Harmonic Bergman projectors on homogeneous trees

In this paper we investigate some properties of the harmonic Bergman spaces $\mathcal A^p(\sigma)$ on a $q$-homogeneous tree, where $q\geq 2$, $1\leq p<\infty$, and $\sigma$ is a finite measure on the tree with radial decreasing density, hence nondoubling. These spaces were introduced by J.~Cohen, F.~Colonna, M.~Picardello and D.~Singman. When $p=2$ they are reproducing kernel Hilbert spaces and we compute explicitely their reproducing kernel. We then study the boundedness properties of the Bergman projector on $L^p(\sigma)$ for $1<p<\infty$ and their weak type (1,1) boundedness for radially exponentially decreasing measures on the tree. The weak type (1,1) boundedness is a consequence of the fact that the Bergman kernel satisfies an appropriate integral H\"ormander's condition.


Introduction
The main focus of this paper is on the projectors associated to the harmonic Bergman spaces on homogeneous trees introduced in [9].The Bergman spaces A p pσq, 1 ď p ă 8, are in some ways the harmonic analogues of the classical holomorphic Bergman spaces on the hyperbolic disk, whereby p-integrability is relative to the reference measure σ on the tree, that is a finite measure with radial density with respect to the counting measure, and where harmonicity is relative to the socalled combinatorial Laplacian.The analogy between the hyperbolic disk and the homogeneous tree inspires many ideas behind our constructions (see [6], [8]).
The space A 2 pσq is, as expected, a reproducing kernel Hilbert space (RKHS) and the problem of understanding the associated projectors hinges on the explicit knowledge of the kernel, an information that we derive by introducing a somewhat canonical basis for A 2 pσq.The core of this contribution is devoted to proving that, for a prototypical class of measures, the extension of the Bergman projector is bounded on L p pσq if and only if p ą 1, and is of weak type (1,1).The results are thus almost faithful reformulations of those that hold true for the holomorphic Bergman spaces on the hyperbolic disk ( [5], [15], [17], [20], and [22]), but many of the key ingredients, first and foremost the explicit determination of the reproducing kernel, call for a rather different approach.
Let X be a homogeneous tree.A function on the tree is said to be harmonic if the mean of its values on the neighbors of any vertex coincides with the value at that vertex.J. Cohen, F. Colonna, M. Picardello, and D. Singman introduce harmonic Bergman spaces on homogeneous trees in [9].They consider a family of reference measures which consists of finite measures that are absolutely continuous with respect to the counting measure and whose Radon-Nikodym derivative σ is a radial strictly positive decreasing function on X.For every 1 ď p ă 8, the harmonic Bergman space A p pσq is the closed subspace of L p pσq consisting of harmonic functions.The requirement for the measure σ to be finite is suggested by the fact that the only harmonic function which is p-integrable with respect to the counting measure is the null function.
In the context of the hyperbolic disk, when p " 2, the weighted Bergman spaces are RKHS, and the holomorphic Bergman kernel is known as well as the properties of the associated projector.Indeed, the extension of the holomorphic Bergman projector to the weighted L p -spaces is bounded if and only if p ą 1, see [17], [20] and [22].Furthermore, it is of weak type p1, 1q, see [5] and [15].In our work, first of all, we show that A 2 pσq is a RKHS for every reference measure σ and we provide an explicit formula for the reproducing kernel K σ in Theorem 11.Since A 2 pσq is closed in L 2 pσq, there exists an orthogonal projection P σ : L 2 pσq Ñ A 2 pσq.We prove that, for a particular class of reference measures, P σ extends to a bounded operator from L p pσq to A p pσq if and only if p ą 1.Moreover, we show that P σ is of weak type (1,1): to do so we use a Calderón-Zygmund decomposition of integrable functions adapted to the measure σ.Notice that the measure σ is not doubling with respect to the standard discrete metric on X, but it turns out to be doubling with respect to the so called Gromov metric (see Section 4).Hence a Calderón-Zygmund theory in this setting holds, and we show that the Bergman kernel satisfies an integral Hörmander's condition related to such theory, so that it is of weak type (1,1).
The measures we focus on are exponentially decreasing radial measures, i.e. they are exponentially decreasing with respect to the distance from o and can be viewed as natural counterparts of the measures involved in the definition of the standard weighted holomorphic Bergman spaces on the hyperbolic disk.The fact that the extension of the projector to the weighted L p -spaces is bounded if and only if p ą 1 follows from the fact that the projector coincides with a particular Toepliz-type operator (see Section 3.4 in [22]).
