Abstract
Abstract. In this paper, we consider the Fefferman-Stein decomposition of Q α (R n ) and give an affirmative answer to an open problem posed by M. Essen, S. Janson, L. Peng and J. Xiao in 2000. One of our main methods is to study the structure of the predual space of Q α (R n ) by the microlocal quantities. This result indicates that the norm of the predual space of Q α (R n ) depends on the micro-local structure in a self-correlation way.

In this paper, we give a wavelet characterization of the predual of Q-space Q α (R n ) without using a family of Borel measures. By this result, we obtain a Fefferman-Stein type decomposition of Q α (R n ). Let R 0 be the unit operator and R i , i = 1, · · · , n, be the Riesz transforms, respectively. In 1972, in the celebrated paper [4] , C. Fefferman and E. M. Stein proved the following result. The importance of Fefferman-Stein decomposition exists in two aspects. On one hand, there is a close relation between the∂−equation and FeffermanStein decomposition. On the other hand, this decomposition helps us get a better understanding of the structure of BMO(R n ) and the distance between L ∞ (R n ) and BMO(R n ). Due to the above two points, Fefferman-Stein decomposition of BMO(R n ) has been studied extensively by many mathematicians since 1970s. We refer the readers to Jones [5, 6] and Uchiyama [16] for further information. In latest decades, Fefferman-Stein decomposition is also extended to other function spaces, for example, BLO, C 0 and V MO. We refer the reader to [1] , [12] and the reference therein.
As an analogy of BMO space, Q-spaces own the similar structure and many common properties (see [2] , [3] and [19] ). It is natural to seek a Fefferman-Stien type decomposition of Q-spaces. For the Q-spaces on unit disk, Nicolau-Xiao [10] obtained a decomposition of Q p (∂D) similar to Fefferman and Stein's result of BMO(∂D) (see [10] , Theorem 1.2). On Euclidean space R n , Essen-Janson-Peng-Xiao [3] introduced Q α (R n ) as a generalization of Q p (∂D). For α ∈ (−∞, ∞), Q α (R n ) are defined as the spaces of all measurable functions with In this paper, we give an affirmative answer to this open problem. Generally speaking, there are two methods to obtain the Fefferman-Stein decomposition of BMO(R n ). In [4] , C. Fefferman and E. M. Stein split BMO functions by an extension theorem, i.e., the Hahn-Banach theorem in functional analysis. In [16] , A. Uchiyama gave a constructive proof of Theorem 1. In this paper, using wavelets, we study the micro-local structure of P α (R n ), the predual of Q α (R n ). By this way, we obtain a Fefferman-Stein type decomposition of Q α (R n ). For the Fefferman-Stein decomposition of Q α (R n ), the difficulties exist in two aspects.
(1) For a function f in P α (R n ), 0 < α < n 2
, the higher frequency party and the lower frequency party make different contributions to the norm f P α . That is to say, P α (R n ) have special micro-local structure. (2) For any function f , Riesz transforms may cause a perturbation on all the range of its frequencies. To obtain Fefferman-Stein type decomposition, we need to control the range of the perturbation.
To overcome the above two difficulties, on one hand, we analyze the micro-local structure of functions in P α (R n ). Such micro-local structure could help us get a wavelet characterization of P α (R n ) without involving a group of Borel measures. On the other hand, we use classical Meyer wavelets to control the range of the perturbation. See also Remark 5.9.
In Section 2, we will give the definition of wavelet basis {Φ
It is well-known that a function g(x)
can be written as a sum
Let g(x) be a function in Besov spaces or Triebel-Lizorkin spaces. Roughly speaking, the norms of g(x) can be determined by the l p (L q )-norm or L p (l q )-norm of {g j (x)}, respectively. See [8] and [15] 
, the situation becomes complicated and we can not use the above ideas. In Section 3, we introduce the micro-local quantities with levels to study the structure of functions in
as the localization of g(x) on Q. Then we restrict the range of frequency by limiting the number of j. In fact, we consider the function
We obtain three micro-local quantities about g t,Q (x) by using some basic results in analysis. See Section 3 for details.
