CONTINUITY PROPERTIES FOR THE MAXIMAL OPERATOR ASSOCIATED WITH THE COMMUTATOR OF THE BOCHNER-RIESZ OPERATOR

In this paper, we obtain some strong and weak type continuity properties for the maximal operator associated with the commutator of the Bochner-Riesz operator on Hardy spaces, Hardy type spaces and weak Hardy type spaces.


Introduction
Let b ∈ BMO(R n ) and T be a standard Calderón-Zygmund singular integral operator, the commutator [b, T ] is defined by Many authors have investigated the properties for [b, T ].A celebrated result of Coifman, Rochberg and Weiss [5] states that the commutator [b, T ] is bounded on L p (1 < p < ∞).Subsequently, Coifman and Meyer [4] observed that the weighted L p (1 < p < ∞) boundedness for [b, T ] can be obtained by the weighted L p estimate with Muckenhoupt A p weight for T .Later, Álvarez, Bagby, Kurtz and Pérez [2] extended the idea of Coifman and Meyer and proved the following result: For a general linear operator T , if 1 < p, q < ∞ and T is bounded on L p (w) for all w ∈ A q , then [b, T ] is bounded on L p (u) for all u ∈ A q .In the case of p = 1, it is a well-known fact that Calderón-Zygmund singular integral operator T is a weak type (1,1) operator and a bounded operator from the standard Hardy space H 1 to L 1 .Fairly recently, Pérez [13] observed the fact that [b, T ] is neither a weak type (1,1) operator nor a bounded operator from H 1 to L 1 .He obtained a weak type L log L inequality and the boundedness from a certain modified Hardy space H 1 b to L 1 for [b, T ].In this paper, we will consider the commutator of Bochner-Riesz operator, let λ and r be two positive numbers, the Bochner-Riesz operator T r λ in R n (n ≥ 2) is defined in terms of Fourier transforms by where f denotes the Fourier transforms of f .It can be written as a convolution operator where B r λ (x) is the kernel of T r λ and B r λ (x) = r −n B λ ( x r ), it is well-known that B λ (x) satisfies the following inequality: 2 ) , for any x ∈ R n and r > 0 and any multi-index β ∈ Z n + .Let b ∈ BMO(R n ), the commutator generated by b and T r λ is defined by The maximal operator associated with T r λ,b is defined by If λ ≥ n−1 2 , Shi and Sun [14] showed that the maximal Bochner-Riesz operator, Combining the above result due to Álvarez, Bagby, Kurtz and Pérez with the result due to Shi and Sun [14], we can easily observe that T * λ,b is bounded on L p (R n ).Hu and Lu [7] further discussed the L p boundedness for We notice the fact that (n − 1) , where 0 < p ≤ 1.However, we do not know whether the operator T * λ,b satisfies the weak type L log L inequality.After we submitted the paper, we learned that Jiang, Tang and Yang [9] proved, independently of us, the similar results in H p b and H p,∞ b when n n+1 < p ≤ 1.Our range in this paper allows to have all 0 < p < 1.
Now, let us recall some notations and definitions.Most of the notations we use are standard.Q denotes a cube with sides parallel to the axes and λQ (λ > 0) denotes the cube Q dilated by λ.For a lo- , where [x] denotes the integer part of x.A tempered distribution f is said to belong to the Hardy type space where a j are (p, b, ∞) atoms and As usual, we define on Let b be a locally integrable function.We say that a tempered distribution f belongs to the weak Hardy type space (2) Each function f k can be decomposed as where the functions b k j satisfy the following properties: We define on the space H p,∞ b (R n ) the following quasinorm For brevity, we will sometimes denote

Weak type (H 1 , L 1 ) estimate
In this section, we establish the weak type (H 1 , L 1 ) estimate for T * λ,b , where H 1 (R n ) is a well-known standard Hardy space.Our main result is the following theorem.
for any α > 0 and any f ∈ H 1 (R n ).
Remark.After the paper is accepted for publication, it has been proved using similar method that T * λ,b is bounded from H 1 to L 1 (see [11]).However, we still do not know if T * λ,b is weak type (1, 1).
To prove our Theorem 1, we first recall the following lemma due to M. Christ [3].Lemma 1.For any α > 0 and any finite collection of dyadic cubes Q and associated positive scalars λ Q , there exists a collection of pairwise disjoint dyadic cubes S such that (2)