In the spirit of the results of [3] and [4] on the disk, one could investigate the boundedness of the Bergman projectors for general reference measures.In [9], [10], [11], the authors introduce and study the optimal measures, a subclass of the reference measures which, roughly speaking, decrease fast as the distance from the origin increases.The exponentially decreasing radial measures are optimal in this sense.The study of the boundedness of the Bergman projector for optimal measures is still work in progress.Another related question is whether the Calderón-Zygmund theory that we develop here could be applied to other operators.
The paper is organized as follows.In the first section we recall the definition of the harmonic Bergman spaces and, for every reference measure, we provide an orthonormal basis of the Hilbert space A 2 pσq.The basis plays a fundamental role in Section 2 in the proof of the two formulae for the kernel of the RKHS A 2 pσq: the first is a recursive formula, while the second is the explicit formula of the kernel given in Theorem 11.In Section 3 we focus on the exponentially decreasing radial measures and state two results characterizing the boundedness of the extension of a class of Toeplitz-type operators inspired by the operators considered in [22] (see Theorem 14 and Theorem 15).As a consequence, in Theorem 17 we show that the extension of the harmonic Bergman projector to the weighted L p spaces is bounded if and only if p ą 1.The last section is devoted to the Calderón-Zygmund decomposition of integrable functions (presented in Proposition 30), the formulation of the Hörmander's type condition, see (32), and the weak type (1,1) boundedness of the Bergman projectors is obtained as byproduct.
In what follows, we shall use the symbol » (À, or Á) between two quantities when the left hand side is equal (smaller than or equal to, or greater than or equal to, respectively) to the right hand side up to the multiplication by a (fixed) positive constant.Furthermore we assume the following convention on sums: the sum is null whenever the starting index is greater than the final index.If Y Ď X, we denote by ½ Y the characteristic function of Y .Finally, we adopt the symbol \ for disjoint unions and txu for the largest integer less than or equal to x P R.
A graph is a pair pX, Eq, where X is the set of vertices and E is the family of unoriented edges, where an edge is a two-element subset of X.If two vertices u, v in X are joined by an edge, they are called adjacent and this is denoted by u " v.A tree is an undirected, connected, cycle-free graph.A q-homogeneous tree is a tree in which each vertex has exactly q `1 adjacent vertices.With slight abuse, we refer to the set of vertices X as the tree itself.We fix an origin o P X.
From now on we consider a q-homogeneous tree X with q ě 2. Given u, v P X, with u ‰ v, we denote by ru, vs the unique ordered t-uple px 0 " u, x 1 , . . ., x t´1 " vq P X t , where tx i , x i`1 u P E and all the x i are distinct.We say that ru, vs is the path starting at u and ending at v. With slight abuse of notation, if ru, vs " px 0 , . . ., x t´1 q we write x i P ru, vs, i P t0, . . ., t ´1u.In particular, if u and v are adjacent, both ru, vs, rv, us P X 2 are oriented, unlike the edge tu, vu P E which is not.A homogeneous tree X carries a natural distance d : X ˆX Ñ N, where for every u, v P X the distance dpu, vq is the minimal length of a path joining u and v.If v P X, then we denote by Spv, nq and Bpv, nq the sphere and the ball centered at v with radius n P N, respectively, i.e., Spv, nq " tx P X : dpv, xq " nu and Bpv, nq " tx P X : dpv, xq ď nu.
It is straightforward to check that We call norm of a vertex v in X its distance from o, i.e. |v| " dpo, vq.We say that a function f on X is radial (with respect to o) if its value at a vertex x P X depends only on |x|.If v ‰ o, then we define the sector of v as the subset T v :" tx P X : ro, vs Ď ro, xsu, and we adopt the convention T o " X.Moreover, we call successors of v the elements of the set spvq " tu P X : u " v, |u| " |v| `1u.Evidently, For every v ‰ o we call predecessor of v and denote by ppvq the only neighbor of v which is not a successor of v; it follows that |ppvq| " |v| ´1.The vertex o is the only one having no predecessors, and spoq " Spo, 1q.This defines the predecessor function p : Xztou Ñ X, and, for every positive integer ℓ, its ℓ-fold composition p ℓ : XzBpo, ℓ ´1q Ñ X is the ℓ-th predecessor function.

