FACTORIZATION OF THE GREEN ' S OPERATOR AND WEAK-TYPE ESTIMATES

1 . Introduction. Let X be a tree, which is to say, a connected graph without loops . The order of a vertex of the tree is the number of edges that meet at the vertex . To avoid some messy, but essentially trivial, complications we will assume throughout that the order of every vertex is at least three . We say that a vertex u is a neighbor of the vertex v if u and v are connected by an edge. When u and v are neighbors we write u v. The set of transition probabilities {p(u, v)} is said to determine a nearest neighbor random walk if 0 _< p(u, v) < 1 for all u, v E X and p(u, v) > 0 if and only if u v . The walk is said to be stochastic if j:,, p(u, v) = 1 for all u in the tree . The transition probabilities determine a transition operator, P, as follows : For f a function defined on the (vertices of the) tree,

We often identify the transition operator with its associated set of transition probabilities.
We assume further that the transition operator P is regular in the sense that there is a positive number 6 such that p(u, v) >_ b whenever uv.This implies that the orders of the vertices are bounded above.
The visiting probability, U(u, v), for u and v in the tree is the probability that a walk starting at u will visit v.There are two common conventions if u = v .The standard convention .isthat U(u, u) is the probability that the walk will visit u at some time in the future, and this is our definition .The other convention is that U(u, u) = 1 .With this in mind we define : { a(u, v) U(u, v), if u 7É v 1, ifu=v .
P is transient if U(u, v) < 1 for all u and v in the tree.We will require that P be strongly transient in the sense that there is a S > 0 so that U(u, v) < 1 -S Supported in part by NSF Grants DMS-8701271 and DMS-8701203 .
whenever uv.In the Appendix to [KPTI it is shown that if there is a positive number y such that p(u, v) < 1/2 -y then P is strongly transient .Let q + 1 -q(u) + 1 be the order of the vertex u .We say that the transition operator P is isotropic if p(u, v) = 1 /(q + 1) whenever uv ; we say that it is symmetric if p(u, v) = p(v, u) for all u and v .A tree is homogeneous if all vertices are of the same order.A transition operator is homogeneous if it is defined on a homogeneous tree and the set of transition probabilities at each vertex are the same.It is easy to see that both an isotropic transition operator on an order bounded tree and a homogeneous symmetric transition operator are always strongly transient .In the sequel we assume that P is strongly transient and we set Thus,0<5<1 .
Throughout this paper functions on the tree X are complex valued, X is supplied with the discrete topology and the atomic measure which assigns mass one to each vertex .We think of the tree as a collection of vertices and of edges as a relationship on the tree.From this point of view X is a locally compact measure space.For a function, f, on the tree, 1/r Ilf11P = (E lf( u )I P UEX lif11 .= sup if(u)1 .

UEX
The Laplacian operator, A, is defined by Af o-u That is, A = P -d.The Green's operator, G, speaking loosely, is the inverse of -A .It is defined by a kernel G(u, v) , the Green's kernel,

VEX 0<p<oo
We determine G as follows: Fix u and v in X.A path in X connecting u to v is a finite sequence of vertices, w = {wo,wl, . . . .wn }, where wo = u, wn = v and for each k = 1, . . ., n, wk_1 is a neighbor of wk .The length of the path w is 2(w) = n .The weight of the path w is n W(w) = 11 p(wk-1, wk).

k=1
A trivial path w = {u} has length zero and weight one .
Definition.G(u, v) W(w) where w ranges over all paths that connect u to v.
Observations .It is immediate that G(u, v) = E' o p(k) (U, v) and so G = o Pk where the p(k) are transition probabilities associated with the transition operator Pk .It is not difficult to see that G(u, v) is the expected number of visits to v of a random walk that starts at u.We note further that P is transient if and only if G(u, v) < oo for all u and v .We do not use any of these observations.For any two vertices, u and v, in X there is a unique path of shortest length that connects u to v.This path is called the geodesic that connects u to v. We denote its length as d(u,v), and observe that d is a metric on X .
General references for matters raised in this introduction are [C], [KPT], and [G] .
