Martingale Type, the Gamlen-Gaudet Construction and a Greedy Algorithm

In the present paper we identify those filtered probability spaces $(\Omega,\, \mathcal{F},\, \left(\mathcal{F}_n\right),\, \mathbb{P})$ that determine already the martingale type of a Banach space $X$. We isolate intrinsic conditions on the filtration $(\mathcal{F}_n)$ of purely atomic $\sigma$-algebras which determine that the upper $\ell^p$ estimates \[ \|f\|_{L^p(\Omega,\, X)}^p\leq C^p\left( \|\mathbb{E} f|\mathcal{F}_0\|^p_{L^p(\Omega,\, X)}+\sum_{n=1}^{\infty} \|\Delta_n f\|^p_{L^p(\Omega,\, X)}\right),\qquad f\in L^p(\Omega,X)\] imply that the Banach space $X$ is of martingale type $p$. Our paper complements \mbox{G. Pisier's} investigation \cite{Pisier1975} and continues the work by S. Geiss and second named author in \cite{Geiss2008}.


Introduction
Let (Ω, F, P) be a probability space.Let (F n ) be an increasing sequence of purely atomic subσ-algebras F. If F is the smallest σ-algebra containing F n , then we say that (Ω, F, (F n ) , P) forms a filtered probability space.Given a Banach space X we let L p (Ω, F, P ; X) denote the Banach space of p-integrable Bochner measurable functions.If the underlying filtration and measure are clear from the context we abbreviate our notation to L p (Ω, X) or L p (X) and L p (Ω) in case X = R.
By way of introduction we recall Burkholder's classical inequality for scalar valued martingales.If 1 < p < ∞ there exist c p > 0, and C p < ∞ such that where ).The estimate (1) implies that for 1 < p < ∞ martingale differences converge unconditionally in L p (Ω, F, P) i.e.
for f ∈ L p (Ω, F, P) and ε n ∈ {−1, 1}.Moreover for 1 < p ⩽ 2 inequality (1) yields the upper ℓ p estimates in term of martingale differences, Passing from scalar valued martingales to vector valued ones we fix a Banach space X.It is well known that neither Burkholder's inequalities nor its consequences (2) and (3) hold true for vector valued martingales in general (see [13]).The validity of ( 2) respectively (3) each define severely restricting isomorphic invariants on the underlying Banach space X.We say that X is a UMD space if there exists C p < ∞ such that for any filtered probability space (Ω, F, (F n ) , P) and any f ∈ L p (X) and ε n ∈ {−1, 1} we have where . By U M D(X, p) we denote the smallest constant for which inequality (4) is satisfied.As pointed out by Pisier the UMD property of a Banach space X is independent of 1 < p < ∞ (see [5,4], [13]).
A Banach space X satisfies martingale type p if there exists C p < ∞ such that any filtered probability space (Ω, F, (F n ) , P) gives rise to the upper ℓ p estimates for f ∈ L p (X) In that case we write X satisfies M T p and by M T p (X) we denote smallest constant C p such that (5) is satisfied.The inequality (3) states that R is a Banach space of martingale type p for any p ∈ (1,2].We let I n denote the collection of pairwise disjoint dyadic intervals of measure 2 −n contained in [0, 1).Thus I ⊆ [0, 1) is a dyadic interval in I n if there exists k ∈ {0, 1, . . ., 2 n − 1} such that We denote by I = We put D = σ ( D n ) and denote by ([0, 1), D, (D n ), dt) the dyadic filtration of the unit interval.Pisier proved in [12] that the dyadic filtration determines already the martingale type of a Banach space X.Indeed in Pisier's article combining Proposition 2.4 b) with Theorem 3.1 b) gives that a Banach space X satisfies martingale type p if and only if there exists C p < ∞ such that ∥f ∥ L p ([0,1), X) ⩽ C p ∥E(f |D 0 )∥ p L p ([0,1), X) + for f ∈ L p ([0, 1), X), where L p ([0, 1), X) = L p ([0, 1), D, dt, X).This equivalence is also presented in Theorem 10.22 of Pisier's recent book on martingales in Banach spaces [13].
We interpret the statement of the above cited Theorem of Pisier in terms of Haar functions.Since 1 < p ⩽ 2 it is clear that for a given f ∈ L p (X) and n ∈ N we have and consequently where the series converges in L p (X). Therefore we may restate (6) as .
Since I n−1 consists of pairwise disjoint dyadic intervals we have Consequently the inequality (6) holds true if and only if Pietsch and Wenzel in their book [11] on orthonormal systems and Banach space geometry introduced the notion of Haar type p of a Banach space X (see specifically [11,Chapter 7]).Recall that a Banach space X is said to satisfy Haar type p if there exists C < ∞ such that for any sequence {x I } I∈I ⊆ X (such that I∈I x I h I converges in L p (X)) the inequality (7) holds true.In that case we say that X satisfies HT p and by HT p (X) we denote smallest constant C such that (7) is satisfied.Thus Pisier's theorem [12, Proposition 2.4 b) and Theorem 3.1 b)](see also [13,Theorem 10.22]) asserts that a Banach space X is of martingale type p iff it is of Haar type p.
Recall that we say (Ω, F, (F n ) , P) is a filtered probability space if (F n ) is an increasing sequence of purely atomic sub-σ-algebras in a probability space (Ω, F, P) and In the present paper we identify precisely all filtered probability spaces (Ω, F, (F n ) , P) that are able to determine the martingale type of Banach space X.We associate explicit intrinsic conditions on the filtration (F n ) which determine that the upper ℓ p estimates (5) imply the martingale type p of Banach space X.As a consequence we obtain the following dichotomy.
Theorem 1.Let 1 < p ⩽ 2. For each fixed (Ω, F, (F n ) , P) the following dichotomy holds true: Either, there exists C > 0 such that for any Banach space X and any f ∈ L p (Ω, F, P, X) or the filtered probability space (Ω, F, (F n ) , P) and the upper ℓ p estimates (8) already determine that the Banach space X is of martingale type p.
Theorem 2 of Section 3 and Theorem 15 of Section 4 are the main results of this paper.The dichotomy formulated in Theorem 1 is a direct consequence thereof.
Next we point out the connection between Haar type and Carleson constants.In [3] the authors fixed a collection of dyadic intervals G ⊂ I and obtained intrinsic conditions on the size of G such that the upper ℓ p estimates for x I ∈ X, I ∈ G, imply that the Banach space X is of Haar type p. Specifically in [3] it was shown that G and the upper ℓ p estimates (9) determine the Haar type of X if and only if the Carleson constant of G is unbounded, that is, The present paper could be viewed as continuation of [3].In [7] the size of collections G ⊂ I , expressed in term of its Carleson constant G was utilized in the solution of the Gamlen-Gaudet problem for dyadic H 1 .

