aa r X i v : . [ m a t h . F A ] A p r Weak Orlicz-Hardy Martingale Spaces
Yong JIAO and Lian WU (Institute of Probability and Statistics, Central South University, Changsha 410075, China)Revised version, April 12, 2013
Abstract
In this paper, several weak Orlicz-Hardy martingale spaces associated withconcave functions are introduced, and some weak atomic decomposition theo-rems for them are established. With the help of weak atomic decompositions,a sufficient condition for a sublinear operator defined on the weak Orlicz-Hardymartingale spaces to be bounded is given. Further, we investigate the dualityof weak Orlicz-Hardy martingale spaces and obtain a new John-Nirenberg typeinequality when the stochastic basis is regular. These results can be regardedas weak versions of the Orlicz-Hardy martingale spaces due to Miyamoto, Nakaiand Sadasue.
The Lebesgue’s theory of integration has taken a center role in modern analysis, whichleads the more extensive classes of function spaces and martingale spaces to naturallyarise. As is well known, as a generalization of L p -space, the Orlicz space was introducedin [2]. Since then, the Orlicz spaces have been widely used in probability, partialdifferential equations and harmonic analysis; see [1, 3, 14, 16, 18], and so forth. Inparticular, Takashi, Eiichi and Gaku very recently studied the Orlicz-Hardy martingalespaces in [11], and using the atomic decomposition they obtained some very interestingmartingale inequalities as well as the dulity, and proved a generalized John-Nirenbergtype inequality for martingale when the stochastic basis is regular. Let us briefly recallthe main results of [11]. Partially supported by the National Natural Science Foundation of China(11001273, 11150110456),the Research Fund for the Doctoral Program of Higher Education of China (20100162120035) andPostdoctoral Science Foundation of China and Central South University.2000
Mathematics subject classification:
Primary 60G46; Secondary 60G42.
Key words and phrases : martingale, weak Orlicz-Hardy space, atomic decomposition, duality, John-Nirenberg inequality. G be the set of all functions Φ : [0 , ∞ ) → [0 , ∞ ) satisfying Φ(0) = 0, lim r →∞ Φ( r ) = ∞ . The Orlicz space L Φ is defined as the collection of all measurable functions f withrespect to (Ω , F , P ) such that E (Φ( c | f | )) < ∞ for some c > k f k L Φ = inf n c > E (Φ( | f | /c )) ≤ o , where E denotes the expectation with respect to F . For q ∈ [1 , ∞ ) and a function φ : (0 , ∞ ) → (0 , ∞ ), the generalized Campanato martingale spaces L q,φ is defined by L q,φ = { f ∈ L q : k f k L q,φ < ∞} , where k f k L q,φ = sup n ≥ sup A ∈F n φ ( P ( A )) P ( A ) Z A | f − E n f | q dP ! /q , (1.1)with the convention that E f = 0 . We refer to the recent paper [13] for the Morrey-Campanato spaces. Denote by G ℓ the set G ℓ = n Φ ∈ G : ∃ c Φ ≥ ℓ ∈ (0 ,
1] s . t . Φ( tr ) ≤ c Φ max { t ℓ , t } Φ( r ) for t, r ∈ [0 , ∞ ) o . Then for Φ ∈ G ℓ and φ ( r ) = r Φ − (1 /r ) , where and in what follows Φ − denotes theinverse function of Φ , the following duality holds, (cid:0) H s Φ (cid:1) ∗ = L ,φ . See Section 2 for the notation H s Φ . Moreover, the John-Nirenberg type inequality holdswhen the stochastic basis is regular, namely, L q,φ are equivalent for all 1 ≤ q < ∞ . It should be mentioned that Miyamoto, Nakai and Sadasue’s results above are exactlythe generalization in [20] due to Weisz when Φ( t ) = t p , 0 < p ≤ M △ -condition were first introduced in [5], and interpolation theoremsand inequalities were proved for these spaces; Liu et al investigated the boundednessof some sublinear operators defined on weak Orlicz-Hardy martingale spaces in [9] andthe first named author studied some embedding relationships between them in [7] in2011; however, the existing results about weak Orlicz-Hardy martingale spaces are allassociated with convex functions. In the present paper we are interested in the case Φis not convex. We denote t qφ ( x ) = 1 φ ( x ) x − /q sup P ( ν< ∞ ) ≤ x k f − f ν k q , where ν is a stopping time and f ν is the stopped martingale. Very differently from(1.1), we define the weak generalized Campanato martingale space w L q,φ as follows.2 efinition 1.1. For q ∈ [1 , ∞ ) and a function φ : (0 , ∞ ) → (0 , ∞ ) , let w L q,φ = n f ∈ L q : k f k w L q,φ = Z ∞ t qφ ( x ) x dx < ∞ o . Then for Φ ∈ G ℓ and φ ( r ) = r Φ − (1 /r ) , we have (cid:0) w H s Φ (cid:1) ∗ = w L ,φ . See Section 2 for the notation of w H s Φ . Furthermore, we by the duality obtain the weaktype John-Nirenberg inequality when the stochastic basis is regular. That is, w L q,φ are equivalent for all 1 ≤ q < ∞ . We note that our theorems can deduce Weisz’s mainresults in [19].It is well known that the method of atomic decompositions plays an important rolein martingale theory; see for example, [4, 6, 21, 22]. In the present paper, the importantstep is to establish the weak atomic characterizations of weak Orlicz-Hardy martingalespaces. To this end, the main difficulty encountered is that there is no similar result toreplace Lemma 3.1 or Remark 3.2 in [11]. Inspired by [19], we adopt a different methodand apply some technical estimates. Particularly, we note that Φ − ( t ) t p are increasing and Φ − ( t ) t q are decreasing on (0 , ∞ ) for Φ ∈ G ℓ with q < ∞ , where p = p Φ − and q = q Φ − denote the lower index and the upper index of convex function Φ − , respectively; seealso Section 2 for the definitions of the lower and upper index.This paper is organized as follows. Section 2 is on preliminaries and notations. Sec-tion 3 is devoted to the weak atomic decompositions of weak Hardy-Orlicz martingalespaces. By the atomic decompositions, a sufficient condition for a sublinear operatordefined on weak Hardy-Orlicz martingale spaces to be bounded is given in Section 4.In Section 5, we deduce the new John-Nirenberg type inequality by duality.We end this section by an open question. For q ∈ [1 , ∞ ) and a function φ : (0 , ∞ ) → (0 , ∞ ), define L q,φ = { f ∈ L q : k f k L q,φ < ∞} , where k f k L q,φ = sup ν φ ( P ( ν < ∞ )) P ( ν < ∞ ) Z { ν< ∞} | f − f ν | q dP ! /q . The supremum is taken all the stopping times ν. Then by the Proposition 2.12 in [11], k f k L q,φ ≤ k f k L q,φ ≤ C Φ k f k L q,φ , (1.2)where φ satisfies φ ( r ) ≤ c Φ φ ( s ) for 0 < r ≤ s < ∞ . Now we denote u qφ ( x ) = 1 φ ( x ) x − /q sup n ≥ sup A ∈F n sup P ( A ) ≤ x Z A | f − E n f | q dP ! /q , where ν is a stopping time and f ν is the stopped martingale. It is natural to define wL q,φ = n f ∈ L q : k f k wL q,φ = Z ∞ u qφ ( x ) x dx < ∞ o .
