Atomic decompositions of Lorentz martingale spaces and applications

In the paper we present three atomic decomposition theorems of Lorentz martingale spaces. With the help of atomic decomposition we obtain a sufficient condition for sublinear operator defined on Lorentz martingale spaces to be bounded. Using this sufficient condition, we investigate some inequalities on Lorentz martingale spaces. Finally we discuss the restricted weak-type interpolation, and prove the classical Marcinkiewicz interpolation theorem in the martingale setting.


Introduction and Preliminaries
The idea of atomic decomposition in martingale theory is derived from harmonic analysis.Just as it does in harmonic analysis, the method is key ingredient in dealing with many problems including martingale inequalities, duality, interpolation and so on, especially for small-index martingale and multi-parameter martingale.As well known, Weisz [8] gave some atomic decompositions on martingale Hardy spaces and proved many important theorems by atomic decompositions; Weisz [9] made a further study of atomic decompositions for weak Hardy spaces consisting of Vilenkin martingale, and proved a weak version of the Hardy-Littlewood inequality; Liu and Hou [5] investigated the atomic decompositions for vector-valued martingale and some geometry properties of Banach spaces were charactered; Hou and Ren [3] considered the vector-valued weak atomic decompositions and weak martingale inequalities; in [10], [11], the authors discussed the operator interpolation by atomic decompositions of weighted martingale Hardy spaces.
In this paper we present three atomic decomposition theorems for Lorentz martingale spaces H s p,q , Q p,q , D p,q .Applying these theorems, a sufficient condition for a sublinear operator defined on the Lorentz martingale spaces to be bounded is given.And then we obtain some continuous imbedding relationships among Lorentz martingale spaces.These are new versions of the basic inequalities in the classical martingale theory.Finally we also give a restricted weak-type interpolation theorem, and obtain the version of classical Marcinkiewicz interpolation theorem in the martingale setting.
Let (Ω, Σ, P ) be complete probability space and f be a measurable function defined on Ω.The decreasing rearrangement of f is the function f * defined by We adopt the convention inf Ø = ∞.The Lorentz space L p,q (Ω) = L p,q , 0 < p < ∞, 0 < q ≤ ∞, consists of those measurable functions f with finite quasinorm f p,q given by It will be convenient for us to use an equivalent definition of f p,q , namely To check that these two expressions are the same, simply make the substitution y = P (|f (x)| > t) and then integrate by parts.
It is well know that if 1 < p < ∞ and 1 ≤ q ≤ ∞, or p = q = 1, then L p,q is a Banach space, and f p,q is equivalent to a norm.However, for other values of p and q , L p,q is only a quasi-Banach spaces.In particular,if 0 < q ≤ 1 ≤ p or 0 < q ≤ p < 1 then f p,q is equivalent to a q− norm.Recall also that a quasi-norm • in X is equivalent to a p− norm, 0 < p < 1, if there exists c > 0 such that for any x i ∈ X, i = 1, ..., n For all these properties, and more on Lorentz spaces, see for example [1,2,4].Holder's inequality for Lorentz spaces is the following, which first appears in work of O'Neil [7], for all 0 < p, q, p 1 , q 1 , p 2 , q 2 ≤ ∞ such that 1 p = 1 p1 + 1 p2 and 1 q = 1 q1 + 1 q2 .Let {Σ n } n≥0 a nondecreasing sequence of sub-σ -fields of Σ such that Σ = Σ n .We denote the expectation operator and the conditional expectation operator relative to Σ n by E and E n , respectively.For Denote by Λ , the set of all non-decreasing, non-negative and adapted r.v.sequences ρ = (ρ n ) n≥0 with ρ ∞ = lim n→∞ ρ n .We shall say that a martingale f = (f n ) n≥0 has predictable control in L p,q if there is a sequence As usually, we define Lorentz martingale spaces(see [1]), If we change the L p,q − norms in above definitions by L p − norms, we get the usual Hardy martingale spaces (see [6]).
Remark.The norms of Q p,q and D p,q are attainable respectively.For example, there exists , which is also called the optimal control.It turns out that Lorentz spaces, as many other quasi-Banach spaces, admit some sort of atomic decomposition.Firstly we give the definition of an atom.
Throughout the paper, we denote the set of integers and the set of nonnegative integers by Z and N , respectively.We write A B if A ≤ cB for some positive constant c independent of appropriate quantities involved in the expressions A and B .