Now we begin to prove Theorem 1.
It is easy to see that the result of Theorem 1 follows from the inequality where C is independent of r, f and α.
For any given f ∈ H 1 (R n ), we have the well-known atomic decompo- We may assume that f is a finite sum Once Theorem 1 is proved for such f .For general f , it is the limit of this kind of f k (in H 1 norm or almost everywhere sense) where f k are finite sums having forms of Q λ Q a Q , and then Theorem 1 follows by a limiting argument.It is convenient for us to assume that each Q (the supporting cube of a Q ) in the given atomic decomposition of f is dyadic and λ Q > 0.
For fixed α > 0 and the finite collection of dyadic cube Q and associated positive scalars λ Q > 0 in the given atomic decomposition of f , by Lemma 1, there exists a collection of pairwise disjoint dyadic cube S such that (1) (2) Thus, we only need to prove the following inequality For fixed cube Similarly we can get These above estimates imply the following inequality By the mean value theorem and the boundedness of B r λ , we have This implies the following estimate We notice the fact that , where M is the well-known Hardy-Littlewood maximal operator.Thus T * λ is of weak type (1, 1) and T r λ is weak type (1, 1) uniformly associated with r.We get This finishes the proof of Theorem 1.

(H p b , L p ) type estimate
In this section we will establish the (H p b , L p ) type estmate for T * λ,b .Our main result is the following theorem.
Proof: By the definition of space H p b (R n ), we only need to prove the following inequality for any (p, b, ∞) atom a Q , where C is a constant independent on a Q and r.
We easily get the following decomposition By the and the Hölder inequality, we get Since 0 < p ≤ 1, we have First, we estimate each J 1j .
This implies that Thus we have in the last inequality, we use the fact that λ > n p − n+1 2 implies that λ − n−1 2 ≥ 0 for 0 < p ≤ 1.Since 0 < p ≤ 1, it is easy to see that there exists a nonnegative integer m such that n n+m+1 < p ≤ n n+m .In this case, we notice the fact that , by the vanishing moments of a Q and (m + 1)-order Taylor expansion of B r λ (x − y) at x − x Q , we have This implies that 2 , Also by the vanishing moments of a Q and (m + 1)-order Taylor expansion of B r λ (x − y) at x − x Q , we have Because of these above estimates and the condition that Now we start to estimate J 2j .
When 0 < r < r Q , we also get and By the vanishing moments of a Q and Taylor formula, we also have and When r ≥ r Q and λ ≥ m + n+1 2 , we also get This implies that Because of above estimates and the condition This completes the proof of Theorem 2.

(H p,∞ b , L p,∞ ) type estimate
In this section, we obtain the (H p,∞ b , L p,∞ ) type estimate for T * λ,b , where L p,∞ is a well-known weak L p space.
Before proving Theorem 3, we need to recall the so-called superposition principle on the weak-type estimates.

Lemma 2. Let 0 < p < 1, if a series of measurable functions {f
Now we begin to prove Theorem 3.
b k j be an atomic decomposition for f as Definition 2.
Let us first observe that This implies that Thus we have Now let A k Q k j be the cube with the same center as Q k j and sides A k times longer, where A k being a positive number to be chosen later and depending on k and p. Denote and Q k j be a cube with center x k j and side-length r k j , we write Let us first estimate K 2 .If c is any complex number, we can write Similar to proof of Theorem 2, there exists a nonnegative integer m such that n n+m+1 < p ≤ n n+m for fixed 0 < p ≤ 1.Now we estimate K 21 and if r ≥ r k j , by the vanishing moments of b k j and the (m + 1)-order Taylor formula of B r λ (x − y) at x − x Q , we have These imply that 2 , if 0 < r < r k j for a fixed k and j, we also have following estimate, and if r ≥ r k,j for a fixed k and j, Thus we can get Because , and By taking the infimum with c ∈ C in the right-hand side of above inequality, we obtain that For fix (k, j) ∈ Γ 1 , we where , by Lemma 2, we obtain we have the following estimate by the vanishing moments of b k j and Taylor formula, Similar to the estimate of M 1 , we get For (k, j) ∈ Γ 2 , if λ ≥ m + n+1 2 , we also obtain , where C is dependent on r, N and the atomic decomposition of f .
Finally, taking the limit as N → ∞, the infimum of C 1 over all possible representations k j b k j = f and the supremum for r > 0 in the left-hand side of above inequality, we complete the proof of Theorem 3.

Definition 1 .
dy.Sometimes a Q denotes an atom in certain Hardy spaces with compact support included in cube Q.For b ∈ BMO(R n ), b * denotes the norm of b on BMO(R n ).Let 0 < p ≤ 1 and b be a locally integrable function.Given a bounded function a, we say that a is a

Fix N = 1 , 2 ,
. . ., and consider N k=−N f k .By a usual limiting argument it is enough to prove that there exists C = C(b, λ) > 0 such that sup α>0