An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.


Introduction
The boundary behaviour of harmonic functions on the upper half plane or the unit disc is well understood for admissible (that is, non-tangential) convergence. A function harmonic on the disc or the half-space is non-tangentially bounded almost everywhere if and only if its area function 2 (where is a Stoltz domain at the boundary point and is Lebesgue measure) is finite for almost every . This is called the Lusin area theorem ( [4,13]; see also [7,15] and references therein).
This equivalence has been established for rank-one symmetric spacess in [9] and for infinite homogeneous trees in [1,6,11] and, with a more probabilistic argument, in [14]. The aim of this paper is to prove an area theorem for product of trees, of the type: existence of admissible limits finiteness of the area function almost everywhere. Our approach adapts to this discrete environment ideas developed in [10,12] for the product of discs and of rankone symmetric spaces (see also [8]). The argument overcomes several complications arising from the use of discrete difference equations instead of the classical identities for the Green function and the Laplacian, and extends the results of [1] valid for one tree. The converse implication, finiteness of this area function on a product existence of admissible limits almost everywhere, cannot hold in full generality in a cartesian product (see Proposition 2.1; for hints on variants of the area theorem suitable to prove the converse implication see Remark 2.2). Thereby we extend the Lusin area theorem proved in [1] for one tree to the product of finitely many trees; for simplicity, we restrict attention to the product of two trees of the same homogeneity. We consider jointly harmonic functions that are non-tangentially bounded (almost) everywhere.
The product of two discs was studied in [12], where the Lusin area theorem was proved for jointly harmonic functions non-tangentially bounded only locally, and then extended to the product of finitely many rank-one symmetric spaces in [10]. The argument of [10,12] estimates the area integral via the Green formula, that holds on bi-discs of finite radius; then a very complicated computation, that makes use of suitable mollifiers, is used to restrict attention to the portion of the bi-discs contained in a truncated admissible domain where is bounded. This local approach is extremely difficult in a discrete setting, where we cannot use derivatives of the mollifiers; we shall consider the local version of our theorem in a future paper.
The motivation of this work is the following. The area theorem has never been stated for higher rank symmetric spaces except in the degenerate case of the product of rank-one spaces like half-planes or discs. An appropriate expression of the area function for nondegenerate higher rank is therefore unknown. It was observed by A. Korányi that it should be easier to find this expression in the combinatorial setting of higher rank buildings of Bruhat-Tits. The degenerate case of higher-rank buildings is the product of homogeneous or semi-homogeneous trees, and the present work is the first step towards this goal; the next step should be the environment of rank-2 affine buildings.

Homogeneous Trees
We adopt most of the terminology of [6]: here is a review. A tree is a connected, simply connected, locally finite graph. With abuse of notation we shall also write for the set of vertices of the tree. We suppose that is homogeneous, that is, every vertex of belongs of distinct vertices such that 0 , and 1 for all ; this sequence is called the geodesic path from to . The integer is denoted by ; is a metric on . We fix a reference vertex ; this induces a partial ordering in : if belongs to the geodesic path from to . Every , , has exactly one neighbor closer to , called the predecessor of . For , the length is defined as . For any vertex and any integer , is the vertex of length in the geodesic from to . The Laplace operator associated with is .

Definition 2 A function
is harmonic if 0 for every . We shall say that is harmonic at if 0 for the vertex . A function is harmonic on a subset of f it is harmonic at every vertex therein.

Restricted Non-Tangential Convergence and the Area Function on a Tree
Let be the set of infinite (one-sided) geodesics starting at . In analogy with the previous notation, for and , is the vertex of length in the geodesic . For the arc generated by is the set . The sets , , form an open base at . Equipped with this topology is compact and totally disconnected. Moreover, let . Then the sets generate a compact topology on . Denote by the set of all the oriented edges in (i.e., ordered pairs of neighbours). For denote by the beginning vertex of and by the ending vertex: . The choice of the reference vertex gives rise to a positive orientation on edges: an edge is positively oriented if is the predecessor of with respect to . The beginning and ending vertices induce two maps and defined as above. These maps induce two different liftings to , and , of any . We shall henceforth write .

Definition 5
The area function of on is the function on defined by 2 1 2 . Definition 6 (admissible regions) Let be a measurable subset of and , or more generally let be a measurable function on . For simplicity let us fix a reference vertex . We define the admissible region (or Stoltz domain) , that is, the set of vertices whose distance from the geodesic ray from to some is at most . For every positive integer the family is a partition of into 1 1 open and closed sets. We define the -isotropic measure on the algebra of sets generated by the sets , by 1 . (2. 2) The measure extends to a regular Borel probability measure on , called harmonic measure,. This is the hitting distribution of the random walk on starting at induced by .