In Section 4, applying the above micro-local analysis of functions in P α (R n ), we give a new wavelet characterization of this space. As the predual of Q α (R n ), P α (R n ) has been studied by many authors. One method is to define the predual space P α (R n ) by a family of Borel measures. See Kalita [7] , Wu-Xie [18] and Yuan-Sickel-Yang [22] . This idea can result in wavelet characterization of the predual space; but the predual space with the induced norm is a pseudo-Banach space. Dafni-Xiao [2] used a method of Hausdorff capacity to study P α (R n ). Peng-Yang [11] defined P α (R n ) by the atoms ( see also [20] ). By these methods, P α (R n ) are Banach spaces; but these authors did not consider the wavelet characterization of P α (R n ). Compared with the former results of [7] , [11] , [20] and [22] , our result owns the following advantage. Let f be a function in P α (R n ). Our wavelet characterization indicates clearly that different frequencies exert different influences to the P α -norm of f . See Theorem 3.4 and Theorem 4.2. To obtain the Fefferman-Stein type decomposition of Q α (R n ), we need such a wavelet characterization of P α (R n ). In Section 5, by the characterization obtained in Section 4 and the properties of Meyer wavelets and Daubechies wavelets, we characterize P α (R n ) associated with Riesz transforms. See Theorem 5.8. Applying this result and the duality between P α (R n ) and Q α (R n ), we obtain a Fefferman-Stein type decomposition of Q α (R n ). We need to point out that our definition of Q α (R n ) is different from the one introduced in [3] . For non-trivial spaces, the scope of α in [3] is restricted to (0, min{1, n 2 }), while the scope in our definition can be relaxed to (0, n 2 ). More importantly, when α ∈ (0, min{1, n 2 }), our definition is equivalent to the one in [3] . So the Fefferman-Stein type decomposition obtained in Section 5 gives a positive answer to Problem 1.1 proposed in [3] .
The rest of this paper is organized as follows. In Section 2, we state some preliminary notations and lemmas which will be used in the sequel. In Section 3, we study the micro-local quantities for the functions in P α (R n ). In Section 4, we obtain a wavelet characterization of P α (R n ) by the micro-local result in Section 3. In Section 5, a Fefferman-Stein type decomposition of Q α (R n ) is obtained. For this decomposition, we need a characterization of P α (R n ) associated with Riesz transforms. We will give this characterization in Section 6.
Preliminaries
In this section, we present some preliminaries on wavelets, functions and operators which will be used in this paper.
Wavelets and classic function spaces.
In this paper, we use realvalued tensor product wavelets; which can be regular Daubechies wavelets or classical Meyer wavelets. For simplicity, we denote by 0 the zero vector in R n . Let E n = {0, 1} n \{0}. Let Φ 0 (x) and Φ ǫ (x), ǫ ∈ E n , be the scale function and the wavelet functions, respectively. If Φ ǫ (x) is a Daubechies wavelet, we assume there exist m > 8n and M ∈ N such that
ǫ (x) has the vanishing moments up to the order m − 1.
For further information about wavelets, we refer the reader to [8] , [17] and [20] .
We list some knowledge on Sobolev spaces and Hardy space. For 1 < p < ∞, we denote by p ′ the conjugate index of p, that is,
See [8] , [15] and [20] for details. Let χ(x) be the characteristic function of the unit cube
, we have the following characterization, see [8] and [20] :
if and only if
, there are only constants in Q α (R n ) defined by (1.1). Now we introduce a new definition of Q-spaces such that the Q-spaces are not trivial for 1 ≤ α ≤ n 2 . For α ∈ R, denote by
is a space defined as the set of all measurable functions f with sup
where the supremum is taken over all cubes Q.
is a space defined as the set of all measurable functions f ∈ Q α (R n ) and satisfying
where the limit are taken over all cubes Q.
, the Q-spaces in Definition 2.3 are not trivial. Further, for other indices α, the corresponding Q α (R n ) coincide with the ones defined in [3] . So Q α (R n ) defined in Definition 2.3 is a generalization of Q-spaces defined by (1.1).
Similar to W s,2 (R n ), Q-spaces can be also characterized by wavelets.
By Propositions 2.2 and 2.5, we may identify a function
with the wavelet coefficients {g
Calderón-Zygmund operators. In this subsection, we introduce some preliminaries about Calderón-Zygmund operators, see [8] , [9] and [14] . For x y, let K(x, y) be a smooth function such that
where N 0 is a large enough constant and N 0 ≤ m.