Atomic decompositions
Now we can present the atomic decompositions for Lorenz martingale spaces.
Theorem 2.1.If the martingale f ∈ H s p,q , 0 < p < ∞, 0 < q ≤ ∞ then there exist a sequence a k of (1, p, ∞)-atoms and a positive real number sequence (μ k ) ∈ l q such that Conversely, if 0 < q ≤ 1, q ≤ p < ∞, and the martingale f has the above decomposition, then f ∈ H s p,q and where the inf is taken over all the preceding decompositions of f .
Proof.Assume that f ∈ H s p,q , q = ∞.Now consider the following stopping time for all k ∈ Z : The sequence of these stopping times is obviously non-decreasing.It easy to see that , and which implies that (a k n ) is a L 2 − bounded martingale so that there exists If n ≤ τ k then a k n = 0 and we get that a k is really a (1, p, ∞) atom and For q = ∞, standard rectifications can be made.
Conversely, if f has the above decomposition, then from s(a k ) ∞ ≤ P (τ k < ∞) − 1 p and we get For 0 < q ≤ 1, q ≤ p < ∞, • p,q is equivalent to a q− norm, which gives the desired result.
Theorem 2.2.If the martingale f ∈ Q p,q , 0 < p < ∞, 0 < q ≤ ∞, then there exist a sequence (a k ) of (2, p, ∞) atoms and a real number sequence Conversely, if 0 < q ≤ 1, q ≤ p < ∞, and the martingale f has the above decomposition, then f ∈ Q p,q and f Qp,q inf( k∈Z where the inf is taken over all the above decompositions.
The stopping times τ k are defined in this case by Let a k and μ k (k ∈ Z) be defined as in the proof of Theorem 2.1.Then for a fixed k , (a k n ) is also a martingale.Since S(f As in Theorem 2.1, we can show that a k is a (2, p, ∞)-atom.Also ( Conversely, if a k is (2, p, ∞)− atom, one can show that a k q H S p,q ≤ 1 q .The rest can be proved similar to Theorem 2.1.
Theorem 2.3.If the martingale f ∈ D p,q , 0 < p < ∞, 0 < q ≤ ∞, then there exist a sequence (a k ) of (3, p, ∞)-atoms and a real number sequence Conversely, if 0 < q ≤ 1, q ≤ p < ∞, and the martingale f has the above decomposition, then f ∈ D p,q and f Dp,q inf( k∈Z where the inf is taken over all the above decompositions.
The proof of Theorem 2.3 is similar to that of Theorem 2.2 and so we omit it.

Bounded operators on Lorentz martingale spaces
As one of the applications of the atomic decompositions, we shall obtain a sufficient condition for a sublinear operator to be bounded from Lorentz martingale spaces to function Lorentz spaces.Applying the condition to M f, Sf and sf , we deduce a series of inequalities on Lorentz martingale spaces.
An operator T : X → Y is called a sublinear operator if it satisfies where X is a martingale space, Y is a measurable function space.
Theorem 3.1.Let T : H s r → L r be a bounded sublinear operator for some 1 ≤ r < ∞.If for all (1, p, ∞)− atoms a, where τ is the stopping time associated with a, then for 0 < p < r, 0 < q ≤ ∞, we have Proof.Assume that f ∈ H s p,q .By Theorem 2.1, f can be decomposed into the sum of a sequence of (1, p, ∞)− atoms.For any fixed y > 0 choose j ∈ Z such that 2 j−1 ≤ y < 2 j and let It follows from the boundedness of T that On the other hand, since Since T is subliear, and thus for all 0 < p < r, T : H s p,∞ → L p,∞ is bounded.Now for any fixed 0 < p < r, we can choose 0 < p 0 , p 1 < r, 0 < θ < 1 satisfying 1 p = 1−θ p0 + θ p1 .From interpolation theorem (see Theorem 5.11 [2]) and the boundedness of sublinear is hereditary for the interpolation spaces, we obtain for 0 < q ≤ ∞ T : On the lines of the proof of Theorem 3.1, we can prove the following Theorems 3.2 and 3.3 by using Theorems 2.2 and 2.3, respectively.Theorem 3.2.Let T : Q r → L r be a bounded sublinear operator for some for all (2, p, ∞)− atoms a, where τ is the stopping time associated with a, then for 0 < p < r, 0 < q ≤ ∞, we have Theorem 3.3.Let T : D r → L r be a bounded sublinear operator for some for all (3, p, ∞)− atoms a, where τ is the stopping time associated with a, then for 0 < p < r, 0 < q ≤ ∞, we have Theorem 3.4.For all martingale f = (f n ) n≥0 the following imbeddings hold: 2) For 0 < p < ∞, 0 < q ≤ ∞, Q p,q → H * p,q , Q p,q → H S p,q , Q p,q → H s p,q D p,q → H * p,q , D p,q → H S p,q , D p,q → H s p,q .Proof.
In fact, we can regard martingale spaces as the subspaces of sequence spaces.Consider the operator bounded for all p ≥ 2. For any fixed p > 2, we can choose p 0 , p 1 > 2, 0 < θ < 1 satisfying which gives H * p,q → H s p,q .By considering Q defined on the sequence space L p (l 2 ), we can similarly prove H S p,q → H s p,q 2) For all 0 Hence s(a k ) = 0 on the set {τ k = ∞}.By Theorem 3.2 and 3.3, the assertion is proved.
Remark.If we put p = q in the above embeddings, Theorem 2.11 in [8] can be deduced.

Restricted weak type interpolation
We say that a sublinear operator T is of Lorentz-s restricted weak-type (p, q) if T maps H s p,1 to L p,∞ .For convenience, we call T as restricted weak-type (p, q).Then we have the next interpolation from one restricted weak-type estimate to another.Theorem 4.1.Let T be of restricted weak-type (p i , q i ) for i = 0, 1 , and Then T is also of restricted weak-type (p, q).
and the proof is complete.Now we show how restricted weak-type estimate can be transferred to strong type.It is also the version of the classical Marcinkiewicz interpolation theorem in the martingale setting(see Theorem 4.13 in [1]).Theorem 4.2.Let T be of restricted weak-type (p i , q i ) for i = 0, 1 , and 1 < p i < ∞, 1 < q i ≤ ∞, q 0 = q 1 .Put Then T is of type (H s p,r , L q,r ), for 0 < r < 1 and r ≤ q.
Proof.For 0 < r < 1 and r ≤ q, we know that • q,r is equivalent to a r− norm, so it is enough to prove T a q,r 1, for all (1, p, ∞)− atoms.Once it is proved then from Theorem 2.1, Now we shall show T a q,r 1.Consider the case q 1 , q 2 < ∞.From the proof of Theorem 4.1, it is easy to see that