3n the time of writing this paper, we do not know if there is a result similar to (1.2).Throughout this paper, Z and N denote the integer set and nonnegative integerset, respectively. We denote by C the positive constant, which can vary from line toline. Let (Ω , F , P ) be a probability space, and {F n } n ≥ be a non-decreasing sequence ofsub- σ -algebras of F such that F = σ ( S n ≥ F n ). The expectation operator and theconditioned expectation operator are denoted by E and E n , resp. For a martingale f = ( f n ) n ≥ relative to (Ω , F , P ; ( F n ) n ≥ ), we denote its martingale difference by df i = f i − f i − ( i ≥
0, with convention f − = 0). Then the maximal function, the quadraticvariation and the conditional quadratic variation of martingale f are defined by M n ( f ) = sup ≤ i ≤ n | f i | , M ( f ) = sup i ≥ | f i | ,S n ( f ) = n X i =0 | df i | ! / , S ( f ) = ∞ X i =0 | df i | ! / ,s n ( f ) = n X i =0 E i − | df i | ! / , s ( f ) = ∞ X i =0 E i − | df i | ! / . The stochastic basis ( F n ) n ≥ is said to be regular, if for n ≥ A ∈ F n , thereexists B ∈ F n − such that A ⊂ B and P ( B ) ≤ RP ( A ), where R is a positive constantindependent of n . A martingale is said to be regular if it is adapted to a regular σ -algebra sequence. This amounts to saying that there exists a constant R > f n ≤ Rf n − for all nonnegative martingales ( f n ) n ≥ adapted to the stochastic basis ( F n ) n ≥ .Recall that G is the collection of all functions Φ : [0 , ∞ ) → [0 , ∞ ) satisfying Φ(0) =0, lim r →∞ Φ( r ) = ∞ . A function Φ is said to satisfy ∆ -condition, denoted by Φ ∈ ∆ ,if there exists a constant C > t ) ≤ C Φ( t ) , ∀ t > . A function Φ :[0 , ∞ ) → [0 , ∞ ) is said to be subadditive if Φ( r + s ) ≤ Φ( r ) + Φ( s ) for all r, s ∈ [0 , ∞ ).We note that all concave functions are subadditive. Let Φ and Φ belong to G , whichare said to be equivalent if there exists a constant C ≥ ( t ) /C ≤ Φ ( t ) ≤ C Φ ( t ) for all t ≥ . Definition 2.1.
Let Φ ∈ G , then the weak Orlicz space wL Φ is defined as the set of allmeasurable functions f with respect to (Ω , F , P ) such that k f k wL Φ < ∞ , where k f k wL Φ := inf n c > t> Φ (cid:16) tc (cid:17) P ( | f | > t ) ≤ o .
4f Φ( t ) = t p , 0 < p < ∞ , then wL Φ = wL p , where the weak L p space wL p consistsof all measurable functions f for which k f k wL p := sup t> tP ( | f | > t ) /p < ∞ . It was proved in [9] that the functional k · k wL Φ is a complete quasi-norm on wL Φ whenΦ ∈ ∆ and Φ is convex. In this paper we are interested in the case Φ is not convex.We assume that Φ is of lower type ℓ for some ℓ ∈ (0 ,
1] and upper type 1, i.e., thereexist a constant c Φ ∈ [1 , ∞ ) and some ℓ ∈ (0 ,
1] such thatΦ( tr ) ≤ c Φ max { t ℓ , t } Φ( r ) for t, r ∈ [0 , ∞ ) . Let G ℓ be the set of all Φ ∈ G satisfying the above inequality. For example, Φ( t ) = t p (cid:0) log( e + t ) (cid:1) q is in G ℓ if 0 < ℓ ≤ p < q ≥ . Let Φ ∈ G ℓ , from [15] we know that Φis equivalent to a concave function in G ℓ . Further, we can verify the functionals k · k wL Φ and k · k wL Ψ are equivalent if Φ , Ψ ∈ G ℓ are equivalent. Therefore, we always assumethat Φ ∈ G ℓ is concave in our theorems below. Thus Φ is subadditive, increasing,continuous and bijective from [0 , ∞ ) to itself when Φ ∈ G ℓ .Obviously, G ℓ ⊂ ∆ . It was shown in [11] that for any concave function Φ ∈ G ℓ , L Φ is a ℓ -quasi norm. Here the functional k · k wL Φ satisfies the following properties:(i) k f k wL Φ ≥
0, and k f k wL Φ = 0 if and only if f = 0;(ii) k cf k wL Φ = | c |k f k wL Φ ;(iii) k f + g k wL Φ ≤ (2 c Φ ) /ℓ · k f k wL Φ + k g k wL Φ ).Indeed, (i) and (ii) can be proved easily. Here we only prove the generalized triangleinequality, namely (iii). Suppose that k f k wL Φ = a , k g k wL Φ = b , a, b >
0, and K =(2 c Φ ) /ℓ . Then ∀ t > (cid:16) tK · a + b ) (cid:17) P ( | f + g | > t ) ≤ Φ (cid:16) tK · a + b ) (cid:17) P (cid:16) | f | > t (cid:17) + P (cid:16) | g | > t (cid:17)! ≤ c Φ · c Φ Φ (cid:16) t a (cid:17) P (cid:16) | f | > t (cid:17) + Φ (cid:16) t b (cid:17) P (cid:16) | g | > t (cid:17)! ≤ . Proposition 2.2.
Let Φ ∈ G ℓ , then L ⊂ L Φ ⊂ wL Φ . Proof.
Suppose that f ∈ L , then Z Ω Φ (cid:16) | f |k f k (cid:17) dP = Z {| f |≤k f k } Φ (cid:16) | f |k f k (cid:17) dp + Z {| f | > k f k } Φ (cid:16) | f |k f k (cid:17) dp ≤ Z {| f |≤k f k } Φ(1) dp + Z {| f | > k f k } c Φ max n | f |k f k , (cid:16) | f |k f k (cid:17) ℓ o Φ(1) dp ≤ Φ(1) + c Φ Φ(1) Z {| f | > k f k } | f |k f k dP ≤ Φ(1) + c Φ Φ(1) . C = max { Φ(1) + c Φ Φ(1) , } , then Z Ω Φ (cid:16) | f | ( C · c Φ ) /ℓ k f k (cid:17) dP ≤ c Φ · C c Φ Z Ω Φ (cid:16) | f |k f k (cid:17) dP ≤ , which means k f k L Φ ≤ ( C · c Φ ) /ℓ k f k . Suppose that f ∈ L Φ and t >
0. ThenΦ (cid:16) t k f k L Φ (cid:17) P ( | f | > t ) ≤ Z {| f | >t } Φ (cid:16) | f |k f k L Φ (cid:17) dP ≤ Z Ω Φ (cid:16) | f |k f k L Φ (cid:17) dP ≤ , which implies L Φ ⊂ wL Φ . The proof is complete. Remark 2.3.