A linear operator T is said to be a Calderón-Zygmund operator in
Remark 2.6. In (2.1), the values of K(x, y) have not been defined for x = y. According to Schwartz kernel theorem, the kernel K(x, y) of a linear continuous operator T is only a distribution in S ′ (R 2n ).
At the end of this subsection, we list a variant result about the continuity of Calderón-Zygmund operators on Sobolev spaces ( see also [12] ).
We have
Predual spaces of
Below we state the standard atoms, the wavelet atoms and the predual spaces of Q α (R n ) which were introduced in [11] and [20] .
where {λ u } u∈Z ∈ l 1 and g u (x) are (α, 2)−atoms.
where {λ u } u∈Z ∈ l 1 and g u (x) are (α, 2)−wavelet atoms.
In fact, P α (R n ) and P α w (R n ) are the same spaces and Calderón-Zygmund operators are bounded on P α (R n ). The proof of the following lemma can be found in [11] and [20] .
Applying the same ideas in [11] , [12] , [13] and [22] , we have the following duality relation.
It is well known that the norms of P 0 (R n ) and P n 2 (R n ) depend only on the L p (l 2 )-norms of function series { f j = Q j f } j∈Z for p = 1 and p = 2, respectively. For the case 0 < α < n 2 , the situation is complicated. In this section, we use wavelets to analyze the micro-local structure of P α (R n ). First, we present a theorem on conditional maximum value in Subsection 3.1. Then we consider the micro-local quantities in Subsection 3.2.
Conditional maximum value for non-negative sequence. For
For a non-negative sequenceg t j,k , we find the maximum value of the following quantities:
ns restricted conditions
There exist (2 n − 1) 0≤s≤t 2 ns elements in G t,n . We can see that f t j,k is a sequence, where the number of nonnegative terms is at most (2 n −1) 
are restricted in a closed domain, so the conclusion is obvious.
Micro-local quantities in
To prove a function g ∈ P α (R n ), we only need to consider sup
with f Q 0 α ≤ 1. However, by this method, we can not know the micro-local structure of g(x) in details.
To delete this shortage, we introduce a new method. Let
We localize g(x) by restricting its wavelet coefficients g ǫ j,k such that Q j,k ⊂ Q. Then we limit the range of frequencies of g(x) and analyze its microlocal information.
For this purpose, we consider the function g t,Q (x) defined in (1.2). For such a g t,Q , the number of (ǫ, j, k) such that g
ns . We study micro-local functions g t,Q in P α (R n ) and obtain three kinds of micro-local quantities. For all t ≥ 0, j ∈ Z and k ∈ Z n , we consider the series
Notice that there is a one-to-one relation between the sequences g t j,k and the function g t j,k (x). Sometimes, we do not distinguish them. For simplicity, we suppose that our functions are real-valued. Let
If f, g and
To compute max
, according to (3.3), we can restrict f to the function 
We denote by By (ii) of Proposition 2.5, we have s, u) ∈ G t,n and (ǫ, s, u) 
According to Theorem 3.3, there exists at least a sequencē 
According to (3.3) and (3. 
Applying (ii) of Proposition 2.5 again, we know that {Q indicate the micro-local characteristics in both the frequency structure and the local structure.
In the rest of this paper, the above three kinds of quantities will be used in the following sections repeatedly. These micro-local quantities tell us the global information of functions in P α (R n ). In Section 4, this idea will be used to get a characterization of P α (R n ) by a group of L 1 functions defined by the absolute values of wavelet coefficients. Such wavelet characterization does not involve the action of a group of Borel measures.
Wavelet characterization of
In recent years, as the predual of Q α (R n ), P α (R n ) was studied by many authors. We refer the reader to [2] , [11] , [20] , [21] and [22] for details. In this section, we give a new wavelet characterization of P α (R n ) by using the micro-local results obtained in Section 3.
In the famous book [8] , Y. Meyer proved that the Hardy space H 1 (R n ) can be characterized by some L 1 functions defined by wavelet coefficients without using a family of Borel measures. Based on the micro-local results in Section 3, we give a similar result. We show that each element in P α (R n ) can be characterized by a group of L 1 functions P s,t,N g(x). In Section 5, this result will be used to get a characterization of P α (R n ) associated with Riesz transformations.