Harmonic functions and harmonic Bergman spaces
Definition 1.Let f be a complex valued function on X.The combinatorial Laplacian of f is defined by We say that We shall call a function harmonic if it is harmonic on X.
It is easy to prove that a function is harmonic if and only if for every v P X and n P N, we have The harmonicity property (2) implies a certain rigidity for the function.In particular, the value of a harmonic function at a vertex y P X "propagates" to every layer of the sector T y , as showed in the following proposition, which is a modified version of [9,Lemma 4.1].In that lemma, the authors show that a function which is harmonic and radial on a sector T y , y P Xztou, is completely described by its values at y and ppyq.We consider a harmonic function on the sector T y , removing the radiality assumption, and we formulate a result for its average on Spo, nq X T y , n ě |y|.We omit the proof since it is an easy adaptation of the proof of [9, Lemma 4.1].Proposition 2. Let y P Xztou.If f : X Ñ C is harmonic on T y , then for every n P N, n ě |y|, we have Furthermore, if f : X Ñ C is radial on T y and satisfies (4) for every n ě |y|, then f is harmonic on T y .
We introduce a technique which allows to extend a function which is harmonic on a ball centered in o to a function harmonic on X.Let n P N and g be a function on X which is harmonic on Bpo, nq.It is easy to see that there are infinitely many ways to extend g to a harmonic function on X which coincides with g on Bpo, n`1q.As we see next, there is however a unique harmonic function g H n on X which is radial when restricted on T y for every y P Spo, n `1q.
Let x P XzBpo, nq.There exists a unique y P Spo, n `1q such that x P T y , and y " p |x|´n´1 pxq (where p 0 " id X ).Since we aim to construct g H n radial and harmonic on T y , by Proposition 2 we have that For simplicity we introduce the notation and we set a ´1 " 0. Hence The function g H n defined above is harmonic on X by Proposition 2 and because Observe that g H n is indeed harmonic on Bpo, nq because a 0 " 1 and a ´1 " 0, yield g H n " g on Bpo, n `1q, and not only on Bpo, nq.Furthermore, the extension g H n is radial on every sector "starting" from a point in Spo, n `1q by construction.

Harmonic Bergman spaces
Homogeneous trees are classically endowed with the counting measure.The main feature of such measure is the invariance under the group of isometries of the tree.When studying spaces of harmonic functions, this measure is however inadequate because the only harmonic function that is p-summable, 1 ď p ă 8, with respect to the counting measure is the null function, as we show in the following statement.o Bpo, 2q

Spo, 3q
Figure 1: The function g is harmonic on Bpo, 2q, that is the set of vertices in the blue area.The function g H 2 is obtained by extending the values of g in Spo, 3q (the green area) along sectors in such a way that g H 2 is harmonic on X and constant on the vertices lying on the same red arc, that is on the "layers" of the sectors.
Proof.Suppose that f is harmonic.We have that since every vertex is neighbor of exactly q `1 other vertices.Hence the unique inequality in the computation above is an equality, so that which means that |f | p is harmonic, too.If f is not the null function, then there exists v P X such that f pvq ‰ 0. Hence by (3), we have which is a contradiction.Hence f " 0.
Since we are interested in Bergman spaces of harmonic functions, the previous proposition leads to consider finite measures on X.In [9], the authors introduce harmonic Bergman spaces with respect to the following class of measures.Definition 4. A reference measure on X is a finite measure that is absolutely continuous with respect to the counting measure and whose Radon-Nikodym derivative σ is a radial strictly positive decreasing function on X.With slight abuse of notation we denote by σ the reference measure, too.Given a reference measure σ on X for every p P r1, 8q the Bergman space A p pσq is the space of harmonic functions on X such that }f } p A p pσq :" Every Bergman space A p pσq is a Banach space and when p " 2, it is a Hilbert space with the scalar product xf, gy A 2 pσq :" If σ is a reference measure on X, and if we denote by σ n the value of σ on the sphere Sp0, nq, then by (1) the total mass of σ is denoted by B σ and its value is Interesting examples of reference measures are the exponentially decreasing radial measures, consisting of the measures having density µ α pxq " q ´α|x| , x P X.
Indeed, µ α is radial, positive and decreasing.Furthermore, we write B α in place of B µα , namely Proposition 6.For every reference measure σ the measure metric space pX, d, σq is nondoubling.