2 .Disk realization of the tree.There are two natural ways to give an orientation to the edges in a tree, the disk realization of the tree and the half-plane realization, which we describe in the next section .
For the disk picture, an arbitrary vertex is selected and is denoted o.It is viewed as the initial point of a random walk that is governed by the transition probabilities {p(u, v)} .A point on the boundary of X, OX, is a semi-infinite geodesic, x = {xo, xl . . . ., xk, . . .}, where xo = o.Note that d(o, xk) = k for all k.Suppose u and v are in X .We say that w is between u and v if w is on the geodesic that connects u to v. If x E áX then w is between x and u if w is between u and xk for all k large enough.Let D = X U OX.If v is between o and u we say that v is above u, or that u is below v.If x = {xo, x, . . . ., xk, . . .} we say that the geodesic {xo, xi . . . . .xk, . . .} connects x to o.It follows that o is above every point in D and if x E CM then x is below xk for all k.
For each vertex u there are q(u) neighbors of u, {uj}, that are below u, except for o that has q(o) -f-1 lower neighbors .For each vertex u, u 7É o, there is a unique vertex u -such that u-u and u -is above u.
We now define a subbase for the topology of D .It consists of all sets N(u), u E X where N(u) = {v : v is below u}.With this topology D is compact, the restriction of the topology to X is discrete, and its restriction to áX is compact .If {xo, xl . . . . .xk, . . .} E áX then N(xk) fl 8X is referred to as an interval of level k.We say that xk is a vertex of leve] k .We write Iu = N(u) fl OX .
We now define the hitting (harmonic) measure, p, on 8X.Denote by Fn the (random) vertex at step n for the random walk determined by P with F o = o.p(I«) = Pr(3ko : Fk E N(u) Vk > ko 1 Fo = o) p extends to a Borel measure on 9X .
Consider the random walk determined by P starting at u and conditioned to remain in N(u) from some step onward.Define a hitting measure, v, for this walk just as we defined p. Then v(Iu) = 1 and for vu, v 5A u-, define the relative forward probability, 7r(u, v) = v(I,) .Clearly We say that we are moving forward if we move to a position below.
By Proposition 2 of [KPT], for u =~o Observe that this formula shows that the probability of being in N(u) from some step onward, conditioning en the event of being in N(u -) from some step onward, is the same as that of the event of starting at u-, conditioning on the event of never returning to u-and being in N(u) from the first step onward.Let A(u) be the probability that a random walk starting at u moves forward en its first step and never returns to u .We see that A(u -) is the denominator in equation ( 1), so that Notation.neighbors of u 7 ri = lr(u,uj)1 result for A(o) follows from the definitions .For u 7É o we use the relationship Lemma 1 .

ao j=1
Using the notation introduced before the statement of Lemma 1 we note that when u 7É o.It is easy to see that if {wo , wl, . . ., w n } is the geodesic connecting o to u then n h(I.) = 117r(wk-1,wk) k=1 If w = {wo , wl, . . ., wn } is the geodesic connecting u to v we define and observe that n (8) a(u,v) = Il a(wkl , wk) .k=1 Observation.It is clear that the regularity and strong transitivity of the walk imply that there is a S > 0 such that whenever u and v are vertices and u = v-then 7r(u, v) > 8. and A(u) ,... 1 for all vertices u.Theorem 2 .If u and v are vertices and u is above v then Proof.. Use equations ( 2)-( 8), gather terms and simplify.
3. Half-plane realization of the tree .We begin with the disk picture .Select a point on óX and denote it by oo.To each other point, y on CM we associate the unique doubly infinite geodesic { . . ., y-2, y_1, yo, yl, y2, . . .} such that {yo, yl, . . .} is cofinal with the geodesic in D that defines y and {yo, y_1, . . .} is cofinal with the geodesic defining oc .For this realization we call the tree Y and the finite part of the boundary is 8Y, H = Y U aY.For each vertex u there is a unique geodesic that connects u to oo ; which is to say a half-infinite geodesic with initial point u that is cofinal with the geodesic in X that defines oo.We say that each vertex on that geodesic is above u, and that v is below w whenever w is above v.The point oc is above every other point in H, and a point on áY is below every vertex that lies on the geodesic that connects it to oo .A crucial (simplifying) difference with the disk picture is that for every vertex, u, on the tree there is a unique vertex, u-, that is a neighbor of u and is above u.