For s ∈ Z and N ∈ N, let
We can see that 
We denote by Q 0,g
By (3.3), we have (4.3) sup
We prove first an approximation lemma on P α (R n ).
Lemma 4.1. For g(x)
For arbitrary δ > 0, there exist s and N such that g −g s,N P α ≤ δ.
Proof. For any 0 < δ <
, according to (α, 2)−wavelet atom decomposition, there exist {λ u } u∈N + ∈ l 1 and a group of (α, 2)−wavelet atoms a u (x) such that g = u∈N + λ u a u (x) and
Further, there exists an integer N δ > 0 such that
Now, for u = 1, · · · , N δ , we consider the atoms
then there exists an integerÑ δ > 0 such that
Since 1 ≤ u ≤ N δ , there exists an integer j δ ∈ Z such that (4.6)
By (4.4) and (4.5), we know that
and g 2,δ (x) P α ≤ 
For t = N, we have According to (4.9)-(4.12), the proof of Theorem 4.2 is completed.
Fefferman-Stein type decomposition of Q-spaces
In this section, by the wavelet characterization obtained in Theorem 4.2, we give a Fefferman-Stein type decomposition of Q α (R n ). In [3] , M. Essen, S. Janson, L. Peng and J. Xiao proved that
We can see that the Hardy space
Further, in the proof of Theorem 5.7, we need some special properties of Daubechies wavelets. Except for Theorem 5.7, we use the classical Meyer wavelets Φ ǫ (x) throughout Section 5 and Section 6. Let Φ 0 and Φ 1 be the classical Meyer wavelets in [8] . The supports of Φ 0 and Φ 1 satisfy the following conditions
).
For tensor product Meyer wavelet satisfying (
For any s ∈ Z and N ∈ N, we set
For all integers t = 0, 1, · · · , N, denote
By Theorem 4.2, we introduce a spaceP α (R n ).
This space is not really new. In fact,
Proof. P 0 (R n ) = H 1 (R n ) is known, so (i) is evident. Now we consider the cases 0 < α <
Hence sup
Conversely, we have
.
Theorem 5.3 tells us that if g ∈ P α (R n ), the P α -norm of g is equivalent to
.
In other words, for 0 < α < n 2
, the higher frequency part T 
The spaces L 1,α (R n ) and L ∞,α (R n ) are defined as follows.
Definition 5.4. Given
The following theorem can be deduced from Proposition 5.1 and Theorem 5.3.
Remark 5.6. For α = 0, we have:
Now we use Daubechies wavelets to prove that
We know that there exist some integer M and Daubechies scale function
where e = (1, 1, · · · , 1).
e) and let
Proof. For j, j ′ ∈ 2N and j j ′ , the supports of Φ(2 j x) and Φ(2 j ′ x) are disjoint. Hence the above f (x) in (5.6) belongs to L ∞ (R n ). The same reasoning gives, for any j ′ ∈ N,
Now we compute the wavelet coefficients of f (x) in (5.6).
For j ′ ≥ 0, by orthogonality of the wavelets, we have
Similarly, for the case j
By the above estimates and the wavelet characterization of Q-spaces,
is smooth in the ball {x : |x| ≤ 2 M−1 }. Applying (5.5), there exists a positive δ > 0 such that for |x| < δ,
We can see that if 2 2 j |x| < δ, then
Hence 
Theorem 2 results in the solving of Fefferman-Stein decomposition of BMO(R n ). The following theorem is a similar result for P α (R n ) and it extends Theorem 2. If α = 0, Theorem 5.8 becomes Theorem 2. We omit the proof for this case. The proof for the cases 0 < α < n 2 is very long. So we only state this result here and postpone the proof to Section 6.
Remark 5.9. For the cases 0 < α < . Their open problems attract a lot of attention and have been studied extensively by many authors. See [2] , [11] , [18] , [20] , [22] and the references therein. In the open problem 8.3 of [3] , the authors ask what would be the suitable subspaces of Q α (R n ) in which FeffermanStein decomposition is valid.
Unlike the situation of
If we want to solve the problem 8.3 by Fefferman-Stein's idea, we need to establish a relation between P α (R n ) and the functions in L 1 (R n ). For this purpose, we should apply much more skills. For example, in Section 3, we consider the micro-local property of P α (R n ). In Subsection 5.1, we construct the space L 1,α (R n ) by Meyer wavelets.