Proof.Let σ be a reference measure.For every n P N, let v n P X be such that |v n | " 2n.Then maxtσpxq : x P Bpv n , nqu " σ n and so On the other side, since o P Bpv n , 2nq, we have by the finiteness of σ.This concludes the proof.
Given a reference measure σ, we introduce the decreasing sequence pb n q nPN which collects some important information on σ.For every n P N, we define The sums are finite because σ is a finite measure on X.We shall use the notation b n instead of b n pσq whenever the measure is clear from the context.The next lemma is a technical result that is very useful in what follows.
Lemma 7. Let n P N and g be a function on X which is harmonic and vanishes on Bpo, nq.Then there exists a constant b 1 n ą 0 such that for every where b n is defined in (8) and x¨, ¨yA 2 pσq in (6).
Remark 8.The constant b 1 n has a structure similar to that of b n , as can be seen in the proof below, but we are not interested in it.
Proof.Observe that, from the fact that g| Bpo,nq " 0 and (5), for every x P X with |x| ą n we have g H n pxq " a |x|´n´1 gpp |x|´n´1 pxqq.Take f P A 2 pσq.Then, by applying Proposition 2 to f , we have `bn f pyq ´b1 n f pppyqq ˘gpyq, as required.

A canonical orthonormal basis of A 2 pσq
The goal of this section is the construction of an orthonormal basis of the space A 2 pσq.
Let us consider the linear spaces For every v P X we set I v " t1, . . ., |spvq|´1u.For every v P X we fix an orthonormal basis te v,j u jPIv of W v with respect to to the scalar product xϕ, ψy Wv " ÿ yPspvq ϕpyqψpyq.
Let v P X and j P I v .We consider the extension by zero to all of X of e v,j , namely, It is immediate to see that E v,j is harmonic on Bpo, |v|q and vanishes on Bpo, |v|q.
We denote the harmonic extension of E v,j by f v,j " pE v,j q H |v| , namely Hence f v,j is harmonic for every v P X and j P I v .Furthermore f v,j is bounded, since for every x P X |f v,j pxq| ď p1 ´q´1 q ´1}e v,j } 8 .
Hence f v,j P A 2 pσq for every reference measure σ.Notice that, upon setting f 0 pxq " 1, the family is independent of the choice of the reference measure σ.
Proposition 9.The family B is a complete orthogonal system in A 2 pσq for every reference measure σ.
Proof.Fix a reference measure σ.The fact that f 0 is orthogonal to every other function of the family follows from the harmonicity of f v,j and (3).Indeed Let us consider v, w P X with v ‰ w.Without loss of generality we may consider two situations: either T v X T w " H or T v Ĺ T w .In the first case f v,j K f w,k for every j P I v and k P I w , because their supports are disjoint.If T v Ĺ T w , then we can suppose that |w| ă |v|.Since f v,j | Bpo,|w|`1q " 0, from Lemma 7 we have It remains to prove the orthogonality in the case v " w.Let j, k P I v be such that j ‰ k.We know that f v,k | Bpo,|v|q " 0, so that by Lemma 7 where we used the fact that supppE v,k q, supppE v,j q Ď spvq and the orthogonality of e v,j and e v,k in W v .
We now show that B is complete.Take g P A 2 pσq such that xg, f y A 2 pσq " 0 for every f P B. We show that g is the null function in A 2 pσq.In particular we prove by induction that g " 0 on every Bpo, mq, m P N. We start by observing that xg, f 0 y A 2 pσq " 0 implies gpoq " 0. Indeed by (3) We assume now g " 0 on Bpo, mq for some m P N. Let v P Spo, mq.Observe that since g is harmonic and gpvq " 0, we have g| spvq P W v .Hence for every j and this implies that gpyq " 0 for every y P spvq and so for every y P Spo, m `1q, namely g vanishes on Bpo, m `1q.The fact that g " 0 follows by induction.
We now fix a measure σ and compute the norm of functions of the family B in A 2 pσq.Evidently, }f 0 } 2 A 2 pσq " B σ .Let v P X and j P I v .By (11), we have Hence the norm of f v,j does not depend on j and coincides with the constant in (8).
is an orthonormal basis of A 2 pσq.

The reproducing kernel of A pσq
In this section we fix a reference measure σ.We show that the Bergman space A 2 pσq is a RKHS and we first obtain a recursive formula for the kernel and we then derive a formula in closed form.Observe that the main ingredient used in the proofs are the harmonic extension and the orthonormal basis defined in the previous section together with the fact that W v are reproducing kernel Hilbert spaces, too.Let z P X.We consider the evaluation functional Φ z : A 2 pσq Ñ C defined by Φ z g " gpzq.Observe that Φ z is a bounded operator.Indeed by the Cauchy-Schwarz inequality Thus A 2 pσq is a RKHS, that is for every z P X there exists K z P A 2 pσq such that xg, K z y A 2 pσq " gpzq, g P A 2 pσq.
Let K : X ˆX Ñ C be the kernel defined by Kpz, xq :" K z pxq.
Since B σ defined in ( 13) is an orthonormal basis of A 2 σ , for every z P X we can write We recall that by (9), for every z P X tv P X : f v,j pzq ‰ 0 for some Hence for every z P X the sum in ( 14) is finite and the decomposition of K z holds true pointwise.
Our goal is to compute K z .To this end, we introduce the auxiliary function Γ : X ˆX ˆX Ñ R which is a parametrization of the family of reproducing kernels for the spaces tW v u vPX .For every pv, z, xq P X ˆX ˆX we set Proof.Since the measure σ is finite and the constant functions are harmonic, K o " 1 Bσ P A 2 pσq.The reproducing property follows from the computations in (10).Now we observe that for every v, z P X such that z P T v and g P A where we used Γpv, z, ¨q| spvq P W v .We now consider the case when |z| " 1.The function K z P A 2 pσq because it is sum of functions in A 2 pσq.We prove the reproducing property.For g P A 2 pσq, by the reproducing formula of K o and ( 16) with v " o, where we used that g is harmonic at o.It remains to consider the case when |z| " m ą 1.We have K z P A 2 pσq since it is the sum of bounded and harmonic functions.For g P A 2 pσq by induction on m and ( 16) with v " v m´1 we have xg, K z y A 2 pσq " ´1 q gpv m´2 q `q `1 q gpv m´1 q `1 b m´1 xg, pΓpv m´1 , z, ¨qq H m´1 y A 2 pσq " ´1 q gpv m´2 q `q `1 q gpv m´1 q `ÿ yPspv m´1 q Γpv m´1 , z, yqgpyq " ´1 q gpv m´2 q `1 q ÿ y"v m´1 gpyq `q ´1 q gpzq ´1 q ÿ yPspv m´1 q y‰z gpyq " gpzq, where we used the fact that g is harmonic at v m´1 .
In Proposition 10 the kernel K z is expressed through a two-step recursive formula.We want to find an explicit formula for K z .
Theorem 11.For every pz, xq P X ˆX Proof.Let z P X and ro, zs " tv t u |z| t"0 .We start by proving that The case z " o follows trivially from Proposition 10 and the convention on sums stated in the Introduction.We prove (18) by induction on m " |z| ě 1.The case m " 1 directly follows from Proposition 10, too.Let m P N, m ą 1 and z P X, with |z| " m, and suppose that (18) holds for every vertex in Bpo, m ´1q.Hence by Proposition 10 we have where we used pq `1qa n´1 ´an´2 " qa n .Hence we proved (18) by induction.Since suppppΓpv t , v t`1 , ¨qq H t q " T vt ztv t u, we have that the t-th term of the sum in (18) does not vanish if and only if x P T vt ztv t u, that is when v t P ro, xs, and hence by (15) where tv t u |z| t"0 " ro, zs.Furthermore, it is clear that K is symmetric, that is Kpz, xq " Kpx, zq.
In the following sections we restrict our attention to the family of the exponentially decreasing radial measures µ α , α ą 1, defined in Example 5.
We shall use the notation L p α and A p α for the Lebesgue and Bergman spaces with respect to µ α , respectively.Furthermore, we denote by K α : X ˆX Ñ R the reproducing kernel of A 2 α .It will be useful to keep track of the weight in the constants introduced in (8), so we denote them by b α,n .In particular observe that in this case there is a relation between the constants: for every n Furthermore we set B α " µ α pXq.Now we show that the kernel K α satisfies an integral condition which will be formalized in Section 4, see (32).

Boundedness of the Bergman projector on L p α
In this section we study the boundedness properties of the extension of the Bergman projector to L p α spaces.For the class of exponentially decreasing radial measures we are able to prove that the extension of the Bergman projector to the relative weighted L p -space is bounded if and only if p ą 1 (see Theorem 17).