Notation, conventions and definitions
Let (Ω, F, (F n ) , P) be a filtered probability space, where F n , n ⩾ 0, is a σ-algebra generated by a finite family of atoms whose expectation appears in the statement of Theorem 1 we may modify the fields F n in such a way that every atom A ∈ A n will be, either split up in strictly smaller atoms in A n+1 , or else never split at any later step m > n.This change in the fields F n will clearly not modify the family n A n of all atoms, nor the union n F n ; also, it will not affect the definition given below for the family E and the value of its Carleson constant E .Let A j , where A j ∈ A n+1 and are ordered in such a way that for every j ∈ {1, . . ., N (A) − 1} (see Figure 1).For every A ∈ A we put A ⋄ := A 1 .
Let R ⊆ n F n be a nested collection, that is for A, B ∈ R such that B ∩ A ̸ = ∅ we have A ⊆ B or B ⊆ A. Given A ∈ F we introduce the following subfamily R is maximal element with respect to inclusion}, and inductively we define for k ∈ {2, 3, . ..} : Notice that collection E satisfies the following property: If B ∈ G k (E, A) we have For any collection of measurable sets R we define the point set covered by R: We will denote by R the Carleson constant of the collection R i.e.

R = sup
In [8] the collection E and its size, expressed in term of its Carleson constant E , was utilized in the classification theorem for martingale H 1 spaces.