If Theorem 5.8 holds, by Theorem 5.7, we could obtain a FeffermanStein type decomposition of Q α (R n ) using Fefferman-Stein's skill in [4] . This result solves Problem 1.1 (Problem 8.3 in [3] ).
Proof. By the continuity of the Calderón-Zygmund operators on Q-spaces,
Now we prove the converse result. Let
The norm of B is defined as
We define
S is a closed subset of B and g 0 → (g 0 , R 1 g 0 , · · · , R n g 0 ) define a norm preserving map from P α (R n ) to S . Hence the set of continuous linear functionals on P α (R n ) is equivalent to the set of continuous linear functionals on S . We know that the dual space of
6. The proof of Theorem 5.8
In [4] , C. Fefferman and E. M. Stein used the Riesz transformations to characterize Hardy space H 1 (R n ) in terms of the L 1 norm. In this section, we prove a similar result for the cases 0 < α < n 2 . First, we prove that
By (ii) of Proposition 2.11, Riesz transforms are bounded on
By (i) of Theorem 5.5, we have
The proof of the converse of (6.1) is cumbersome. We divide the proof into four parts which are given in Subsections 6.1-6.4, respectively. 6.1. A lemma. We first prove the following lemma.
Proof. (i) By applying wavelet characterization of Hardy spaces and the orthogonality properties of the Meyer wavelets, we have
(ii) P α (R n ) is a Banach space, hence we have
To prove the inequality on the left side, denote
We denote by F g the set
By (ii) of Proposition 2.5, we have
Hence, by (3.3) and (6.2), we can get max g j P α , g −g j P α ≤ g P α .
(iii) By the definition of the norm of g(x) in L 1,α (R n ), for any s ∈ Z, N ∈ N and s − N ≤ j ≤ s, there exists j 0 such that 0 ≤ j 0 ≤ N and
If j < s − j 0 , we apply (i) of this lemma to get the desired assertion. If j ≥ s − j 0 , we apply (ii) of this lemma to get the desired assertion.
6.2. The proof of converse part. For the converse of (6.1), it is sufficient to prove that ∀s 1 ∈ Z, N 1 ≥ 1 and
Owing to (5.2), there exists {g
Due to (6.4) , to estimate the L 1,α −norm of 
which is a rearrangement of the set {1, · · · , n} such that
If the dimension n = 1, then τ = 1. If the dimension n > 1, τ can be any number between 1 and n. That is to say, according to (6.7), there exist n possibilities for u i s,N . In fact, by induction, we can transfer the cases τ > 2 to the case τ = 2. Hence we only give the proofs for τ = 1 and τ = 2.
6.3. The proof of τ = 1. For τ = 1, we denote
We divide the proof into three subcases. By (6.5), we have g
Further, for i = 1, · · · , n, we have
Hence, by the equation (6.8), for i = 1, · · · , n, For i = 1, · · · , n, we know that
Define h i (x), i = 1, 2, 3, 4, as
where the sequences {h ǫ,i j,k }, i = 1, 2, 3, 4, are four arbitrary sequences satisfying the condition h i L ∞ ≤ 1.
We consider
By (6.6 ) and the definition of t
For i = 1, · · · , n, we have That is to say, g s 1 ,N 1 (x) satisfies the conditions (6.12) and (6.13). By (6.5) and (6.6), g Putting the four parts together, we complete the proof of (6.3) for τ = 1.
6.4. The proof of τ = 2. For τ = 2, we denote        t Further, by (6.14) and Lemma 6. Then we divideg˜s 1 ,Ñ 1 (x) into three functions as Subcases 6.3.2 and 6.3.3. Becauseg˜s 1 ,Ñ 1 (x) satisfies (6.17), (6.18) and (6.19) , we obtain the estimate (6.15) by a similar proof.
Step 2. For Subcases 6.4.2 and 6.4.4, we decompose g s 1 ,N 1 (x) into three functions like what we did for Subcases 6.3.2 and 6.3.3. By a similar proof, we get the estimate (6.3).
Step 3. For Subcase 6. By the above three steps, we can obtain the estimate (6.3) for the case τ = 2. The proof of Theorem 5.8 is thus completed.

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