Littlewood–Paley Theory for Triangle Buildings

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Introduction
Let ( , F, π) be a σ -finite measure space. A sequence of σ -algebras (F n : n ∈ Z) is a filtration if F n ⊂ F n+1 . Given f a locally integrable function on by E[ f |F n ], we denote its conditional expectation value with respect to F n . Let M * and S denote, respectively, the maximal function and the square function defined by where d n f = f n − f n−1 . The Hardy and Littlewood maximal estimate (see [8]) implies that from where it is easy to deduce that for p ∈ (1, ∞] For the square function, if p ∈ (1, ∞), then there is C p > 1 such that The inequality (1.2) goes back to Paley [12], and has been reproved in many ways, for example, [2][3][4]7,10]. Its main application is in proving the L p -boundedness of martingale transforms (see [2]), that is, for operators of the form T f = n∈Z a n d n f where (a n : n ∈ Z) is a sequence of uniformly bounded functions such that a n+1 is F n -measurable. In 1975, Cairoli and Walsh (see [5]) have started to generalize the theory of martingales to two-parameter cases. Let us recall that a sequence of σ -fields (F n,m : n, m ∈ Z) is a two-parameter filtration if This lead to the introduction of other (smaller) classes of martingales (see [19,20]). In particular, in [5], Cairoli  Under (F 4 ), the result obtained by Jensen, Marcinkiewicz, and Zygmund in [9] implies that the maximal function is bounded on L p ( ) for p ∈ (1, ∞]. In this context, the square function is defined by |d n,m f | 2 1/2 (1.6) where d n,m denote the double difference operator, i.e. In [11], it was observed by Metraux that the boundedness of S on L p ( ) for p ∈ (1, ∞) is implied by the one parameter Littlewood-Paley theory. Also the concept of a martingale transform has a natural generalization, that is, T f = n,m∈Z a n,m d n,m f where (a n,m : n, m ∈ Z) is a sequence of uniformly bounded functions such that a n+1,m+1 is F n,m -measurable.
In this article, we are interested in a case when the condition (F 4 ) is not satisfied. The simplest example may be obtained by considering the Heisenberg group together with the non-isotropic two parameter dilations δ s,t (x, y, z) = (sx, t y, st z).
Since in this setup the dyadic cubes do not posses the same properties as the Euclidean cubes, it is more convenient to work on the p-adic version of the Heisenberg group. We observe that this group can be identified with 0 , a subset of a boundary of the building of GL(3, Q p ) consisting of the points opposite to a given ω 0 . The set 0 has a natural two-parameter filtration (F n,m : n, m ∈ Z) (see Sect. 2 for details). The maximal function and the square function are defined by (1.5) and (1.6), respectively. The results we obtain are summarized in the following three theorems.
Theorem C If (a n,m : n, m ∈ Z) is a sequence of uniformly bounded functions such that a n+1,m+1 is F n,m -measurable, then the martingale transform Let us briefly describe methods we use. First, we observe that instead of (F 4 ) the stochastic basis satisfies the remarkable identity (2.2). Based on it, we show that the following pointwise estimate holds proving the maximal theorem. Thanks to the two-parameter Khintchine's inequality, to bound the square function S, it is enough to show Theorem C. To do so, we define a new square function S which has a nature similar to the square function used in the presence of (F 4 ). Then, we adapt the technique developed by Duoandikoetxea and Rubio de Francia in [6] (see Theorem 3). This implies L p -boundedness of S. Since S does not preserve the L 2 norm, the lower bound requires an extra argument. Namely, we view the square function S as an operator with values in L p ( 2 ) and take its dual. As a consequence of Theorem 3 and the identity (4.7), the latter is bounded on L p . Finally, let us comment on the behavior of the maximal function M * close to L 1 . Based on the pointwise estimate (1.7), in view of [8], we conclude that M * is of weaktype for functions in the Orlicz space L(log L) 3 . To better understand the maximal function M * , we investigate exact behavior close to L 1 . This together with weighted estimates is the subject of the forthcoming paper. It is also interesting how to extend Theorems A, B and C to higher rank and other types of affine buildings.
Let H be the family of affine hyperplanes, called walls, To each wall H j;k , we associate r j;k the orthogonal reflection in a, i.e.
Set r 1 = r 1;0 , r 2 = r 2;0 and r 0 = r 0;1 . The finite Weyl group W 0 is the subgroup of GL(a) generated by r 1 and r 2 . The affine Weyl group W is the subgroup of Aff(a) generated by r 0 , r 1 and r 2 .
Let C be the family of open connected components of a \ H ∈H H . The elements of C are called chambers. By C 0 , we denote the fundamental chamber, i.e.
The group W acts simply transitively on C. Moreover, C 0 is a fundamental domain for the action of W on a (see e.g. [1, VI, §1-3]). The vertices of C 0 are {0, λ 1 , λ 2 }. The set of all vertices of all C ∈ C is denoted by V ( ). Under the action of W , V ( ) is made up of three orbits, W (0), W (λ 1 ), and W (λ 2 ). Vertices in the same orbit are said to have the same type. Any chamber C ∈ C has one vertex in each orbit or in other words one vertex of each of the three types.
The family C may be regarded as a simplicial complex by taking as the simplexes all non-empty subsets of vertices of C, for all C ∈ C. Two chambers C and C are iadjacent for i ∈ I if C = C or if there is w ∈ W such that C = wC 0 and C = wr i C 0 . Since r 2 i = 1 this defines an equivalence relation. The fundamental sector is defined by Given λ ∈ P and w ∈ W 0 the set λ + wS 0 is called a sector in with base vertex λ. The angle spanned by a sector at its base vertex is π/3.

The Definition of Triangle Buildings
For the theory of affine buildings, we refer the reader to [13]. See also the first author's expository paper [14], for an elementary introduction to the p-adics, and to precisely the sort of the buildings which this paper deals with.
A simplicial complex X is anÃ 2 building, or as we like to call it, a triangle building, if each of its vertices is assigned one of the three types, and if it contains a family of subcomplexes called apartments such that 1. Each apartment is type-isomorphic to , 2. Any two simplexes of X lie in a common apartment, 3. For any two apartments, A and A , having a chamber in common, there is a type-preserving isomorphism ψ : We assume also that the system of apartments is complete, meaning that any subcomplex of X type-isomorphic to is an apartment. A simplex C is a chamber in X if it is a chamber for some apartment. Two chambers of X are i-adjacent if they are i-adjacent in some apartment. For i ∈ I and for a chamber C of X , let q i (C) be equal to It may be proved that q i (C) is independent of C and of i. Denote the common value by q, and assume local finiteness: q < ∞. Any edge of X , i.e., any 1-simplex, is contained in precisely q + 1 chambers. It follows from the axioms that the ball of radius one about any vertex x of X is made up of x itself, which is of one type, q 2 + q + 1 vertices of a second type, and a further q 2 + q + 1 vertices of the third type. Moreover, adjacency between vertices of the second and third types makes them into, respectively, the points and the lines of a finite projective plane.
A subcomplex S is called a sector of X if it is a sector in some apartment. Two sectors are called equivalent if they contain a common subsector. Let denote the set of equivalence classes of sectors. If x is a vertex of X and ω ∈ , there is a unique sector denoted [x, ω] which has base vertex x and represents ω.
Given any two points ω and ω ∈ , one can find two sectors representing them which lie in a common apartment. If that apartment is unique, we say that ω and ω are opposite, and denote the unique apartment by [ω, ω ]. In fact, ω and ω are opposite precisely when the two sectors in the common apartment point in opposite directions in the Euclidean sense.

Filtrations
We fix once and for all an origin vertex O ∈ X and a point ω 0 ∈ . Choose O so that it has the same type as the origin of . Let S 0 = [O, ω 0 ] be the sector representing ω 0 with base vertex O. By 0 , we denote the subset of consisting of ω's opposite to ω 0 . For purposes of motivation only, we recall that if X is the building of GL(3, Q p ), then 0 can be identified with the p-adic Heisenberg group (see Appendix 1 for details). Let A 0 be any apartment containing S 0 . By ψ, we denote the type-preserving isomorphism between A 0 and such that ψ(S 0 ) = −S 0 . We set ρ = ψ • ρ 0 where ρ 0 is the retraction from X to A 0 . With these definitions, ρ : X → is a typepreserving simplicial map, and for any ω ∈ 0 the apartment [ω, ω 0 ] maps bijectively to with ω 0 mapping to the bottom (of Fig. 1) and ω mapping to the top.
For any vertex x of X , define the subset E x ⊂ 0 to consist of all ω's such that By F λ , we denote the σ -field generated by sets E x for x ∈ X with ρ(x) = λ. There are countably many such x, and the corresponding sets E x are mutually disjoint, and hence, F λ is a countably generated atomic σ -field.
Let denote the partial order on P where λ μ if and only if λ − μ, α 1 ≤ 0 and λ − μ, α 2 ≤ 0. If we draw and orient as in Fig. 1, then λ μ exactly when μ lies in the sector pointing upward from λ.
Proof Choose any vertex x so that ρ(x) = μ. Because λ μ, there is a unique vertex y in the sector [x, ω 0 ] so that ρ(y) = λ. For any ω ∈ E x , the apartment [ω, ω 0 ] contains x, and hence, it contains [x, ω 0 ], which hence contains y. This establishes that E x ⊆ E y . In other words, each atom of F μ is a subset of some atom of F λ . Hence, each atom of F λ is a disjoint union of atoms of F μ .
In fact, Proposition 2.1 says that (F λ : λ ∈ P) = F iλ 1 + jλ 2 : i, j ∈ Z is a two parameter filtration. Let All σ -fields in this paper should be extended so as to include π -null sets. A depends only on that part of the apartment which retracts to a certain "lower" half-plane with boundary parallel to λ 2 (respectively λ 1 ).
We note that the Cairoli-Walsh condition (F 4 ) introduced in [5] is not satisfied, i.e.
Instead of (F 4 ), we have
The ball in X of radius 1 around x has the structure of a finite projective plane. In Fig. 2, the spot marked x is for vertices of X which retract via ρ to λ. Recall that E x is an atom of the σ -field F λ . The spot marked p 1 is for vertices retracting to λ + λ 1 ; the spot marked l is for vertices retracting to λ + λ 2 ; the spot marked l 1 is for vertices retracting to λ + λ 1 − λ 2 ; etc. In the ball of radius 1 around x, only x itself retracts to the spot marked x. The line-type vertex known as l 0 is the only vertex in the ball retracting to its spot; q line-type vertices retract to the same spot as l 1 ; the remaining q 2 line-type vertices retract to the spot marked l. Likewise, p 0 is the unique point-type vertex of the ball retracting to its spot; q point-type vertices retract to the spot marked p; q 2 retract to the same spot as p 1 . It follows that where p runs through the point-type vertices of the ball, l runs through the line-type vertices of the ball, and ∼ stands for the incidence relation. We have Therefore, we obtain which finishes the proof of (2.1). Applying one more average to the next to the last expression of (2.4), we get For any line l p 0 , there are q points p such that p ∼ l and p l 0 and among them there is exactly one incident to l 1 . Hence, in the last sum, each line l p 0 appears q − 1 times. Thus, we can write The following lemma describes the composition of projections on the same level. (2.5) Proof We carry out the proof for k ≥ j ≥ 0. For any ω ∈ 0 , there is a connected chain of vertices ( is a connected chain of vertices and that ρ( , the edges between the x i 's and the triangles pointing downward from those edges to ω 0 . Referring to Fig. 3, the extra triangle pointing downward from the first edge has vertices x 0 , x 1 , and y 0 . Note that Proceeding one step at a time, one may verify that the restriction of ρ to B is an injection and that B and ρ(B) are isomorphic complexes. By basic properties of affine buildings, one knows it is possible to extend B to an apartment. Any such apartment will retract bijectively to , and will be of the form form [ω, ω 0 ] where ω is the equivalence class represented by the upward pointing sectors of the apartment. Moreover, using the definition of π one may calculate that The important point is that the measure of the set depends only on the level of λ and the length of the chain.
Basic properties of affine buildings imply that any apartment containing x 0 and x k contains the entire chain. Hence, Fix x 0 . Proceeding one step at a time, one sees there are q k connected chains which is the same thing.
Consider E λ E μ . If λ μ then the product is equal to E λ ; similarly if μ λ. If λ and μ are incomparable, the following lemma allows us to reduce to the case where λ and μ are on the same level.

Lemma 2.4 Suppose λ ∈ P and
for i, k ∈ N. Then for any locally integrable function f on 0  Proof We first prove (2.6) for i = 1 and k = 1.
Because Fig. 2 to fix the notation, and note that if p 1 retracts to λ, then x retracts to λ and p to μ. One calculates: Next consider the case i = 1, k > 1. Set μ = μ+λ 1 , ν = μ+λ 1 −λ 2 and ν = ν +λ 1 (see Fig. 4). Since F μ is a subfield of F μ , we have Thus, applying Lemma 2.3, we obtain where in the last step we have used the case k = 1. Now apply induction on k and Lemma 2.3 again to get To extend to the case i > 1, use induction on i and observe that The proof of (2.8) is analogous, starting with the case i = 1, k = 0. Identity that (2.6) can be read as E μ E λ = E μ E λ . The expectation operators are orthogonal projections with respect to the usual inner product, and taking adjoints gives To be more precise, one takes the inner product of either side of (2.7) with some nice test function, applies self-adjointness, and reduces to (2.6). Likewise, (2.9) follows from (2.8).

. Then for any locally integrable function f on
Proof Suppose k ≥ 0. By Lemma 2.4 for any j ≥ 0, we have So if g is F λ+ j λ 2 -measurable and compactly supported, then The test functions g which we use are sufficient to distinguish between one F i,∞ -measurable function and another. Since E i,∞ E μ f and E λ E μ f are both F i,∞measurable, the proof is done.

Maximal Functions
The natural maximal function M * for a locally integrable function f on 0 is defined by In addition, we define two auxiliary single-parameter maximal functions

Lemma 3.1 Let λ ∈ P and k ∈ N. For any non-negative locally integrable function f on
Proof We may assume λ = 0. Let us define (see Fig. 5) We show Hence, By repeated application of Lemma 2.4, we have which finishes the proof of (3.1). By iteration of (3.1), we obtain Proof Inequalities (3.2) are two instances of Doob's well-known maximal inequality for single parameter martingales (see e.g. [15]). To show (3.3), consider a non-negative f ∈ L p ( 0 , F μ ). Fix λ ∈ P. Since f ∈ L p ( 0 , F μ ) for any μ μ we may assume μ λ. Let By Lemma 3.1, If λ = iλ 1 + jλ 2 , then repeated application of Lemma 2.5 gives By taking the supremum over λ ∈ P, we get Hence, by (3.2), we obtain (3. 3) for f ∈ L p ( 0 , F μ ). Finally, a standard Fatou's lemma argument establishes the theorem for arbitrary f ∈ L p ( 0 ).

Square Function
Let f be a locally integrable function on 0 . Given i, j ∈ Z, we define projections Note that L i (respectively R j ) is the martingale difference operator for the filtration F i,∞ : i ∈ Z (respectively F ∞, j : j ∈ Z ). For λ = iλ 1 + jλ 2 , we set The following development is inspired by that of Stein and Street in [17]. We start by defining the corresponding square function.
We will also need its dual counterpart Moreover, on L 2 ( 0 ) square functions S and S preserve the norm.

Proof
Since Hence, S preserves the norm. For p = 2, we use the two-parameter Khintchine inequality (see [12]) and bounds on single parameter martingale transforms (see [2,15,18]). Let ( i : i ∈ Z) and ( j : j ∈ Z) be sequences of real numbers, with absolute values bounded above by 1. For N ∈ N, we consider the operator which may be written as a composition L N R N where Since by Burkholder's inequality (see [2,15]) the operators R N and L N are bounded on L p ( 0 ) with bounds uniform in N , we have Setting r k to be the Rademacher function, by Khintchine's inequality, we get |i|,| j|≤N which is bounded by f p L p . Finally, let N approach infinity and use the monotone convergence theorem to get For the opposite inequality, we take f ∈ L p ( 0 )∩ L 2 ( 0 ) and g ∈ L p ( 0 )∩ L 2 ( 0 ) where 1/ p + 1/ p = 1. By polarization of (3.4) and the Cauchy-Schwarz and Hölder inequalities, we obtain Given a set {v λ : λ ∈ P} of vectors in a Banach space, we say that λ∈P v λ converges unconditionally if, whenever we choose a bijection φ : N → P, exists, and is independent of φ.
Equivalently, we may ask that for any increasing, exhaustive sequence (F N : N ∈ N) of finite subsets of P, the limit The following proposition provides a Calderón reproducing formula. Let For f ∈ L p ( 0 ) and g ∈ L p ( 0 ), where 1/ p + 1/ p = 1, we have (3.5) In particular,

Theorem 3
Let (T λ : λ ∈ P) be a family of operators such that for some δ > 0 and p 0 ∈ (1, 2) Then for any p ∈ ( p 0 , 2] the sum λ∈P T λ converges unconditionally in the strong operator topology for operators on L p ( 0 ).
Proof First, recall that the Cotlar-Stein Lemma (see e.g. [16]) states that (3.7) implies the unconditional convergence of λ∈P T λ in the strong operator topology on L 2 ( 0 ). Let (F N : N ∈ N) be an arbitrary increasing and exhaustive sequence of finite subsets of P. For N > 0, we set By (3.6), (3.7) and interpolation, each T μ is bounded on L p for p ∈ [1, 2] and the same holds for the finite sum V N . We consider f ∈ L p ( 0 ) for p ∈ ( p 0 , 2). By Proposition 3.2 and Theorem 2, we Finally, by change of variables, we get Assuming there is δ p > 0 such that we can estimate (3.11) Theorem 2, Proposition 3.2 and (3.11) imply that the V M are uniformly bounded on L p . For the proof of (3.10), we consider an operator T defined for f ∈ L p π, 2 (P) by Since D λ L 1 →L 1 1 and T μ L 1 →L 1 1, we have Also, by (3.8), we can estimate Therefore, using interpolation between L 1 π, 1 (P) and L 2 π, 2 (P) we obtain that there is δ > 0 such that Because |D λ g| L * R * (|g|), and because Theorem 1 says that L * and R * are bounded on L p 0 , we know that (D λ : λ ∈ P) is bounded on L p 0 (π, ∞ (P)). Of course the same holds for D λ+γ +γ : λ ∈ P . Hence, by (3.9), we get Next, interpolating between L p 0 (π, p 0 (P)) and L p 0 (π, ∞ (P)) gives a δ > 0 such that Finally, interpolating between L p 0 π, 2 (P) and L 2 π, 2 (P) , we obtain (3.10).
To complete the proof, we are going to show that (V N f : N ∈ N) is a Cauchy sequence in L p ( 0 ). Let us consider g ∈ L p ( 0 ) ∩ L 2 ( 0 ). Setting and using the log-convexity of the L q -norms, we get Lp .
Since (V N g : N ∈ N) converges in L 2 ( 0 ) and is uniformly bounded on Lp( 0 ) it is a Cauchy sequence in L p ( 0 ). For an arbitrary f ∈ L p ( 0 ) use the density of g's as above. We have Thus, (V N f : N ∈ N) also converges, and this finishes the proof of the theorem.

Double Differences
The martingale transforms are expressed in terms of double differences defined for a martingale f = ( f λ : λ ∈ P) as

Martingale Transforms
The following proposition is our key tool.

Martingale Square Function
For a martingale f = ( f λ : λ ∈ P) there is the natural square function defined by Although S does not preserve L 2 norm, we have Theorem 5 For every p ∈ (1, ∞) there is C p > 0 such that Proof We start from proving the identity Let us notice that Therefore, consecutively we have Hence, by Lemma 2.2, which together with (4.8) implies (4.7). Next, we consider an operator T defined for a function f ∈ L p ( 0 ) by We also need an operator T acting on g ∈ L p ( 0 ) as We observe that by two-parameter Khinchine's inequality and Theorem 4 we have The dual operator T : L p π, 2 (Z 2 ) → L p ( 0 ) is given by Since T g ∈ L p π, 2 (Z 2 ) , by (4.7) and Theorem 4, Therefore, by Cauchy-Schwarz and Hölder inequalities and since T f L p ( 2 ) = S f L p the proof is finished.
Finally, the method of the proof of Theorem 3, together with Theorems 4 and 5 shows the following Theorem 6 Let (T λ : λ ∈ P) be a family of operators such that for some δ > 0 and p 0 ∈ (1, 2) Then for any p ∈ ( p 0 , 2] the sum λ∈P T λ converges unconditionally in the strong operator topology for the operators on L p ( 0 ).

Appendix: About 0 and Heisenberg Group
In some cases 0 can be identified with a Heisenberg group over a nonarchimedean local field. Let us recall, that F is a nonarchimedean local field if it is a topological field 1 that is locally compact, second countable, non-discrete and totally disconnected.
Since F together with the additive structure is a locally compact topological group it has a Haar measure μ that is unique up to multiplicative constant. Observe that for each x ∈ F, the measure μ x (B) = μ(x B) is also a Haar measure. We set where B is any measurable set with finite and positive measure. By O = {x ∈ F : |x| ≤ 1}, we denote the ring of integers in F. We fix π ∈ p − p 2 , where p = x ∈ F : |x| < 1 .
We are going to sketch the construction of a building associated to GL(3, F). For more details, we refer to [14]. A lattice is a subset L ⊂ F 3 of the form where {v 1 , v 2 , v 3 } is a basis of F 3 . We say that two lattices L 1 and L 2 are equivalent if and only if L 1 = aL 2 for some nonzero a ∈ F. Then X , the building of GL(3, F), is the set of equivalence classes of lattices in F 3 . For x, y ∈ X there are a basis {v 1 , v 2 , v 3 } of F 3 and integers j 1 ≤ j 2 ≤ j 3 such that (see [14,Proposition 3.1]) x = Ov 1 + Ov 2 + Ov 3 , and y = π j 1 Ov 1 + π j 2 Ov 2 + π j 3 Ov 3 .