Infinite Carleson constant and Haar type
Recall that to a filtered probability space (Ω, F, (F n ) , P) we associated the collection E in equation (11) and its Carleson constant E in equation (13).
Theorem 2. Let (Ω, F, (F n ) , P) be a filtered probability space, where each F n is purely atomic.Let 1 < p ⩽ 2 and E = ∞.Let X be a Banach space.If there exists T p > 0 such that for any then the space X is of Haar type p. Moreover where HT p (X) is the Haar type p constant of X.
Remark 3. The constant on the right hand side of (15) reflects the constant in Lemma 4 H7).This refers also to the factor on the right side in (16).
Under the hypothesis E = ∞ we will show that span{1 A } A∈E supports/contains systems of functions equivalent to the initial segments of the Haar system in the Bochner-Lebesgue space L p (Ω, X).Our approach may be divided into three separate steps as follows.First in Lemma 4 we design a "greedy algorithm" to construct a family of proto-Haar functions H A associated to the collections of pairwise disjoint atoms G k (E, A).Our greedy algorithm ensures that the following variational estimates hold true (16) Definition 7 and Lemma 11 form the second part of the proof, where we explain the significance of the Gamlen-Gaudet construction in determining Haar type of a Banach space.Finally in Lemma 14 we show that E = ∞ implies that E supports all initial segments of a generalized Haar system.
We organize the proof in such a way that it is a direct consequence of Lemma 11 and Lemma 14.

The Greedy Algorithm determines the selection of signs
In this section we design the greedy algorithm that forms the main novelty of the present paper.It is employed in the proof of Lemma 4 were we construct those proto-Haar functions H A for which the variational estimates (16) hold true.
holds true then there exists a function H A such that

H2) supp H
l=m and functions (H l ) ∞ l=m such that for every l ⩾ m we have 1.H l satisfies properties H1), H2), H4), H7), We put , where Y l,j ∈ A l+1 are disjoint atoms and are ordered according to their size (see (10)).Since there exists an index s such that Therefore there exists −1 < c l+1 ⩽ 1 such that We define Properties of sets Y l , U l , V l are obvious by the construction.Function H l+1 clearly is F l+1measurable.In fact properties H1), H2), H4) of H l+1 are obvious from the definition of H l+1 .Only property H7) demands more attention.
Observe that the sequence of functions (H n ) forms a martingale with respect to the filtration (F n ).Indeed functions H l and H l+1 are equal outside of the set Y l ∈ A l and Recall that (U l ), (V l ) are sequences of increasing sets.It follows from this and (18) that Now we will prove the existence of an index t such that H t also satisfies H5) and H6).Then we will put H A := H t .By (17) there exists a finite family of disjoint atoms Since the family is finite we can find t such that {B j } s j=1 ⊂ F t−1 .Since Y t ∈ A t and {B j } ⊂ A ∩ F t−1 either there exists j such that Y t ⊂ B j or Y t and B j are disjoint for every j.If Y t ⊂ B j then ⩽ 2 −k P(A) Otherwise Y t is disjoint from the union of sets B j and by ( 19) This proves that H t satisfies H5).Now for H6) we only need to observe that from the construction the following estimate is satisfied: Remark 5.The proto-Haar function H A was constructed by means of the greedy algorithm presented in the proof of Lemma 4. As a critical consequence the variational norm (16) of the function H A is bounded.See Lemma 4 condition H7).We point out that the complexity of our construction can not be reduced entirely.This became especially clear in view of the following example communicated to us by Maciej Rzeszut.We are grateful for his permission to present it here.We fix 0 < ε < 1 2 .There exists a function H[0,1] such that for the dyadic filtration it satisfies H1)-H6) but it fails to satisfy H7).We select N 0 ∈ N such that for any N > N 0 We shall employ Khintchine's inequality to see that where D n denotes the σ-algebra generated by I n .We write for a given function g.Recall also In summary each of the functions f N = sgn(Θ N ) satisfies H1)-H6) and for any constant C in H7) condition H7) is violated for N large enough.

Gamlen and Gaudet determine Haar type
In this subsection for any n, similarly as in [3], we introduce a family of conditions, which will imply Haar type.Recall that I n = {I : I is dyadic and |I| ⩾ 2 −n }.
Definition 7.For a filtered probability space (Ω, F, (F n ) , P) we say that the corresponding collection E supports all initial segments of a generalized Haar system if there exists a constant K > 0 such that for any n ∈ N, δ > 0 and ε = δ2 −n−1 there are a family of collections {B I } I∈I n , a family of functions {g I } I∈I n−1 and an atom A 0 ∈ E satisfying the following conditions: A3) The elements of B I are pairwise disjoint for every A8) For any atom A ∈ A S and any m ∈ {1, . . ., S} there exists at most one dyadic interval We will denote that interval by I(A, m) (Remark 10 below restates a combinatorially dual picture of this condition).