Product of Trees and Statement of the Area Theorem
Let now 1 , 2 be homogeneous trees with reference vertices 1 , 2 respectively, 1 2 1 2 measurable and let be a measurable function (we shall see in Section 2.4 that, without loss of generality, may be assumed to be a constant integer; for simplicity, it is convenient to regard as a constant throughout this paper). For every , the tube (or bi-cone) of width is 1 2 . The admissible region of width over a subset is .
Definition 7 When applied to the first or second variable of functions defined on , the Laplacian is denoted by , 1 2. A function on 1 2 is called bi-harmonic (or jointly harmonic) if 1 2 0. Without loss of generality, we restrict attention to real valued bi-harmonic functions.
By abuse of notation, we shall denote again by the product measure 1 2 on of the harmonic measures in each tree.

Definition 8
We have defined the gradient on a tree (Definition 3), hence also the gradients on each variable of the product of two trees, denoted by 1  Our goal is to extend to the context of the cartesian product of two trees the following theorem, that is a particular case of the results of [1]: If is a bounded harmonic function on , for every fixed 0, for almost every . Our extension to , inspired by statements valid on products of disks or half-spaces [3,8,12] or of symmetric spaces [10], is the following:

Lusin Area Theorem for bi-harmonic functions Let a bi-harmonic function on that is non-tangentially bounded almost everywhere, that is, for almost all there exists such that
sup .
Then, for every 0, there is a subset such that and for almost every the area sum 12 2 is finite. If is uniformly bounded, then the area sum is finite -almost everywhere. 1 2 . If is constant on one of its two variables and unbounded in the other, the area function vanishes identically, hence the conclusion of the Main Theorem holds trivially, although is unbounded. Remark 2.2 Despite of the previous Proposition, some form of an inverse Lusin area theorem for the bi-disc is known, due to J. Brossard, who showed in [2] that for a biharmonic function 1 2 , the finiteness over a bi-cone of the area integral introduced by P. and M.P. Malliavin in [12] is actually equivalent to the existence of a finite limit when 1 2 tends to the boundary inside any smaller bi-cone of the function 1 2 This fact, and a stimulating presentation due to R.F. Gundy [8], suggest that, for a bounded biharmonic function on a bi-tree, non-tangential convergence and finiteness of the area sum should be almost everywhere equivalent provided that the area sum of [12] is replaced by a suitably larger one. This will be the subject of a future paper.

Reduction to Uniformly Bounded Bi-Harmonic Functions
We claim that, for every bi-harmonic function on that is non-tangentially bounded almost everywhere and every 0, there is a constant 0 and a subset with such that is bounded in . Indeed, observe that we are assuming that, for almost all , sup . (2.4) This means that, up to a null set, is the union of subsets of the type for all and for some .
It follows that, given any non-negative integer , there exists a closed subset with 1 and constants , such that on . This proves the claim.
If is uniformly bounded, then ; hence the last statement of the Theorem follows from the first.

Green Kernel
Definition 9 Let us regard the nearest-neighbor isotropic transition operator of Definition 1 as a kernel: . Then its operator powers are given by 1 . The Green kernel is defined as 0 . (3.1) The Green function singular at the vertex is defined as , and it is easy to see [5, (4.49)] that 1 . (3. 2) The Green function on 1 2 is now 1 2 , that is

Example 3.4 For a vertex in a tree and
, the Poisson kernel normalized at in a tree is the harmonic function defined as follows. Let be the bifurcation index, that is the number of edges in common between the finite geodesic path and the infinite geodesic starting at ; similarly, let be the number of edges in common between the geodesics and . Then it is easy to see [5]  is finite. In particular, the (unrestricted) area sum is finite almost everywhere. It actually follows from Eq. 3.5 that the assumption that be bounded is unnecessary; anyway it is not restrictive for our results, in view of the uniformization procedure of Section 2.4.

Gradient and Laplacian
For all functions , on the homogeneous tree let us introduce the bilinear form If and are functions on the edges of , the bilinear form 1 1 , for each fixed , is a (real) inner product that we denote by ; its associated norm satisfies 2 1 1 2 . (3.8) In particular, the norm in Eq. 2.1 is of this type. In what follows we shall consider this inner product on two copies 1 and 2 of the tree : for clarity, we shall denote the bilinear forms on each of the two copies by 1 and 2 , respectively.

Definition 10
If is defined on the vertices of a tree , let 2 .
Let us write and note that 2 2 . (3.9) The following useful facts follow immediately from the definition of (Definition 1) and of (Definition 3). and, if is bounded, then 2 .

The Green Formula in One Tree
Definition 11 (Boundary) The boundary of a finite subset is .
The Green formulas are well known in the continuous setup. In the discrete context of a tree, the following interesting analogue has been observed in [6].

Identities for Discrete Difference Operators in One Variable
The following statement is easily verified. The last identity in its part follows from Eq. 3.9. .
The following statements are immediate consequences of the definitions of gradient and mean: Corollary 4.2 2 2 .
As a consequence we have the Green identities already observed in [6]: