ ν.
Proof. Integrating both the potentials IM p and IV p with respect to both the measures, we can get any of the following Recalling that we write a s = sgn(a)|a| s , with some algebra we get It is clear that a s − b s has the same sign as a − b for every a, b ∈ R, s > 0. It follows that the general term of the above series is positive, from which M ≡ V on E(T ).
It is clear from the dual definition of capacity that if a property holds c p -a.e. then it also holds µ-a.e. with respect to any charge µ of finite energy. Hence, the above result can be rewritten in the following slightly less general but more natural form.
As a consequence, we have a partial converse of the properties of equilibrium measures given in Theorem 1.
Corollary 4.4. Let µ be a Borel measure on ∂T such that E p (µ) < ∞ and IM p = 1 c p -a.e. on E = supp(µ). Then µ is the p-equilibrium measure for E.
These uniqueness results can be re-interpreted in terms of functions in place of measures. The following Sobolev space naturally arises from the space of charges of finite energy, In fact, given a charge µ on ∂T , E p (µ) = M p p p = ∇IM p p p , so that the following Proposition is self evident.  If the tree T is spherically symmetric, by Proposition 4.1 we know that in the linear case p = 2 we don't need the energy condition g, h ∈ W 1,2 (T ) in the statement.

4.2.
Boundary values. The uniqueness results we presented above apply to functions that admit boundary values c p −almost everywhere. A priory is not obvious which functions have this property. Here we prove that in fact all the functions in the Sobolev spaces for which we have uniqueness results indeed admit boundary values c p −almost everywhere. Other results of this kind can be found in [7]. To prove the existence of boundary values we follow an approach exploiting Carleson measures, which was already presented in [5] for the linear case p = 2. We include here the argumentation adapted for general Sobolev spaces for completeness.
We say that µ is a measure on T := V (T ) ∪ ∂T if µ V (T ) is a function on vertices and µ ∂T is a measure on the boundary. Observe that if µ V (T ) = I * (µ| ∂T ) then it defines a measure which is not finite. Definition 4.7. We say that a Borel measure µ on T is a Carleson measure for W 1,p if there exists a constant C(µ) > 0 such that for all g ∈ W 1,p These measures have been widely studied and characterized (even in the weighted case), (see for example [4], [2] and [6]). In [4] it is shown that condition (6) can be reformulated purely in terms of the measure µ. In fact, it is shown that it is equivalent to Denote by µ CM the best possible constant in (7), which for µ fully supported on ∂T reduces to Observe that if µ is the equilibrium measure for some set E ⊆ ∂T , by Lemma 3.2 it follows that, for every edge α, E p,α (µ)/M (α) ≤ 1, with equality for α = ω. Hence, µ CM = 1 and c p (E) = µ(E)/ µ CM . On the other hand, for any µ supported in E ⊆ ∂T we have the bound E p (µ) ≤ µ CM µ(E), from which µ(E) where the last inequality follows from the fact that the measure µ/E p (µ) p−1 is admissible. We have derived the following expression of p-capacity in terms of Carleson measures of W 1,p spaces (8) c p (E) = sup µ(E) µ CM : supp(µ) ⊆ E .
The following proposition shows that the Fatou's set of a W 1,p function differs from the boundary of the tree at most for a set of null capacity. The argument is taken by [5], where the result is proved for p = 2. Proof. For g ∈ W 1,p (that without loss of generality we normalize to g(o) = 0), define the sequence of functions g * n := I |∇g|χ |α|≤n . It is clear that g * n is pointwise non-decreasing and it extends to the boundary by continuity, being eventually constant. By monotonicity we have that the function g * (ζ) = lim n g * n (ζ) is well defined for every ζ ∈ V (T ) ∪ ∂T . Moreover, we have the uniform bound g * 1,p ≤ g 1,p . Now, let µ be a Carleson measure for W 1,p . By Fatou's Lemma we have ∂T g * (ζ) p dµ(ζ) ≤ lim inf n ∂T g * n (ζ) p dµ(ζ) = lim inf n |α|=n ∂Tα g * n (ζ) p dµ(ζ) = |α|=n g(e(α)) p M (α) ≤ C(µ) g 1,p .
This implies that g * ∈ L ∞ (dµ), and since g * is a bound for the radial variation of g along geodesics, by Dominated Convergence Theorem we deduce that g admits radial limit µ-a.e. on ∂T for every Carleson measure µ. In particular, the equilibrium measure µ E of the set E = ∂T \ F (g) is a Carleson measure since µ E CM = 1, from which follows that the radial limit exists c p -a.e.