A9) For any
A10) For every I ∈ I n we have Remark 9. Note that the condition A 9) implies that for 1 < p < 2 and θ p = 2 − p we have Indeed Hölder's inequality yields where Remark 10.In fact our functions {g I } satisfy the following conditions.Given an atom A ∈ A S and given two dyadic intervals I ⊊ J there exist disjoint integer intervals N I ⊆ {1, . . ., S} and Now we turn to showing that conditions A 1) − A 10) imply Haar type.
Lemma 11.Let 1 < p ⩽ 2. Assume that the collection E supports all initial segments of a generalized Haar system.Then the Banach space X satisfies inequality (14) iff X has Haar type p.
Proof.Let n ∈ N and 0 < δ < 1.We fix a finite sequence (x J ) J∈I n−1 of vectors from Banach space X.In view of Definition 7 there are collections B J ⊊ E with J ∈ I n , functions g J with J ∈ I n−1 and A 0 ∈ E satisfying A 1)-A 10).
For fixed dyadic interval I of length 2 −n we choose The function f and each of the functions g J , where J ∈ I n−1 , are constant on any of the sets Observe that on the probability space ( Ũ, (1 − δ) −1 dt) the system (g J ) J∈I n−1 has the same joint distribution as the usual Haar basis (h J ) J∈I n−1 on the unit interval.Therefore On the other hand one can easily express the integral of f in terms of the function f .
Given j ∈ {1, 2, . . ., S} we write and for fixed J ∈ I n−1 we put Recall that I(A, j) is defined by A 8).We use ( 14) to obtain In summary we get The above estimate no longer depends on the choice of the collection {B I }.Since δ was an arbitrary number from interval (0, 1) we let δ → 0: This means that X has Haar type p with constant Recall that Pisier proved in [12, Proposition 2.4 b) and Theorem 3.1 b)] that a Banach space of Haar type p is also a Banach space of martingale type p.This means that for any filtered probability space (Ω, F, (F n ) , P) the inequality ( 5) is satisfied.

Carleson-Garnett condensation lemma
Here we want to show that E = ∞ implies that E supports all initial segments of a generalized Haar system.In the proof of this fact Lemma 4 plays a crucial role.In order to show that condition (17) from Lemma 4 is satisfied we use that for collections with infinite Carleson constant the following version of the condensation lemma holds true (see [2, Ch.X : Lemma 3.2 and p. 414]).
Proof.Let us fix auxiliary notation.We define families of sets (g j (A)) j∈N by g 0 (A) = {A}, and g l (A) = G kl (R, A) for l > 0. Clearly g l (A) * ⊂ A. The family g l (A) consists of pairwise disjoint sets.The sets g m (B) * are pairwise disjoint for B ∈ g l (A).Thus g m (B) and P(g l+m (A) * ) = B∈g l (A) Now we will prove an auxiliary lemma.
Lemma 13.Let ε ∈ (0, 1), let p, m > 0 be integers and suppose that Proof.Let a 0 := P(A) and a 1 := P(g m+1 (A)) ⩽ a 0 .By the assumptions, we have that By the definition of U, we know that Using ( 27), ( 26) and (28) we have Since t 0 + u 0 = P(g 1 (A) * ) ⩽ P(A) = a 0 it follows that Since R = ∞, there exists a set A such that: B⊂A B∈R
By [9, Lemma 3.1.4]there exists A 0 ⊂ A, A 0 ∈ R with We put T 0 = {A 0 }.Using Lemma 13 we define inductively The definition of the families T j allows one to apply inductively Lemma 13, the result of Lemma 13 at one step giving the assumption for the next step.By the construction of T j conditions (i), (ii) are satisfied.For A ∈ T j−1 by Lemma 13 we get Therefore condition (iii) is satisfied for ε = ε 2 .Since we have space for error we modify T j to be finite families.
We are ready for the next step towards the proof of Theorem 2.
Lemma 14.If E = ∞ then E supports all initial segments of a generalized Haar system with constant K = 4.
Proof.Fix 1 2 > ε > 0 and n ∈ N. We choose k and ε such that 2 −k < ε < ε 2 .We apply the Lemma 12 for ε, k and R = E.We obtain a set A 0 ∈ E and families T j ⊆ E, where j ∈ {0, 1, . . ., n} satisfying conditions (i)-(iv) of Lemma 12.We define B [0,1] = {A 0 } and g [0,1] = H A 0 .Now we define B I and g I inductively with respect to canonical ordering.Fix I such that |I| = 2 −l .Assume that B I and g I are already defined.Recall that I + denotes the left half of I and I − the right one.We define Note that T l+1 ⊆ A .Hence B I + , B I − ⊆ A and by Lemma 12 (ii),(iii) the assumptions of Lemma 4 are satisfied for every atom A ∈ T l+1 .Thus for every A ∈ T l+1 the function H A is well defined.We put It is easy to observe that properties A 1) − A 5) are satisfied.By Lemma 12 iv) and Lemma 4 H3) we know that properties A 6), A 7) hold.Now we prove A 8).We fix an atom A ∈ A S and m ∈ {1, . . ., S}.We show that for any pair of distinct dyadic intervals I, J ∈ I n−1 we have

Note that
for some atom C ∈ A .Note that V is non empty set.Let B 2 be a maximal element of family V with respect to inclusion.There exists a unique l satisfying B 2 ∈ A l .If l < m we have Indeed by Lemma 4 H4) we have for l ⩽ m In summary for fixed A ∈ A S and m ∈ {1, . . ., S} there is at most one dyadic interval I satisfying To obtain A 9) observe that sets in B I are pairwise disjoint.Thus by Lemma 4 H7) we get Now we turn to proving A 10), starting with the right hand side of inequality (23).Clearly we have P(B [0,1] * ) = P(A 0 ).
We inductively assume that the fixed interval I satisfies We prove that inequality (30) holds for I + and I − .We consider the estimates for I + .By A 3), A 5) and Lemma 4 H6) we have we have verified the right hand side of inequality (30).We put l = − log 2 |I|.Observe that by Lemma 4 H6) and Lemma 12 iii) we have for any A ∈ T l .Therefore For any small enough ε > 0 we choose ε > 0, ε > 0 such that for every q ∈ {0, 1, . . ., n} we have In view of (31) this gives us the left hand side of (23) in A 10).
Proof of Theorem 2. Since E = ∞.By Lemma 14 the assumptions of Lemma 11 are satisfied with constant K = 4. Lemma 11 implies that the Banach space X is of Haar type p and by (25) where T p is defined in (14).

Finite Carleson constant and upper L p estimates
Let F n be a purely atomic σ-algebra for n ∈ N. Recall that for the filtered probability space (Ω, F, (F n ) , P) we have defined a collection E associated with it (see (11)).Recall also that the Carleson constant of E is given by J⊆I J∈E P(J).
In the last section we investigated the case where the Carleson constant of E is infinite.The main result there is Theorem 2 which asserts that in case E = ∞, the inequality (14) implies non-trivial martingale type of the Banach space X.
In the present section we consider filtrations (F n ) for which the Carleson constant of E is finite.We will exploit the observation going back to Carleson and Garnett [1] that E can be decomposed as and for any A ∈ E j the following inequality holds true The main result of this section is Theorem 15 below which asserts that the inequality ( 14) is satisfied for any Banach space X provided that E < ∞.
Theorem 15.Let (Ω, F, (F n ) , P) be a filtered probability space, where each F n is purely atomic.Let 1 < p ⩽ 2 and E < ∞.For any Banach space X and any f ∈ L p (Ω, X) the following inequality holds: where Remark 16.The constant T p in the Theorem 15 is not optimal.It reflects our approach, which is based on the disjointification procedure of Carleson and Garnett.

Definitions and auxiliary results
In this subsection we concentrate on special cases of inequality (32).We first determine an algebraic basis for the space of martingale differences associated to a single atom.Recall that in opening paragraph of Section 2 we had modified the fields F n in such a way that every atom A ∈ A n will be, either split up in strictly smaller atoms in A n+1 , or else never split at any later step m > n.
Fix n ∈ N and A ∈ A n .Recall that A = N (A) j=1 A j , where A j ∈ A n+1 are ordered in such a way that For any j ∈ {2, . . ., N (A)} we put The functions {k j=2 form an algebraic basis for the space of martingale differences restricted to A, that is, for the finite dimensional space We put k Ω = 1 Ω .Observe that by (33) and the definition of E (11) for every atom A ∈ E the function k A is well defined and that {k A j : j ∈ {2, . . ., N (A)}, A ∈ A } ∪ {1 Ω } is just an alternative listing of {k B : B ∈ E}.Note that Ω ∈ E and that for 1 ⩽ p < ∞ we have span{k B : B ∈ E} We have N (A) = 4, and the functions {k A 2 , k A 3 , k A 4 } (defined by (33)) form an algebraic basis of the space of martingale differences restricted to A.
Recall that for A ∈ A we defined A 1 by ( 10) and we put A ⋄ = A 1 .Now we let and finally we define the collection Figure 3: The figure depicts the graph of ϕ A .
Remark 17.In the following list of remarks we comment on the relation between the collections A , B, C and E. 17.1) Given K ∈ B there exist (unique) n ∈ N ∪{0} and A(K) ∈ A n such that K = A(K) ☼ .In that case there exists uniquely defined collection consisting of pairwise disjoint atoms On the other hand for A ∈ E and B < ∞ we have  ⩽ (1 + B ) P(A).
Now we consider nested collections with finite Carleson constants and we review the disjointification lemma originally due to Carleson and Garnett [1].(Also see [9].) Lemma 19.Let R be a nested collection of measurable sets such that R < ∞ and M R = ⌊4 R + 1⌋.There are families {R In consequence we obtain a bound on the size of subsequent generations Equation (41) asserts that the sequence P(G k (E i , A) * ) P(A) −1 decreases at geometric rate uniformly with respect to A ∈ R. As a result we will be able to prove that forms an almost disjointly supported sequence of functions.See Lemma 21 and Lemma 23 below.
Here we say that the sequence of functions is almost disjointly supported if it satisfies conditions (42) and ( 43) below.Lemma 20 links almost disjointly supported sequences of functions to martingale type estimates.
Lemma 20.Let X be a Banach space, g j : (Ω, Σ, P) → X, j ∈ N denote a sequence of measurable functions and 1 ⩽ p < ∞.Assume that there exists a sequence {a k } k∈N ∈ ℓ 1 (N) and a family of measurable sets {D k } k∈N such that and, for k, j ∈ N, ) is a family of pairwise disjoint sets and Recall that D k+j = G k+j (R i , Ω) * .Therefore (47) and ( 45) yield assumptions of Lemma 20 with g j , D j and a k = 2 Recall that in the above argument we have assumed that Ω / ∈ R i .If Ω ∈ R i it suffices to change the definitions of g j and D j as follows: where j ∈ N, and repeat the above argument using these changes.
The above lemma gives an estimate for the part of the function which is "contained" in the collection E. We have similar bounds for the part "outside of the collection" i.e. linear span of the functions {1 A ⋄ } A∈A .
Recall that for the collection E we have defined the corresponding collection B in (36).
Lemma 22.Let 1 < p ⩽ 2. Let R ⊂ B and X be a Banach space.For any choice of n ∈ N ∪{0}, Proof.We put For any A ∈ R we define Hence we have Now we fix a set B ☼ ∈ G 1 (R, A).Note that for ω ∈ B ☼ we have Integrating over B ☼ gives us Observe that supp g 1 ⊂ Ω and for j ⩾ 2 we know that supp g j ⊂ G j−1 (B i , Ω) * and that the function g j is constant on every set A ☼ ∈ G j+1 (B i , Ω) (see Figure 5).
In order to see that (56) holds true it suffices to observe that for B ☼ ∈ G j+1 (B i , Ω) we have ⩽ 2 −k P(B ☼ ).
(For a more detailed verification of (56) see (47) above.)Therefore applying Lemma 20 with Let X be a Banach space.We define X p (R) to be the closure of span{x I d I : x I ∈ X, I ∈ R} in L p (Ω×[0, 1], F ⊗D, P ⊗λ , X).Let the norm on X p (R) be the one induced by L p (Ω×[0, 1], X).
The proof given in [3] yields that the following dichotomy holds for the class of X p (R) spaces: • If R < ∞, then there exists C( R , p) > 0 such that for any Banach space X, where x I ∈ X, I ∈ R and the series I∈R x I d I converges in L p (Ω × [0, 1], X).
• If R = ∞, then the validity of the estimate implies that X is of Haar type p.
Review of Maurey's isomorphism: Let (Ω, F, (F n ), P) be a filtered probability space.Let E ⊂ F be defined by (11).As shown in [10] (see also [9,Section 4.1]), Maurey's proof of [6, Prop 4.6, Lemma 4.10] can be extended to the vector valued case as follows: if X satisfies the UMD property then there exists an isomorphism T : L p (Ω, F, P ; X) → X p (E) such that for f ∈ L p (Ω, F, P ; X), x I d I .
Conversely if I∈Gn(E,Ω) x I d I .
Finite Carleson constant: Fix a filtered probability space (Ω, F, (F n ), P) such that E < ∞.Fix 1 < p < ∞.Choose f ∈ L p (Ω, F, P ; X) and let x I be such that In view of [3] X satisfies HT p and M T p .