As before we set N(u) = {v E H : v is below u}.This defines a subbase for a topology on H that is locally compact .The restriction of the topology to Y is discrete and its restriction to áY is locally compact .Again we set Iu = N(u) n DY.Just as each vertex in the disk has a level we can define a leve] for each vertex in the half-plane .Select a reference vertex e, and consider the geodesic {yo, y_1, y_2 . . . .} that connects e to oo.Then we set the leve] of e as 0 and each vertex, u, on the geodesic has level -d(e, u).Observe that every vertex on the tree is below some vertex on that half-infinite geodesic.Using the rule: the leve] of u is the level of u -plus one, a level for each vertex is defined .
As in Section 2 we can define the functions : A(u) and V(u,v) .To define A(u) we no longer need a special case.For all vertices u Furthermore, we may define the relative forlvard probabilities (10) 7r (u, v) = p(u, v) (1 -a(v, u))/A(u) whenever u = v-.Since the boundary is not compact we may no longer define the M(Iu) as probabilities, but we will be able to use conditional probabilities.We proceed by choosing a reference vertex, vo.For example, one may choose vo = e .Set p(Ivo) = 1.Extend u to the tree by the rule: f«u) = h(Iu-)7r(u,u-), extends to a Borel measure on áY.As in Section 3: Observation .Theorem 2 and Theorem 3 have the same forro but the values of the function A are defined differently in the two realizations .
We note that p,(IJ/p(Iu) is the probability that a random walk that is conditioned to eventually stay in N(u), is eventually in N(v).More informally: it is the_ probability that the random walk hits the boundary in I given that it will hit the boundary in Iu .
We also note that A(u) -1.
4. Factorization of the Laplacian and the Green's operator .We begin with the half-plane ; the extension to the disk will be given in Section 6. below .Recall that every vertex is both above and below itself.If we need to rule out this possibility we will use the locution strict1y below or stricly aboye.
We define the kernels for operators S and T. (11 Observe that lis(u, u) = 1 and IiT(u, u) = 1 -a(u, u-) for all vertices u .We let, whenever the series converges absolutely.
For the next definitions and the theorem that follows we use the notation introduced just before Lemma 1 and the formula for A(u) in equation ( 5) .We introduce two difference operators .For g a function defined on the tree: Theorem 4. For ¢ny function g defned on the tree Solving for f we obtain At the fourth step we used equation ( 4).This completes the proof.
We show next that A -is the inverse of T and that -0+ is the inverse of S on appropriate domains .Suppose f is in the domain of T. It is easy to see that Since the coefficients that define Tf(u) are non-negativo and sum to one, 20°( and so 2P, 0 < p -< oo) is in the domain of T. Thus on 2P, 0 < p < oo, 0-T=I.
There is an a such that 0 < ca(u, u-) <_ á < 1 for all u.Consequently if v is (1) as k -> oo.This certainly holds if f is bounded and so: Theorem 5 .0is the inverse of T on ~P, 0 < p < oo.

k=0 j=1 h(I.) k=0
Observation .There is a S, 0 < S < 1 such that 7r(u -, u) _< 1 -S for all u .This follows from the regularity and strong transience of P. Lemma 7 .If f is in £P, 0 < p < oo, then f is in ¡he domain of S.
Proof. .Suppose first that 1 < p < oo. Thus, When 0 < p < 1, t?P C P, and so for all p, 0 < p < oo, £P is in the domain of S.
For f in the domain of S we see that q(u) where the uj are the lower neighbors of u.That is, for all f E QP, 0 < p < oo.For such an f we have, using equation ( 13), Let {xkjl} be the q(xkj) lower neighbors of xkj .
for all vertices u and v on the tree .
We now define the multiplier transform A and its inverse R.
It will be convenient to set R(u) = 1/A(u) .
Proposition 10.The operators A and R are bounded on £P, 0 < p < oo .
Proof.This is immediate from the fact that A(u) -1.