It was proved in [16] that the Orlicz space L Φ has absolute continuousnorm when Φ ∈ ∆ , that is, lim P ( A ) → k f χ A k L Φ = 0 , ∀ f ∈ L Φ . But not every element in wL Φ has absolute continuous quasi norm in spire of Φ ∈ ∆ .For instance, let Ω = (0 , and P be the Lebesgue measure on Ω . Consider wL p ( < p < ∞ ) and function f ( x ) = x − /q . A simple computation shows that f ∈ wL p when q ≥ p , and f has absolutely continuous norm in wL p when q > p , but it has notwhen q = p . Definition 2.4.
Let w L Φ be the set of all f ∈ wL Φ having the absolute continuousquasi norm defined by w L Φ := n f ∈ wL Φ : lim P ( A ) → k f χ A k wL Φ = 0 o . It is easy to check w L Φ is a linear space. Moreover w L Φ is a closed subspace of wL Φ when Φ ∈ G ℓ . Indeed, suppose that f n ∈ w L Φ , and k f n − f k wL Φ → n → ∞ .Then f ∈ wL Φ and k f χ A k wL Φ ≤ (2 c Φ ) /ℓ · k f n χ A k wL Φ + k ( f n − f ) χ A k wL Φ ) ≤ (2 c Φ ) /ℓ · k f n χ A k wL Φ + k f n − f k wL Φ ) . Since f n ∈ w L Φ has absolute continuous norm and k f n − f k wL Φ → n → ∞ , weget lim P ( A ) → k f χ A k wL Φ = 0 , namely, f ∈ w L Φ , which means w L Φ is closed. Moreover, L ⊂ L Φ ⊂ w L Φ whenΦ ∈ G ℓ . The following is an extension of Lebesgue controlled convergence theorem, whichwill be used to describe the quasi norm convergence (See Remark 3.2).6 roposition 2.5. (See [10], Theorem 3.2) Let Φ ∈ G ℓ . f n , g ∈ w L Φ and | f n | ≤ g . If f n converges to f almost everywhere, then lim n →∞ k f n − f k w L Φ = 0 . In order to describe our results, we need the lower index and upper index of Φ. LetΦ ∈ G , the lower index and the upper index of Φ are respectively defined by p Φ = inf t> t Φ ′ ( t )Φ( t ) , q Φ = sup t> t Φ ′ ( t )Φ( t ) . It is well known that 1 ≤ p Φ ≤ q Φ ≤ ∞ if Φ is convex and 0 < p Φ ≤ q Φ ≤ Lemma 2.6.
Let Φ be concave and q Φ − < ∞ . Denote p = p Φ − , q = q Φ − . Then Φ − ( t ) t p , Φ( t ) t /q are increasing on (0 , ∞ ) and Φ − ( t ) t q , Φ( t ) t /p are decreasing on (0 , ∞ ) . Proof.
It is easy to see that Φ − is convex. Thus 1 ≤ p ≤ q < ∞ . From [16], weobtain that Φ − ( t ) t p is increasing on (0 , ∞ ) and Φ − ( t ) t q is decreasing on (0 , ∞ ). Replacing t with Φ( t ), we immediately get that Φ( t ) t /q is increasing on (0 , ∞ ) and Φ( t ) t /p is decreasingon (0 , ∞ ).We now introduce the weak Orlicz-Hardy martingale spaces. Denote by Λ the col-lection of all sequences ( λ n ) n ≥ of non-decreasing, non-negative and adapted functionswith λ ∞ = lim n →∞ λ n . As usual, the weak Orlicz-Hardy martingale spaces are definedas follows: wH Φ = { f = ( f n ) n ≥ : k f k wH Φ = k M ( f ) k wL Φ < ∞} ; wH S Φ = { f = ( f n ) n ≥ : k f k wH S Φ = k S ( f ) k wL Φ < ∞} ; wH s Φ = { f = ( f n ) n ≥ : k f k wH s Φ = k s ( f ) k wL Φ < ∞} ; w Q Φ = { f = ( f n ) n ≥ : ∃ ( λ n ) n ≥ ∈ Λ , s.t. S n ( f ) ≤ λ n − , λ ∞ ∈ wL Φ } , k f k w Q Φ = inf ( λ n ) ∈ Λ k λ ∞ k wL Φ ; w D Φ = { f = ( f n ) n ≥ : ∃ ( λ n ) n ≥ ∈ Λ , s.t. | f n | ≤ λ n − , λ ∞ ∈ wL Φ } , k f k w D Φ = inf ( λ n ) ∈ Λ k λ ∞ k wL Φ . Remark 2.7.
We can get the Orlicz-Hardy martingale space H s Φ when k s ( f ) k wL Φ isreplaced by k s ( f ) k L Φ in the definition above. In order to describe the duality, we define w H s Φ = { f = ( f n ) n ≥ : s ( f ) ∈ w L Φ } . It is easy to see w H s Φ is a closed subspace of wH s Φ . Similarly, we have w H Φ and w H S Φ ,which are closed subspaces of wH Φ and wH S Φ , respectively. efinition 2.8. A measurable function a is said to be a w-1-atom ( or w-2-atom,w-3-atom, resp.) if there exists a stopping time ν such that (a1) a n = E n a = 0 if ν ≥ n , (a2) k s ( a ) k ∞ < ∞ (or k S ( a ) k ∞ < ∞ , k M ( a ) k ∞ < ∞ resp.). Now we define the weak Orlicz-Hardy spaces associated with weak atoms .
Definition 2.9.
Let Φ ∈ G ℓ with ℓ ∈ (0 , . We define wH s Φ ,at ( wH S Φ ,at , wH Φ ,at resp.)as the space of all f ∈ wL Φ which admit a decomposition f = X k ∈ Z a k a.e. with for each k ∈ Z , a k is a w-1-atom (w-2-atom, w-3-atom resp.) and satisfying s ( a k ) ( S ( a k ) , M ( a k ) resp.) ≤ A · k ) for some A > , and inf n c > k ∈ Z Φ (cid:16) k c (cid:17) P ( ν k < ∞ ) ≤ o < ∞ , where ν k is the stopping time corresponding to a k .Moreover, define k f k wH s Φ ,at ( k f k wH S Φ ,at , k f k wH Φ ,at resp . ) = inf inf n c > k ∈ Z Φ (cid:16) k c (cid:17) P ( ν k < ∞ ) ≤ o where the infimum is taken over all decompositions of f described above. Recall that for q ∈ [1 , ∞ ) and a function φ : (0 , ∞ ) → (0 , ∞ ), w L q,φ := n f ∈ L q : k f k w L q,φ = Z ∞ t qφ ( x ) x dx < ∞ o , where t qφ ( x ) = 1 φ ( x ) x − /q sup P ( ν< ∞ ) ≤ x k f − f ν k q , and ν is a stopping time. Then we have Proposition 2.10. If ≤ q ≤ q < ∞ , then k f k w L q ,φ ≤ k f k w L q ,φ . Proof.