In analogy with the operators studied by Zhu in Section 3.4 of [22], we introduce two families of operators.For any real parameters a, b and for c ą 1, we define the integral operators We prove two results that imply the boundedness properties of the Bergman projectors.Theorem 14 is devoted to the study of the boundedness of S a,b,c and T a,b,c on weighted L p -spaces for p ą 1; the case p " 1 needs different arguments and for this reason is treated apart in Theorem 15.The two theorems are the analogues of Theorem 3.11 and Theorem 3.12 in [22], respectively.The proofs of both theorems are postponed to Subsection 3.1.Theorem 14.Let α ą 1, c ą 1 and 1 ă p ă 8.The following conditions are equivalent: (i) the operator S a,b,c is on L p α ; (ii) the operator T a,b,c is bounded on L p α ; (iii) the parameters satisfy c ď a `b, ´pa ă α ´1 ă p pb ´1q .
Theorem 15.Let α ą 1 and c ą 1.The following conditions are equivalent: (i) the operator S a,b,c is bounded on L 1 α ; (ii) the operator T a,b,c is bounded on L 1 α ; (iii) the parameters either satisfy c " a `b, ´a ă α ´1 ă b ´1, or satisfy c ă a `b, ´a ă α ´1 ď b ´1.
We state a corollary which is simply a reformulation of the previous theorems when c " a `b.(ii) the operator T a,b,a`b is bounded on L p α ; (iii) the parameters satisfy ´pa ă α ´1 ă p pb ´1q .
Let β ą 1.Since A 2 β is a closed subspace of L 2 β , there exists an orthogonal projection Observe that by the reproducing property of K β,z " K β pz, ¨q, z P X, we can write the projection P β f of f P L 2 β as follows where we used the orthogonality of P β .Hence we can rewrite P β as the integral operator on L 2 β associated to the reproducing kernel K β , that is It is then natural to investigate whether the restriction of P β to L p β is bounded.Furthermore, when 1 ď p ă 2 one has L 2 β Ĺ L p β and we shall study whether P β admits a bounded extension to L p β .A more general question that we want to answer is whether the integral operator K α β , α ą 1, with kernel K β pz, xqq pα´βq|x| with respect to the measure µ α , that is extends to a bounded operator from L p α to A p α .The following result answers the above questions.
In particular, P α is bounded on L p α if and only if p ą 1.
Remark 18.It is worthwhile observing that the unboundedness of P α on L 1 α may be seen directly with the following example.We make use of Lemma 23 that will be proved in the next subsection.For every n P N, we fix a vertex v n in Spo, nq, and define f n pxq " ½ tvnu pxqq α|x| , x P X.
Clearly, }f n } L 1 α " 1 and f n P L 2 α .Hence, P α f n pzq " K α pz, v n q and by Lemma 23 This shows that P α does not admit a bounded extension to L 1 α .
As a direct application of Theorem 17, we deduce the following result on the dual of Bergman spaces.
Corollary 19.Let 1 ă p ă 8 and α ą 1.Then where 1 ă p 1 ă 8 is such that }f } A p α , for every f P A p α so that g defines a functional in pA p α q ˚.Conversely, for Φ P pA p α q ˚, by the Hahn-Banach theorem, there exists Φ P pL p α q ˚such that Φ| A p α " Φ and }Φ} pA p α q ˚ě } Φ} pL p α q ˚.Then, there exists h P L p 1 α such that Φpf q " Φpf q " xf, hy Hence Φ corresponds to P α h P A 2 α under the pairing (23).

Proofs of Theorems 14 and 15
This subsection is devoted to the proofs of Theorems 14 and 15, splitting them up in various steps.In both statements it is obvious that (i) implies (ii).For the rest of the section α, a, b, c denote real parameters with c ą 1.

Proof that (ii) implies (iii)
In this subsection we suppose that the operator T a,b,c is bounded on L p α and we deduce necessary conditions on the parameters a, b, c, α.Proof.Consider, for x P X, f pxq " q ´R|x| with R P R such that R ą max " 1 ´α p , 1 ´b* .