The theorems of this section suggest that G = TSR.
w Proof: G(u, v) is the sum, over all paths from u to v, of the probabilities of traveling such paths.A(v) is the probability of the event of moving from v towards the boundary and never returning to v .So G(u, v)A(v) is the sum of the probabilities for traveling certain paths, where the sum is over all paths that start from u go to v and then move toward the boundary, never to retum to v .The kernel for TS is KTS(u, v) = EKT(u, w) Ks(w, v)   w where w ranges over all vertices that are above both u and v, since otherwise the summand is zero.Let w0 be the lowest such point and let wk+l = wk , k = 0, 1, 2, . . . .Then 00 KTS(u, v) = 1, hT(u, wk)Ks(wk, v) k=0 00 h(IV) («(u, wk) -a(u, wk+1)) ~(I ) k=0 wk Each path from u to v will pass through w0 and will continue to some wk, but not to wk+l , k = 0, l, 2, . . . .The kth term of the expansion ( 14) involves just those paths and an examination of the definition of p shows that M(I )/p(Iw,, ) is precisely the probability that a random walk starting at wk will pass through v on its way to the boundary never to retum to v. Since a path goes to the boundary with probability one this establishes the equality.
Remarks.The crucial fact in this development is the factorization of the Laplacian which is presented in Theorem 4 without any motivation.This result is a reformulation of results presented in [KPT] .In that paper a correspondence is established between harmonic functions on the tree; that is functions F such that OF = 0, and functions f that are boundary martingales associated with P; which in the language of this paper means that 0+ f = 0.The correspondence is established by the relation F = Tf.
Corollary 12. G = TSR on QP, 0 < p < oo . 5. Boundedness results .If E is a finite subset of the tree let DEI denote the cardinality of the set E.
Lemma 13.Suppose that E is a finite subset of the tree .Then there is a set F, F C E, ¡Fi > (1/2)1EJ, such that SXF(x) <_ C where C is a positive constant that is independent of E.
Proof. .We follow, as a model, the proof of Proposition 3 in [RT] .We say that v is a descendant of u if v is strictly below u, and that v is a descendant of u that derives from a lower neighbor uj of u if v = uj or is a descendant of uj.For E a finite subset of the tree let aLE be the lower boundary of E which we define as the subset of vertices in E that do not have descendants from each of its lower neighbors .(16) ¡Intl < ¡Brch f1 El < ¡Brchi .

Notice that FACTORIZATION ON THE GREENS OPERATOR 199
Assume for now that there is a vertex, x, in E such that every vertex in E is below x.In this case we say that E is triangular.Let Z be the subtree of Y that contains x and all of its descendants .An end of E is a vertex in E that has no descendente in E. End is the set of ends of E. A vertex in Z is a branch point of E if it has at least two descendants in E. Brch is the set of branch points .Since q(u) > 2 the number of ends of E is greater than or equal to one plus the number of branch points .Since every end is in the lower boundary An interior point of E is a point in É that is not in the lower boundary.Int is the set of interior points of E. An interior point has descendants from all of its lower neighbors and so it is a branch point .Thus, JEI = 19LE1 -}-¡Inti .
It follows from ( 15) and ( 16) that 1áLEI > ¡Brchi > lInti, and so JOLE1 > á ¡El .Since E can be written as a finite union of pairwise disjoint triangular sets in such a way that the lower boundary of E is the union of the lower boundaries of the pieces we see that ~aLE1 > ¡El for any finite á set .(The first piece is a highest point in E and all of its descendants in E .Then take from the remaining points a highest point and all of its descendants .Continue in this way until all points in E are exhausted .) Let F be the lower boundary of E. In order to estimate SXF(x) we first observe that for all vertices with a level below the lowest level of a vertex in E we have SXF(x) = 0, and for vertices on the same level as the lowest vertex (or vertices) of E we have that SXF(x) = 1 or 0 depending on whether or not x E F. Recall (see the last Ene of 3) that 7r(u -, u) >_ S > 0 for all vertices u.Suppose SXF(x) _< 1/8 for all x on a given level.This is certainly true for all vertices at or below the lowest level of a vertex in E. Take a vertex x on the next higher level .q SXF(x) = XF(x) + 5: 7rjSXF(xj), j=1 where the xj are the q lower neighbors of x and 7rj = 7r(x, xj).There are two cases to consider.If x 0 F 1: Wj S XF( X j) :5 E j=1 j=1 If x E F then at leas¡ one xjo has no descendant in F and SXF(xjo) = 0. Thus, This completes the proof.