By H¨older’s inequality, t q φ ( x ) = 1 φ ( x ) x − /q sup P ( ν< ∞ ) ≤ x (cid:16) E ( | f − f ν | q χ ( ν < ∞ )) (cid:17) /q ≤ φ ( x ) x − /q sup P ( ν< ∞ ) ≤ x (cid:0) E ( | f − f ν | q (cid:1) /q P ( ν < ∞ ) (1 − q /q )(1 /q ) ≤ φ ( x ) x − /q sup P ( ν< ∞ ) ≤ x k f − f ν k q = t q φ ( x ) , which shows the proposition. 8 Weak Atomic Decompositions
We now are in a position to prove the weak atomic decomposition of the weak martin-gale Orlicz-Hardy spaces.
Theorem 3.1.
Let Φ ∈ G ℓ with ℓ ∈ (0 , and q Φ − < ∞ . Then f ∈ wH s Φ if and only ifthere exist a sequence of w-1-atoms { a k } k ∈ Z and corresponding stopping times { ν k } k ∈ Z such that (i) f n = P k ∈ Z E n a k a.e. , ∀ n ∈ N ; (ii) s ( a k ) ≤ A · k for some A > and inf n c > k ∈ Z Φ (cid:16) k c (cid:17) P ( ν k < ∞ ) ≤ o < ∞ . Moreover, k f k wH s Φ ≈ inf inf n c > k ∈ Z Φ (cid:16) k c (cid:17) P ( ν k < ∞ ) ≤ o , (3.1) where the infimum is taken over all the preceding decompositions of f . Consequently, wH s Φ = wH s Φ ,at with equivalent quasi norms. Proof.
Assume that f = ( f n ) n ≥ ∈ wH s Φ . For k ∈ Z and n ≥
0, let ν k = inf { n : s n +1 ( f ) > k } , a kn = f ν k +1 n − f ν k n Then it’s clear that { ν k } is nondecreasing and that for any fixed k ∈ Z , a k = ( a kn ) n ≥ is a martingale. Further we can see s ( f ν k ) = s ν k ( f ) ≤ k and X k ∈ Z a kn = X k ∈ Z ( f ν k +1 n − f ν k n ) = f n , a.e.for all n ≥
0. Since s ( f ν k ) ≤ k , we have s ( a k ) = (cid:16) ∞ X n =1 E n − | da kn | (cid:17) / = (cid:16) ∞ X n =1 E n − | df ν k +1 n − df ν k n | (cid:17) / = (cid:16) ∞ X n =1 E n − | df n χ { ν k 0. For ν k ≥ n , a kn = f ν k +1 n − f ν k n = 0 . So a k is a w-1-atom and (i) holds. Since { ν k < ∞} = { s ( f ) > k } , for any k ∈ Z we haveΦ k k f k wH s Φ ! P ( ν k < ∞ ) = Φ k k f k wH s Φ ! P ( s ( f ) > k ) ≤ . Thus inf n c > k ∈ Z Φ (cid:16) k c (cid:17) P ( ν k < ∞ ) ≤ o ≤ k f k wH s Φ < ∞ . The main part of the proof is the converse. Suppose that there exists a sequence ofw-1-atoms such that (i) and (ii) hold. Let M = inf n c > k ∈ Z Φ (cid:16) k c (cid:17) P ( ν k < ∞ ) ≤ o . Without loss of generality, we may assume that A = 2 N A , where N A ≥ A > 0, there exist N A ≥ A ≤ N A ). For any λ > 0, choose j ∈ Z such that2 j ≤ λ < j +1 . Now let f n = ∞ X k = −∞ a kn = j − X k = −∞ a kn + ∞ X k = j a kn := g n + h n ( n ∈ N ) . (3.2)Then we have s ( f ) ≤ s ( g ) + s ( h ) by the sublinearity of s ( f ). And thus P ( s ( f ) > Aλ ) ≤ P ( s ( g ) > Aλ ) + P ( s ( h ) > Aλ ) . By (ii) we obtain s ( g ) ≤ j − X k = −∞ s ( a k ) ≤ j − X k = −∞ A · k ≤ A · j ≤ Aλ. So P ( s ( g ) > Aλ ) = 0. Since a kn = E n a k = 0 on the set { ν k ≥ n } , thus s ( a k ) = 0 on { ν k = ∞} . Denote p = p Φ − , q = q Φ − , then by Lemma 2.6 we haveΦ (cid:16) Aλ M (cid:17) P (cid:0) s ( f ) > Aλ (cid:1) ≤ Φ (cid:16) AλM (cid:17) P (cid:0) s ( h ) > Aλ (cid:1) ≤ Φ (cid:16) AλM (cid:17) ∞ X k = j P ( ν k < ∞ ) ≤ Φ (cid:16) AλM (cid:17) ∞ X k = j (cid:16) k M (cid:17) . A = 2 N A , we obtain 2 j ≤ N A + j ≤ Aλ = 2 N A λ < N A + j +1 , where N A ≥ 0. Thus, Φ (cid:16) Aλ M (cid:17) P ( s ( f ) > Aλ ) ≤ Φ (cid:16) AλM (cid:17) N A + j X k = j (cid:16) k M (cid:17) + Φ (cid:16) AλM (cid:17) ∞ X k = N A + j +1 (cid:16) k M (cid:17) = I + II . Now let’s estimate I and II respectively. Using Lemma 2.6 again, we getI = N A + j X k = j Φ (cid:16) AλM (cid:17) Φ (cid:16) k M (cid:17) ≤ N A + j X k = j (cid:16) Aλ k (cid:17) p ≤ N A + j X k = j (2 N A + j +1 − k ) p ≤ p ( N A +1) − − p = C and II = ∞ X k = N A + j +1 Φ (cid:16) AλM (cid:17) Φ (cid:16) k M (cid:17) ≤ ∞ X k = N A + j +1 (cid:16) Aλ k (cid:17) q ≤ ∞ X k = N A + j +1 (cid:16) N A + j +1 − k (cid:17) q = 11 − − q = C . Let C = C + C . It’s easy to see that C > 1. ThusΦ Aλ ( C · c Φ ) /ℓ M ! P (cid:0) s ( f ) > Aλ (cid:1) ≤ c Φ C · c Φ Φ Aλ M ! P (cid:0) s ( f ) > Aλ (cid:1) ≤ c Φ C · c Φ · C = 1 . And so we obtain k f k wH s Φ ≤ ( C · c Φ ) /ℓ · M. (3.3)Consequently, (3.1) holds. The proof of Theorem 3.1 is complete. Remark 3.2. If f ∈ w H s Φ in Theorem 3.1, then not only (i) and (ii) hold, but alsothe sum P nk = m a k converges to f in wH s Φ as m → −∞ , n → ∞ . Indeed, n X k = m a k = n X k = m ( f ν k +1 − f ν k ) = f ν n +1 − f ν m . y the sublinearity of s ( f ) we have k f − n X k = m a k k wH s Φ = k s ( f − f ν n +1 + f ν m ) k wL Φ ≤ k s ( f − f ν n +1 ) + s ( f ν m ) k wL Φ ≤ (2 c Φ ) /ℓ · (cid:18) k s ( f − f ν n +1 ) k wL Φ + k s ( f ν m ) k wL Φ (cid:19) . Since s ( f − f ν n +1 ) = s ( f ) − s ( f ν n +1 ) , then s ( f − f ν n +1 ) ≤ s ( f ) , s ( f ν m ) ≤ s ( f ) and s ( f − f ν n +1 ) , s ( f ν m ) → a.e. as m → −∞ , n → ∞ . Thus by Proposition 2.5, wehave k s ( f − f ν n +1 ) k wL Φ , k s ( f ν m ) k wL Φ → as m → −∞ , n → ∞ , which means k f − P nk = m a k k wH s Φ → as m → −∞ , n → ∞ . Further, for k ∈ Z , a k = ( a kn ) n ≥ is L bounded, hence H s = L is dense in w H s Φ . Recall that if ( F n ) n ≥ is regular, then for any non-negative adapted sequence γ =( γ n ) n ≥ and λ ≥ k γ k ∞ , there is a stopping time ν such that { γ ∗ > λ } ⊂ { ν < ∞} , γ ∗ ν ≤ λ, P ( ν < ∞ ) ≤ RP ( γ ∗ > λ )(see [8]). Moreover, if λ ≤ λ , then we can take two stopping times ν λ and ν λ suchthat ν λ ≤ ν λ . Therefore, if ( F n ) n ≥ is regular, we get the atomic decompositions for wH S Φ and wH Φ . Theorem 3.3. Let Φ ∈ G ℓ with ℓ ∈ (0 , and q Φ − < ∞ . Then, if ( F n ) n ≥ is regular,we have wH S Φ = wH S Φ ,at with equivalent quasi norms ; wH Φ = wH Φ ,at with equivalent quasi norms. Moreover, if f ∈ w H S Φ (or w H Φ resp.), the sum P nk = m a k converges to f in wH S Φ (or wH Φ resp.) as m → −∞ , n → ∞ . Proof. The proof shall be given for wH S Φ , only, since it is just slightly different fromthe one for wH Φ . Let f ∈ wH S Φ . Then for sequence S n ( f ) and k ∈ Z , take stoppingtimes ν k such that { S ( f ) > k } ⊂ { ν k < ∞} , S ν k ( f ) ≤ k , P ( ν k < ∞ ) ≤ RP ( S ( f ) > k )and ν k ≤ ν k +1 , ν k ↑ ∞ . Still define a kn = f ν k +1 n − f ν k n , then a k = ( a kn ) n ≥ is a martingaleand S ( a k ) = (cid:16) ∞ X n =1 | da kn | (cid:17) / = (cid:16) ∞ X n =1 | df n χ { ν k 0. For ν k ≥ n , a kn = f ν k +1 n − f ν k n = f n − f n = 0 . So a k is a w-2-atom and12 n = P k ∈ Z a kn a.e. Since P ( ν k < ∞ ) ≤ RP ( S ( f ) > k ). Then let C = max { R, } , wehaveΦ k ( c Φ C ) /ℓ k f k wH S Φ ! P ( ν k < ∞ ) ≤ c Φ · c Φ C Φ k k f k wH S Φ ! · RP ( s ( f ) > k ) ≤ Φ k k f k wH S Φ ! P ( s ( f ) > k ) ≤ , which meansinf n c > k ∈ Z Φ (cid:16) k c (cid:17) P ( ν k < ∞ ) ≤ o ≤ ( c Φ C ) /ℓ k f k wH S Φ < ∞ . Conversely, suppose that f ∈ wH S Φ ,at . Let M = inf n c > k ∈ Z Φ (cid:16) k c (cid:17) P ( ν k < ∞ ) ≤ o . Without loss of generality, here we also assume that A = 2 N A , where N A ≥ 0. For any λ > 0, choose j ∈ Z such that 2 j ≤ λ < j +1 . Define f N in the same way as in (5.2),then similarly, we obtain S ( g ) ≤ j − X k = −∞ S ( a k ) ≤ Aλ. And since a kn = E n a k = 0 on the set { ν k ≥ n } , S ( a k ) = 0 on { ν k = ∞} . Thus,Φ (cid:16) Aλ M (cid:17) P ( S ( f ) > Aλ ) ≤ Φ (cid:16) AλM (cid:17) P ( S ( h ) > Aλ ) ≤ Φ (cid:16) AλM (cid:17) ∞ X k = j P ( ν k < ∞ )Dealing with the last inequality in the same way as in Theorem 3.1, we obtain k f k wH S Φ ≤ CM. Further, if f ∈ w H S Φ , just as Remark 3.2, by Proposition 2.5 we get that the sum P nk = m a k converges to f in wH S Φ as m → −∞ , n → ∞ . The proof of Theorem 3.2 iscomplete. Theorem 3.4. Let Φ ∈ G ℓ with ℓ ∈ (0 , and q Φ − < ∞ . Then w Q Φ = wH S Φ ,at with equivalent quasi norms ; w D Φ = wH Φ ,at with equivalent quasi norms. roof. The proof of Theorem 3.4 is similar to that of Theorem 3.1. So we just sketchthe outline and omit the details. Suppose that f = ( f n ) n ≥ ∈ w Q Φ (or f = ( f n ) n ≥ ∈ w D Φ resp.). Let ν k = inf { n : λ n > k } , where ( λ n ) n ≥ is the sequence in the definitionof w Q Φ (or w D Φ , resp.) For k ∈ Z and n ≥ 0, we still define a kn = f ν k +1 n − f ν k n .Then, in the same way as in Theorem 3.1, we can prove that there exists A > S ( a k ) ≤ A · k (or M ( a k ) ≤ A · k resp.), and that k f k wH S Φ ,at ≤ C k f k w Q Φ (or k f k wH Φ ,at ≤ C k f k w D Φ , resp.).For the converse part, assume that f = ( f n ) n ≥ ∈ wH S Φ ,at (or f = ( f n ) n ≥ ∈ wH Φ ,at resp.), and let λ n = P k ∈ Z χ ( ν k ≤ n ) k S ( a k ) k ∞ (or λ n = P k ∈ Z χ ( ν k ≤ n ) k M ( a k ) k ∞ resp.). Then ( λ n ) n ≥ is a non-negative, non-decreasing and adapted sequence with S n +1 ( f ) ≤ λ n (or M n +1 ( f ) ≤ λ n resp.). For y > 0, choose j ∈ Z such that 2 j ≤ y < j +1 . Then λ ∞ = λ (1) ∞ + λ (2) ∞ with λ (1) ∞ = P j − k = −∞ χ ( ν k ≤ n ) k S ( a k ) k ∞ and λ (2) ∞ = P ∞ k = j χ ( ν k ≤ n ) k S ( a k ) k ∞ (or, λ (1) ∞ = P j − k = −∞ χ ( ν k ≤ n ) k M ( a k ) k ∞ and λ (2) ∞ = P ∞ k = j χ ( ν k ≤ n ) k M ( a k ) k ∞ resp.). Similar to the argument of (3.3) (replacing s ( g ) and s ( h ) by λ (1) ∞ and λ (2) ∞ , resp.), we obtain k f k w Q Φ ≤ C k f k wH S Φ ,at (or k f k w D Φ ≤ C k f k wH Φ ,at resp.).Theorem 3.3 and Theorem 3.4 together give the following corollary. Corollary 3.5. Let Φ ∈ G ℓ with ℓ ∈ (0 , and q Φ − < ∞ . If ( F n ) n ≥ is regular, then w Q Φ = wH S Φ , w D Φ = wH Φ . As one of the applications of the atomic decompositions, we shall obtain a sufficientcondition for a sublinear operator to be bounded from the weak martingale Orlicz-Hardy spaces to wL Φ spaces. Applying the condition to M ( f ), S ( f ) and s ( f ), wededuce a series of martingale inequalities.An operator T : X → Y is called a sublinear operator if it satisfies | T ( f + g ) | ≤ | T f | + | T g | , | T ( αf ) | ≤ | α || T f | , where X is a martingale spaces, Y is a measurable function space. Theorem 4.1. Let ≤ r ≤ and T : L r (Ω) → L r (Ω) be a bounded sublinear operator.If P ( | T a | > ≤ CP ( ν < ∞ ) (4.1) for all w-1-atoms, where ν is the stopping time associated with a and C is a positiveconstant, then, for Φ ∈ G ℓ with q Φ − < ∞ and /p Φ − < r , there exists a positiveconstant C such that k T f k wL Φ ≤ C k f k wH s Φ , f ∈ wH s Φ . roof. Assume that f ∈ wH s Φ . By Theorem 3.1, f can be decomposed into the sumof a sequence of w-1-atoms. For any fixed λ > 0, choose j ∈ Z such that 2 j − ≤ λ < j and let f = ∞ X k = −∞ a k = j − X k = −∞ a k + ∞ X k = j a k := g + h. It follows from the sublinearity of T that | T f | ≤ | T g | + | T h | , so P ( | T f | > λ ) ≤ P ( | T g | > λ ) + P ( | T h | > λ ) . In Theorem 3.1, we have proved that s ( a k ) ≤ A · k for some A > s ( a k ) = 0 onthe set { ν k = ∞} . Denote p = p Φ − , q = q Φ − . Remember that k a k k r ≤ C k s ( a k ) k r , ≤ r ≤ . (4.2)It results from Lemma 2.6 that k g k r ≤ j − X k = −∞ k a k k r ≤ C j − X k = −∞ k s ( a k ) k r ≤ C j − X k = −∞ k P ( ν k < ∞ ) r . Since T is bounded on L r (Ω), thenΦ (cid:16) λ k f k wH s Φ (cid:17) P ( | T g | > λ ) ≤ Φ (cid:16) λ k f k wH s Φ (cid:17) k T g k rr λ r ≤ C Φ (cid:16) λ k f k wH s Φ (cid:17) k g k rr λ r . By the estimate of k g k r above, we getΦ (cid:16) λ k f k wH s Φ (cid:17) P ( | T g | > λ ) ≤ C Φ (cid:16) λ k f k wH s Φ (cid:17) j − X k = −∞ k P ( ν k < ∞ ) r λ ! r = C Φ (cid:16) λ k f k wH s Φ (cid:17) j − X k = −∞ k Φ (cid:16) k k f k wHs Φ (cid:17) r P ( ν k < ∞ ) r λ Φ (cid:16) k k f k wHs Φ (cid:17) r ! r ≤ C j − X k = −∞ k Φ (cid:16) λ k f k wHs Φ (cid:17) r λ Φ (cid:16) k k f k wHs Φ (cid:17) r ! r . p = p Φ − < r , we obtainΦ (cid:16) λ k f k wH s Φ (cid:17) P ( | T g | > λ ) ≤ C j − X k = −∞ (cid:16) k λ (cid:17)(cid:16) λ k (cid:17) pr ! r = Cλ p − r · ∞ X k =1 − j (cid:16)(cid:16) (cid:17) − pr (cid:17) k ! r ≤ Cλ p − r · (2 j − ) r − p ≤ C . Taking C I = (cid:0) c Φ max { C , } (cid:1) /ℓ , thenΦ (cid:16) λC I k f k wH s Φ (cid:17) P ( | T g | > λ ) ≤ c Φ c Φ max { C , } Φ (cid:16) λ k f k wH s Φ (cid:17) P ( | T g | > λ ) ≤ c Φ c Φ max { C , } C ≤ . On the other hand, since | T h | ≤ P ∞ k = j | T a k | , we getΦ (cid:16) λ k f k wH s Φ (cid:17) P ( | T h | > λ ) ≤ Φ (cid:16) λ k f k wH s Φ (cid:17) P ( | T h | > ≤ Φ (cid:16) λ k f k wH s Φ (cid:17) ∞ X k = j P ( | T a k | > ≤ C Φ (cid:16) λ k f k wH s Φ (cid:17) ∞ X k = j P ( ν k < ∞ )= C ∞ X k = j Φ (cid:16) λ k f k wHs Φ (cid:17) Φ (cid:16) k k f k wHs Φ (cid:17) Φ (cid:16) k k f k wH s Φ (cid:17) P ( ν k < ∞ ) ≤ C ∞ X k = j Φ (cid:16) λ k f k wHs Φ (cid:17) Φ (cid:16) k k f k wHs Φ (cid:17) ≤ C ∞ X k = j (cid:16) λ k (cid:17) q ≤ C . Taking C II = (cid:0) c Φ max { C , } (cid:1) /ℓ , thenΦ (cid:16) λC II k f k wH s Φ (cid:17) P (Φ( | T h | ) > λ ) ≤ . Since T is sublinear,Φ (cid:16) λ C I + C II ) k f k wH s Φ (cid:17) P ( | T f | > λ ) ≤ Φ (cid:16) λC I k f k wH s Φ (cid:17) P ( | T g | > λ )+ Φ (cid:16) λC II k f k wH s Φ (cid:17) P ( | T h | > λ ) ≤ . k T f k wL Φ ≤ C k f k wH s Φ , f ∈ wH s Φ . The proof is complete. Remark 4.2. Similarly, if T : L r (Ω) → L r (Ω) is a bounded sublinear operator, ≤ r < ∞ , and (4.1) holds for all w-2-atoms (or w-3-atoms). Then for Φ ∈ G ℓ with q Φ − < ∞ and p Φ − < r , there exists a constant C > such that k T f k wL Φ ≤ C k f k w Q Φ , f ∈ w Q Φ ( or k T f k wL Φ ≤ C k f k w D Φ , f ∈ w D Φ ) . In this case, we do not need to restrict ≤ r ≤ ; in fact, (4.2) is replaced by k a k k r ≤ C k S ( a k ) k r (or k a k k r ≤ C k M ( a k ) k r ), which always holds for ≤ r < ∞ by the Burkholder-Davis-Gundy inequalities. Theorem 4.3. Let Φ ∈ G ℓ with ℓ ∈ (0 , and q Φ − < ∞ . Then for all martingales f = ( f n ) n ≥ the following inequalities hold: k f k wH Φ ≤ C k f k wH s Φ , k f k wH S Φ ≤ C k f k wH s Φ ; (4.3) k f k wH Φ ≤ C k f k w Q