Since Rp ą 1 ´α we have that f P L p α and for every z P X T a,b,c f pzq " q ´a|z| ÿ xPX K c pz, xqq ´pb`Rq|x| " q ´a|z| `8 ÿ n"0 n"0 q ´pb`Rqn #Spo, nqK c pz, oq by (3) applied to the harmonic function K c pz, ¨q.Hence, since R ą 1 ´b, Now observe that T a,b,c f P L p α implies ÿ zPX q ´pap`αq|z| " 1 `q `1 q `8 ÿ n"1 q p1´ap´αqn ă `8, which holds if and only if ´pa ă α ´1, as required.
From now on we write, for 1 ď p ă 8 }e v,j } p " ¨ÿ yPspvq |e v,j pyq| p ‚1{p , v P X, j P I v .
Proposition 21.Let 1 ď p ă 8.If T a,b,c is bounded on L p α , then a `b ě c.
For every v P Xztou and j P I v , we define g v,j pxq " f v,j pxqq ´R|x| , x P X, where f v,j P B are defined in (9).Since R ą 1´α p , we have that g v,j P L p α .Thus, ´b.Now we use the decomposition ( 14) of K c,z on the orthonormal basis of A 2 c and obtain where we use the orthogonality of B and (12).The norm of T a,b,c g v,j in ÿ yPspvq |e v,j pyq| p " a p n´|v|´1 q n´|v|´1 }e v,j } p p .
For simplicity, for every s P R and 1 ď p ă 8, we put Cps, pq :" `8 ÿ m"1 q p1´sqm´1 a p m´1 , which is finite whenever s ą 1.The above computation yields where Cpap `α, pq converges because ap `α ą 1, by Proposition 20.Furthermore, q ´pRp`αqn a p n´|v|´1 q n´|v|´1 }e v,j } p p " }e v,j } p p q ´pRp`αq|v| `8 ÿ m"1 q p1´pRp`αqqm´1 a p m´1 " }e v,j } p p CpRp `α, pqq ´pRp`αq|v| , where CpRp `α, pq Ñ 1 when R Ñ `8.From the boundedness of T a,b,c and by (20), it follows that for every v P Xztou: which is bounded if and only if c ď a `b.α .It is easy to see that T å,b,c gpxq " q ´pb´αq|x| ÿ zPX K c px, zqgpzqq ´pa`αq|z| " T b´α,a`α,c gpxq, g P L p 1 α .

Calderón-Zygmund decomposition
In this section, we discuss a Calderón-Zygmund decomposition of functions in L 1 α and we formulate the integral Hörmander's condition for kernels on the tree which guarantees the weak type (1,1) boundedness of integral operators which are bounded on L 2 α .As byproduct, we have that P α is of weak type (1,1) for every α ą 1.
By Proposition 6 the measure metric space pX, d, µ α q is nondoubling.We now introduce the Gromov distance ρ, see [2], [18], and show that the measure metric space pX, ρ, µ α q is doubling.For every u, v P X define ρpv, uq " # 0, if u " v; e ´|v^u| , if v ‰ u.
The previous result leads to a Calderón-Zygmund decomposition for integrable functions on the tree at a level t P R `sufficiently large w.r.t the L 1 α -norm of the function.
Proposition 30.Let f P L 1 α and t ą }f } L 1 α {µ α pXq.There exist two families Q and F of disjoint sets of the form Q k,m such that, if we denote by Ω and F the disjoint union of all the sets in Q and F , respectively, the following properties hold: (i) X " Ω \ F ; (ii) |f pzq| ď t for every z P F ; (iii) there exist g, b : X Ñ C and C ą 0 such that f " g `b, supp b Ď Ω, and }g} 2 Proof.For every v P Spo, 1q we consider the decomposition of the sector T v given by Lemma 29.We define two families of subsets Q v and F v following the steps below.Starting from Q k,m " Q 0,0 " T v : 1) if 1 µ α pQ k,m q ÿ zPQ k,m |f pzq|q ´α|z| ą t, then we put Q k,m P Q v and we stop.Otherwise, 2a) if #Q k,m " 1 then Q k,m P F v and we stop; 2b) if #Q k,m ą 1 then for each set in the family Q k,m`1 Y tQ kq`j,m`1 : j P 1, . . .qu we repeat the procedure, starting from 1).We define F v , otherwise.