We now define the formal adjoints of R, S, and T. R* = R. S* and T* are defined as the operators with kernels : This completes the proof .
Theorem 14 .S* is of weak type (1,1).That is, suppose Mat f E 0 and s qÉ 0 .Then Where C is a positive constan¡ that does not depend on f or s .
Proo£ Clearly we may assume that f(x) > 0 for all x and that IIf1I1 = 1 .Since f E Pl we see that E = {x : S* f(x) > s} is finite.Choose the set F as in Lemma 13. s 1 ¡El < J S*f -XF = f f -SXF < ó .
Proof..By Theorem 14 S* is of weak-type (1,1), and by Lemma 15 it is bounded on Q°°; an application of the Marcinkiewicz Interpolation Theorem completes the argument for S*.The result for S follows by duality.
Proof. .S and its inverse -0+ are both bounded on ~P .Observation .G is bounded on 2P, 1 < p < oo if and only if T is bounded on 2P .This follows from G = TSR and T = -C*AO+ .
Proof.. G* = RS*T* .We use Theorem 14, Lemma 15, and the fact that R is a bounded function.Suppose f E Ql and s > 0.

IIRll~llflllS
Theorem 19.There is a po , finite and positive, such that G is bounded on U,po<p<oo .
Proof.. From our remark above we only need to prove our result for T. Notice that the case p = oo is Lemma 6, so we may assume that p is positive and finite .Our proof will use Schur's Lemma (see [RW ;Prop. 3.50]) .
Let O(u) = ák , where k is the level of u and e is a positive number that is chosen below .Observe that KT(u, v) < IXd(,'v) if u is above v and is zero otherwise .Choose r such that 1 -1-1 = 1.M'e want to show: p r E KT(u,v) Notation.Let q + 1 be the maximum order of a vertex in the tree.
Corollary.23 .Under the conditions of Theorem 20, if 1 < p < oo G maps £P onto itself.
We set andifn :~0 It is easy to check that T and A-are inverse to each other on the class of all functions on the tree.We see that T is defined by the kernel KT where With A defined as in Section 3 we find that A+¿~, f(x) = R(x)Of (x) .
We find again that G = TSR and the results of Sections 4 and 5 carry over almost without change .
7. Examples .The simplest example is an isotropic walk on a homogeneous tree of degree q + 1, q > 2, in the half-plane realization .By symmetry we see that 7r(u-, u) -7r = 1 /q for all vertices u.Similarly, a -a(u, u -) for all u.Using equation ( 4) we have 1 a = + q a2 q+1 q+1 from which it follows that a = 1/q .From equation ( 5) we see that A(u) = (q -1)/(q + 1) for all vertices u .We also see that except for a factor of (q -1 )lq the operators S and T are ádjoints .The Green's kernel is given by q 1 d(u,v) G(u, v) = _l q+ 1 () which follows by a trival application of Theorem 11 .Since (GX{xo})(x) _ G(u, xo ) we see that the Green's operator is not bounded on Ql .
An example that arises from a group is a homogeneous tree with a symmetric anisotropic walk.For a specific example take a group with identity e and three generators a, b, and c; with the relations a2 = b2 = c2 = e.Let Y be the Cayley graph of the group.It is a tree that is homogeneous of order three.Each vertex corresponds to a reduced word and the three edges at each vertex correspond to the right multiplication by one of the three generators.Assign to each edge p(g), g = a, b, or c, as the case may be, where p(a) -}p(b) +p(c) = 1, and each of the p's is positive.For any such assignment of probabilities the associated walk is regular, strongly transient, and strongly reversible .