∞ n=1 I
n .Let D n be the σ-algebra generated by I n .For a dyadic interval I ∈ I we let I + ⊊ I, respectively I − denote the left, respectively the right, half of I. Thus I + , I − are dyadic intervals satisfying I + ∩ I − = ∅ and I + ∪ I − = I.We define the Haar function h I : [0, 1) → {−1, 0, 1} by putting h I = 1 I + − 1 I − .

Figure 1 :
Figure 1: Here we put A = A 1 ∪ A 2 ∪ A 3 ∪ A 4 and N (A) = 4. Atoms are lined up according to their size from left to right.In all our figures we adhere to the convention illustrated above.
I∈I N h I (ω), I N = {I : I is dyadic and |I| ⩾ 2 −N } and {h I } I∈I N is the initial segment of Haar system.Consider the {0, 1, −1} valued function

A5)
For any I ∈ I n−1 we have g I : Ω → [−1, 1], supp g I ⊆ B I * and B I + * ⊆ {g I = 1}, B I − * ⊆ {g I = −1}, where I − is the right half of I and I + is the left half of I. A6) There exists S ∈ N such that the elements of B I are F S measurable for every I ∈ I n .A7) Functions g I are F S measurable for any I ∈ I n−1 .

By A 4 )
and A 5) it is enough to consider pairs of dyadic I, J satisfying I ⊊ J.It suffices to consider A ⊆ B I * .Next determine B 1 such that B 1 ∈ B I and A ⊆ B 1 .Let V = {B ∈ A : B 1 ⊆ B and g J is constant on B}.

Figure
Figure 2: GivenA ∈ A n , the figure depicts A 1 , A 2 , A 3 , A 4 ∈ A n+1 such that A = A 1 ∪ A 2 ∪ A 3 ∪ A 4 .We have N (A) = 4, and the functions {k A 2 , k A 3 , k A 4 } (defined by (33)) form an algebraic basis of the space of martingale differences restricted to A.
C ∈ G 1 (E, A) such that C ∈ A n there exists a unique D ∈ A n−1 such that C ⊆ D ☼ , and hence for any B ⊊ C we have B ⊊ D ☼ .Moreover by Remark 17.1) there exist {D j : 2 ⩽ j ⩽ N (D)} ⊂ A n such that D ☼ = N (D) j=2 D j .There exists unique m ∈ {2, . . ., N (D)} such that C = D m .By Remark 17.3) C ⊊ A implies D ☼ ⊊ A, and hence D j ⊊ A for any j ∈ {2, . . ., N (D)}.Moreover D ☼ ∈ G 1 (B, A) and D j ∈ G 1 (E, A) for any j ∈ {2, . . ., N (D)}.It follows

Figure 4 :
Figure 4: The picture highlights the different role of the atoms in A .Let A ∈ A n and R = B.The pairwise disjoint red coloured intervals depict A ☼ ,A ⋄☼ , A ⋄⋄☼ etc.They form G 1 (R, A).The intersection of the decreasing green coloured intervals forms the set T A .

T
∆ n f ∈ span{x I d I : I ∈ G n (E, Ω), x I ∈ X}, where ∆ n f = E(f F n ) − E(f F n−1 ).Specifically if f ∈ L p (Ω, F, P ; X) and T f = I∈E x I d I then T ∆ n f = I∈Gn(E,Ω)