We denote by Ω and F the (disjoint) union of all the subsets in Q and F , respectively.The sets Ω and F clearly satisfy (i) and (ii).We prove that, for every For every Q P Q we put m ą 0, v P Spo, 1q, Theorem 31.Fix α ą 1 and let K : X ˆX Ñ C be a kernel satisfying the Hörmander's condition (32) with respect to µ α .If the integral operator defined on functions f P L 2 α by Kf pzq " ÿ xPX Kpz, xqf pxqq ´α|x| is bounded on L 2 α , then K is of weak type (1,1).Furthermore, K admits a bounded extension K on L p α , for every 1 ă p ă 2.
The following result is obtained as byproduct of Proposition 13 and Theorem 31.It is a discrete counterpart of the result for (unweighted and holomorphic) Bergman spaces on the hyperbolic disk obtained in [15].

Corollary 16 .
Let 1 ď p ă 8 and α ą 1.If a, b P R are such that a `b ą 1, then the following conditions are equivalent: (i) the operator S a,b,a`b is bounded on L p α ;

Proposition 20 .
Let 1 ď p ă 8.If T a,b,c f P L p α for every f P L p α , then ´pa ă α ´1.

Proposition 22 .
Let 1 ă p ă 8.If T a,b,c is bounded on L p α , then α ´1 ă ppb ´1q.Proof.The boundedness of T a,b,c on L p α is equivalent to the boundedness of the adjoint operator T å,b,c on L p 1 since a `b ě c and, by (25), b ´α ą γp.Similarly when γ " c´a´α p .In conclusion, (24) holds and by Schur's test the operator S a,b,c is bounded on L p α pXq.Notice that Proposition 26 shows that (iii) implies (i) in Theorem 14. Proposition 27.If a `b ě c and ´a ă α ´1 ă b ´1, when c " a `b; ´a ă α ´1 ď b ´1, when c ă a `b, then S a,b,c is bounded on L 1 α .Proof.Let f P L 1 α .We suppose c ‰ a `α and we observe that, since a `α ą 1, by Lemma 25 }S a,b,c f } L 1 α " ÿ zPX ˇˇˇˇÿ xPX |K c pz, xq|f pxqq ´b|x| ˇˇˇˇq ´pa`αq|z| ď ÿ xPX |f pxq|q ´b|x| ÿ zPX |K c pz, xq|q ´pa`αq|z| À ÿ xPX |f pxq|q ´b|x| p1 `q´pa`α´cq|x| q À ÿ xPX |f pxq|q ´α|x| " }f } L 1 α , tz, xu Ď T y for some y P spvq;We now show that Γpv, ¨, ¨q is the reproducing kernel of W v , namely that for z P spvq we have ϕpzq " xϕ, Γpv, z, ¨qy Wv , ϕ P W v .W v .It is easy to see that Γpv, z, ¨q is harmonic on Bpo, |v|q so that we can consider the harmonic extension pΓpv, z, ¨qq H |v| , which is bounded by construction.Indeed from the definition of harmonic extension we have for every x P T v ztvu Proposition 10.Let z P X and ro, zs " tv t u Observe that Γ is symmetric in the second and third variables.Furthermore, Γpv, z, ¨q is the null function if z R T v ztvu and whenever z P T v ztvu we have supppΓpv, z, ¨qq " T v ztvu.Moreover, the values of Γpv, z, ¨q on T v ztvu are completely determined by the values on spvq, as the value of Γpv, z, ¨q at x P T v ztvu is equal to the value at p |x|´|v|´1 pxq P spvq.zv Figure 2: Partial representation of the function Γpv, z, ¨q on T v .The value of Γpv, z, ¨q at the vertices in the red area is #spvq´1#spvq , while in the blue area is ´1 #spvq .Clearly, Γpv, z, vq " 0. and it vanishes elsewhere.We recall that if z R T v , then Γpv, z, ¨q " pΓpv, z, ¨qq H |v| is the null function.|z| t"0 .The kernel K z is , we have Remark 12.The confluent of two vertices z, x P X is the common vertex of ro, xs and ro, zs farthest from o, denoted by z ^x.It is possible to see that the value of the kernel K at pz, xq P X ˆX depends only on the values of |x|, |z| and |z ^x|.Indeed, from (17) and the fact that Γpv, z, xq does not vanish if and only if v P ro, zs X ro, xs " ro, z ^xs, we have the sum of |f v,j | p on the sphere Spo, nq vanishes for every n ď |v|.If n ą |v|, then the sum is on Spo, nq X T v and if z P T v is such that |z| " n then p |z|´|v|´1 pzq is the unique vertex in spvq such that z lies in its sector.