Let us now construct a simple example for which the Green's operator fails to be bounded on £P, 1 < p < oc.Take a tree that is homogeneous of order three in its half-plane realization .At each vertex there is one neighbor directly above it and two directly below .To the edge going up assign the probability .4 and to the two edges going down split the difference and give each the probability .3.What makes this example work is that 1/3 < 0.4 < 1/2 .If the "up" probability is greater than or equal to 1/2 the walk fails to be transient and if it is less than 1/3 the Green's operator is bounded on h .The case where it is equal to 1/3 gives the isotropic walk.Take the probabilities as we gave them.From equation (4) we have the equation a= .4+.6a2where a is the visiting probability associated with an upward transition .From this equation we see that a is equal to 2/3 .The argument of Theorem 19 shows that T, and so also G, is bounded on QP if p > log 3/ log(3/2) = po , which is about 2.7.Testing T against the characteristic function of a vertex we find that the result is sharp and that T fails to be bounded on QP if p < po.A more careful analysis shows that T is of restricted weak type (po , po ).(See [SW ;p .197]for definitons.) 8. Comments and questions .The principal boundedness results in this paper follow from an isoperimetric inequality expressed in Lemma 13, which then leads to the weak-type estimate stated in Theorem 14.This circle of ideas is closely related to the results obtained by Gerl in [G], where he obtains U boundedness results starting from another isoperimetric inequality.While the lower boundary described in the proof of Lemma 13 is not the same as the boundary used by Gerl (and others in combinatoric graph theory) there must be Glose connections and those connections should be studied .The notion of a lower boundary was introduced in [RT] in a different setting.The possibility that it could lead to boundedness results for the Green's operator carne to mind when we learned of Gerl's work in [G] .
In [RT] Lemma 13 in the setting of the dyadic martingale was used to compute the K-functional for the spaces CMd, the space of discrete Carleson Measures, and Po, the space of functions with finite support .In this special case the dyadic martingale is the boundary martingale of an isotropic random walk on a tree that is homogeneous of degree three ; the nodes of the graph can be viewed as the index set for the dyadic intervals of R. Lemma 13 allows the extension of the interpolation results to the boundary martingale of any strongly transient random walk on a non-homogeneous tree.In this more general setting the nodes of the tree represent intervals in a nested system of intervals more general than the dyadic system .
In Theorem 19 we find a po such that the Green's operator is bounded on QP if p > po.This po depends on q + 1, the maximum order of a vertex, and á, the "maximum visiting probability", á = sup a(u, u -) .For isotropic random walks on a homogeneous tree of order q + 1, q and a are constants, a = 1lq, and this leads to po = 1.For isotropic random walks on an order bounded tree we know that po = 1 but the argument of Theorem 19 leads to a gróss overestimate of po .In [L] Lyons uses the notion of an average branching number for a random walk on a tree where the branching number at a vertex u is q(u) in our notation .Lyons shows in a variety of problems that this average branching number behaves like q when q is a constant .Lyons' results suggest that there might be an "average visting probability" as well as an average branching number and that the infimum of these values of p for which the Green's operator is bounded on U could be computed from these averages.
Consider now the situation when the random walk is strongly transient and strongly reversible.In this situation the Green's operator maps QP onto £P if 1 < p < oo and it is never bounded on £1 .For if it were bounded on P then T would be bounded on Q1, which implies that S is bounded on £°°.But S is never bounded on Q°°a s we saw in our remarks following the proof of Lemma 6.This raises the problem of describing the class of integrable functions, f, on the tree such that Gf is also integrable .We can define a kind of Hardy space, H = {f : f E £1 , Gf E Q 1 }, lifIIH = jif111 -f-IiGf111.It is easy to see that this norm is equivalent to an "atomic norm" in the sense that there are functions called atoras and f E H if and only if f = 1: Akak(x) where the ak are atoms and 1: JAki < oo where jif IIH -inf 1: ¡Akl over all such representations.This is not very satisfactory since the definition of "atom" is restrictive (an atom being the Laplacian of a point mass) .One would want a less restrictive definition of an atom as well as a maximal function characterization before one would have a satisfying theory.This line of thought also suggests that there should be a BMO theory.We only remark that the tree as a measure space with discrete measure and the natural metric induced by geodesic distance is not a space of homogeneous type, in the sense of Coifman and Weiss [CW] .For spaces of homogeneous type Coifman and Weiss constructed a Hardy space theory.