Φ , k f k wH S Φ ≤ C k f k w Q Φ , k f k wH s Φ ≤ C k f k w Q Φ ; (4.4) k f k wH Φ ≤ C k f k w D Φ , k f k wH S Φ ≤ C k f k w D Φ , k f k wH s Φ ≤ C k f k w D Φ ; (4.5) C − k f k w D Φ ≤ k f k w Q Φ ≤ C k f k w D Φ . (4.6) Moreover, if {F n } n ≥ is regular, then wH S Φ = w Q Φ = w D Φ = wH Φ = wH s Φ . Proof. First we show (4.3). Let f ∈ wH s Φ . The maximal operator T f = M f issublinear. It’s well know that T is L -bounded. If a is a w-1-atom and ν is thestopping time associated with a , then {| T a | > } = { M ( a ) > } ⊂ { ν < ∞} andhence (4.1) holds. Since Φ ∈ G ℓ , the condition p Φ − < − . Thus it follows from Theorem 4.1 that k f k wH Φ = k T f k wL Φ ≤ C k f k wH s Φ . Similarly, considering the operator T f = Sf we get the second inequality of (4.3).Next we show (4.4) and (4.5). Choose r such that p Φ − < < r < ∞ . Noticingthat the operator M f , Sf and sf are L r bounded. Taking T f = M f , Sf or sf , resp.By Remark 4.2, we get (4.4) and (4.5).To prove (4.6), we use (4.4) and (4.5). The method used below is the same as theproof of Theorem 3.5 in [17]. Assume that f = ( f n ) n ≥ ∈ w Q Φ , then there exists anoptimal control ( λ (1) n ) n ≥ such that S n ( f ) ≤ λ (1) n − with λ (1) ∞ ∈ wL Φ . Since | f n | ≤ f ∗ n − + λ (1) n − , 17y (4.4) we have k f k w D Φ ≤ C ( k f k wH Φ + k λ (1) ∞ k wL Φ ) ≤ C k f k w Q Φ . On the other hand, if f = ( f n ) n ≥ ∈ w D Φ , then there exists an optimal control ( λ (2) n ) n ≥ such that | f n | ≤ λ (2) n − with λ (2) ∞ ∈ wL Φ . Notice that S n ( f ) ≤ S n − ( f ) + 2 λ (2) n − . Using (4.5) we can get the other side of (4.6).Further, suppose that {F n } n ≥ is regular. Then for any martingale f = ( f n ) n ≥ , wehave | df n | ≤ R − E n − | df n | (see [21], pp 31, Proposition 2.19). Thus S n ( f ) ≤ r R − s n ( f ) . Since s n ( f ) ∈ F n − , by the definition of w Q Φ we have k f k w Q Φ ≤ k s ( f ) k wL Φ = k f k wH s Φ . Using (4.4), (4.6) and Corollary 3.5, we conclude that wH S Φ = w Q Φ = w D Φ = wH Φ = wH s Φ . In this section, we investigate the dual of weak martingale Orlicz-Hardy spaces andgive a new John-Nirenberg theorem. Theorem 5.1. Let Φ ∈ G ℓ with ℓ ∈ (0 , , q Φ − < ∞ and φ ( r ) = 1 / ( r Φ − (1 /r )) . Then ( w H s Φ ) ∗ = w L ,φ . Proof. Let g ∈ w L ,φ , then g ∈ H s . Define l g ( f ) = E (cid:16) ∞ X n =1 df n dg n (cid:17) , f ∈ H s . From Theorem 3.1, there is a sequence of w-1-atoms a k and corresponding stoppingtimes ν k , where k ∈ Z , such that s ( a k ) ≤ k +1 , Φ k k f k wH s Φ ! P ( ν k < ∞ ) ≤ df n = ∞ X k = −∞ da kn a.e.for all n ∈ N . The last series also converges to df n in H s -norm. Hence l g ( f ) = ∞ X n =1 ∞ X k = −∞ E ( da kn dg n ) . Applying the H¨older inequality and the definition of weak atoms, we get that | l g ( f ) | ≤ ∞ X k = −∞ E (cid:16) ∞ X n =1 | da kn | χ { ν k 0, choose j ∈ Z suchthat 2 j ≤ λ < j +1 and define the martingales f N , g N and h N , respectively, by f Nn = N X k = − N a kn , g Nn = j − X k = − N a kn and h Nn = N X k = j a kn . (5.1)Then we have Φ( s ( f )) ≤ Φ( s ( g )) + Φ( s ( h )) by the sublinearity of s ( f ) and Φ( t ). Andthus P (Φ( s ( f N )) > λ ) ≤ P (Φ( s ( g N )) > λ ) + P (Φ( s ( h N )) > λ ) . Since k s ( g N ) k ≤ j − X k = − N k s ( a k ) k ≤ j − X k = − N (2 − k ) / Φ − (2 k ) , P ( s ( g N ) > Φ − ( λ )) ≤ − ( λ )) k s ( g N ) k ≤ − ( λ )) j − X k = − N (2 − k ) / Φ − (2 k ) ! = j − X k = − N (2 − k ) / Φ − (2 k )Φ − ( λ ) ! ≤ j − X k = − N (2 − k ) / (cid:16) k λ (cid:17) p ! = λ − p (cid:16) j − X k = − N (2 p − ) k (cid:17) ≤ C λ − . In the last inequality above, we used 1 / < ≤ p ≤ q < ∞ , which results from thatΦ − is a convex function. Denote C I = (2 c Φ max { C , } ) /ℓ , thenΦ (cid:16) Φ − ( λ ) C I (cid:17) P ( s ( g N ) > Φ − ( λ )) ≤ c Φ c Φ max { C , } λP ( s ( g N ) > Φ − ( λ )) ≤ . Noticing that a kn = 0 on { ν k ≥ n } and P ( ν k < ∞ ) ≤ − k , we get P ( s ( h N ) > Φ − ( λ )) ≤ N X k = j P ( ν k < ∞ ) ≤ N X k = j − k = 2 − j ≤ λ − . Denote C II = (8 c Φ ) /ℓ , thenΦ (cid:16) Φ − ( λ ) C II (cid:17) P ( s ( h N ) > Φ − ( λ )) ≤ . Let C = 2 q max { C I , C II } , thenΦ (cid:16) Φ − (2 λ ) C (cid:17) P ( s ( f N ) > Φ − (2 λ )) ≤ Φ (cid:16) Φ − (2 λ ) C (cid:17)(cid:16) P ( s ( g N ) > Φ − ( λ )) + P ( s ( h N ) > Φ − ( λ )) (cid:17) ≤ Φ (cid:16) q Φ − ( λ )2 q C I (cid:17) P ( s ( g N ) > Φ − ( λ ))+ Φ (cid:16) q Φ − ( λ )2 q C II (cid:17) P ( s ( h N ) > Φ − ( λ )) ≤ 12 + 12 = 1 , k f N k wH s Φ ≤ C . Since l ( f N ) = E ∞ X n =1 df Nn dg n = E ∞ X n =1 N X k = − N da kn dg n = N X k = − N E ∞ X n =1 | dg n − dg ν k n | (2 k ) / Φ − (2 k ) k S ( g − g ν k ) k = N X k = − N (2 − k ) / − (2 k ) k g − g ν k k , then C k l k ≥ l ( f N ) = N X k = − N φ (2 − k ) (2 − k ) − / k g − g ν k k . Taking over all N ∈ N and the supremum over all of such stopping times such that P ( ν k < ∞ ) ≤ − k , k ∈ Z , we obtain k g k w L ,φ = Z ∞ t φ ( x ) x dx ≤ C ∞ X k = −∞ t φ (2 − k ) ≤ C k l k . The proof of Theorem 5.1 is complete.To obtain the new John-Nirenberg theorem, we first prove two lemmas. Lemma 5.2. Let Φ ∈ G ℓ with ℓ ∈ (0 , , q Φ − < ∞ and φ ( r ) = 1 / ( r Φ − (1 /r )) . If {F n } n ≥ is regular, then ( w H Φ ) ∗ = w L ,φ . Proof. Let g ∈ w L ,φ and define l g ( f ) = E ( f g ) , f ∈ L ∞ . Then | l g ( f ) | = | E ( f g ) | = | ∞ X k = −∞ E ( a k ( g − g ν k )) |≤ ∞ X k = −∞ k a k k ∞ k g − g ν k k ≤ ∞ X k = −∞ k +1 k g − g ν k k ≤ k f k wH Φ ∞ X k = −∞ k k f k wH Φ k g − g ν k k . A k = 1 . Φ (cid:16) k k f k wH Φ (cid:17) , we get | l g ( f ) | ≤ k f k wH Φ ∞ X k = −∞ φ ( A k ) A − k k g − g ν k k ≤ k f k wH Φ ∞ X k = −∞ t φ ( A k ) ≤ C k f k wH Φ k g k w L ,φ Conversely, suppose that l ∈ ( w H Φ ) ∗ . Since L is dense in w H Φ , there exists g ∈ L ⊂ L such that l ( f ) = E ( f g ) , f ∈ L ∞ . Let ν k be the stopping times satisfying P ( ν k < ∞ ) ≤ (Φ(2 k )) − ( k ∈ Z ). For k ∈ Z ,define h k = sign( g − g ν k ) , a k = 2 k ( h k − h ν k k ) . It is easy to see that each a k ( k ∈ Z ) is a w-3-atom. Thus, by Theorem 3.3, if f N isagain defined by (5.1), then k f N k wH Φ ≤ C. Therefore C k l k ≥ | l ( f N ) | = | E ( f N g ) | = | N X k = − N E ( a k g ) | = | N X k = − N k E (( h k − h ν k k ) g ) | = | N X k = − N k E ( h k ( g − g ν k )) | = N X k = − N k k g − g ν k k = N X k = − N φ (cid:0) k ) (cid:1) (cid:16) k ) (cid:17) − k g − g ν k k Taking over all N ∈ N and the supremum over all of such stopping times such that P ( ν k < ∞ ) ≤ (Φ(2 k )) − , k ∈ Z , we obtain k g k w L ,φ = Z ∞ t φ ( x ) x dx ≤ C ∞ X k = −∞ t φ (cid:16) k ) (cid:17) ≤ C k l k . The proof of Lemma 5.2 is complete. Lemma 5.3. Let Φ ∈ G ℓ with ℓ ∈ (0 , , q Φ − < ∞ and φ ( r ) = 1 / ( r Φ − (1 /r )) . Thenfor q ∈ [1 , ∞ ) , if {F n } n ≥ is regular, we have ( w H Φ ) ∗ = w L q,φ . Proof. If g ∈ w L q,φ and l g ( f ) := E ( f g ) , f ∈ L q ′ , q ′ = q/ ( q − | l g ( f ) | = | E ( f g ) | ≤ C k f k wH Φ k g k w L ,φ ≤ C k f k wH Φ k g k w L q,φ . Conversely, suppose that l ∈ ( w H Φ ) ∗ . Since L q ′ ⊂ L ⊂ L Φ ⊂ w L Φ , L q ′ can beembedded continuously in w H Φ . Thus there exists g ∈ L q such that l equals l g on L q ′ .Let ν k be the stopping times satisfying P ( ν k < ∞ ) ≤ − k ( k ∈ Z ). Define h k = | g − g ν k | q − sign( g − g ν k ) k g − g ν k k q − q , a k = Φ − (2 k )(2 k ) − /q ′ ( h k − h ν k k ) . Then k h k k q ′ = 1 and a k = 0 on the set { ν k = ∞} . For λ > 0, choose j ∈ Z such that2 j ≤ λ < j +1 . Define f N , g N and h N ( N ∈ N ) again by (5.1), then k M ( g N ) k q ′ ≤ k g N k q ′ ≤ j − X k = − N k a k k q ′ ≤ j − X k = − N Φ − (2 k )(2 k ) − /q ′ and P ( M ( g N ) > Φ − ( λ )) ≤ − ( λ )) q ′ k M ( g N ) k q ′ q ′ ≤ q ′ (Φ − ( λ )) q ′ j − X k = − N Φ − (2 k )(2 k ) − /q ′ ! q ′ ≤ q ′ j − X k = − N (2 − k ) /q ′ Φ − (2 k )Φ − ( λ ) ! q ′ ≤ C j − X k = − N (2 − k ) /q ′ (cid:16) k λ (cid:17) p ! q ′ = C · λ − q ′ p (cid:16) j − X k = − N (2 p − q ′ ) k (cid:17) ≤ Cλ − . The last inequality holds since 1 /q ′ < ≤ p . Applying the method used in Theorem5.1, we conclude that k f N k wH Φ ≤ C . Consequently, C k l k ≥ | l ( f N ) | = | N X k = − N E ( a k g ) | = | N X k = − N Φ − (2 k )(2 k ) − /q ′ E (( h k − h ν k k ) g ) | = | N X k = − N Φ − (2 k )(2 k ) − /q ′ E ( h k ( g − g ν k )) | = N X k = − N Φ − (2 k )(2 k ) − /q ′ k g − g ν k k q = N X k = − N φ (2 − k ) (2 − k ) − /q k g − g ν k k q N ∈ N and the supremum over all of such stopping times such that P ( ν k < ∞ ) ≤ − k ( k ∈ Z ), we obtain k g k w L q,φ = Z ∞ t qφ ( x ) x dx ≤ C ∞ X k = −∞ t qφ (2 − k ) ≤ C k l k . The proof of the theorem is complete.We finally formulate the weak version of the John-Nirenberg theorem, which directlyresults from Lemma 5.2 and Lemma 5.3. Theorem 5.4. If there exists Φ ∈ G ℓ with q Φ − < ∞ such that φ ( r ) = r Φ − (1 /r ) for all r ∈ (0 , ∞ ) , and {F n } n ≥ is regular. Then w L q,φ spaces are equivalent for all ≤ q < ∞ . Remark 5.5. Considering Φ( t ) ≡ , we obtain the John-Nirenberg theorem, Corollary8 in [19] due to Weisz. 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