The Boundedness of Commutators of Singular Integral Operators with Besov Functions

In this paper, we prove the boundedness of commutator generated by singular integral operator and Besov function from some L to Triebel-Lizorkin spaces.


Introduction
Let K be a Calderón-Zygmund singular integral kernel being of the form where Ω ∈ C ∞ (Σ n−1 ) is homogeneous of degree 0 and The Calderón-Zygmund singular integral operator T is defined by The fractional integral operator I α of order α , 0 < α < n, is the convolution operator with the kernel ω α |x| α−n .In [3], a well known result of Coifman, Rochberg and Weiss states that the commutator of singular integral operator is bounded on some L p , 1 < p < ∞, if and only if b ∈ BMO .In 1982, Chanillo [1] introduced the commutator of fractional integral operator [b, I α ] : and proved a similar result when T is replaced by the fractional integral operator.In [5], Janson investigated the commutator of singular integral operator and fractional integral operator with Lipschitz functions.In 1995, Paluszynski [6] proved that [b, T ] is bounded from L p to F β,∞ p , the Triebel-Lizorkin space, if and only if b ∈ ∧β , where ∧β is the homogeneous Lipschitz space which is the set of functions satisfing where Δ k h denotes the k− th difference operator (see [6]).A generalization of Lipschitz sapce is Besov space ∧p,q β (R n ) defined by the set of all functions satisfying For the properties of Besov functions we refer [9].In 2006, Y. Chen and B. Ma in [2] introduced the commutator of singular integral operators and Besov functions and proved that [b, T ] k is bounded from some L d to certain L r .In this paper, we investigate the boundedness of commutators generated by singular integral operators and Besov functions from Lebesgue integrable spaces to Triebel-Lizorkin spaces, as well as commutators of fractional integral operators.We simply denote T b by [b, T ] and I α b by [b, I α ].Our results are stated below.
From the results stated above it can be seen that if p, q → ∞ then all the theorems are the ones obtained by Paluszynski in [6].The proofs of our theorems are based on an estimate similar to the mean oscillation of a Besov function over cubes.We shall provide it as a lemma in the next section and then prove our theorems.

Some Lemmas and Proofs of Results
In this section, we get an estimate of the oscillation of Besov function b on cubes.Using the property of Besov function, we discuss the boundedness of the commutator.In the following, let Q be a cube and denote by f Q the mean of f over Q , i.e., Proof.Fix a cube Q, and set ∧ = {t = y − z : y, z ∈ Q} .So for any t ∈ ∧, we have |t| ≤ C|Q| 1/n .Using Hölder's inequality , we first estimate Since p/q > 1 , using Hölder's inequality on z , we obtain Therefore we obtain the second inequality in (2.1).
With regard to the first inequality in (2.1), using the second one, together with Hölder inequality, we have Thus, the part (a) of Lemma 2.2 is proved.Now we consider the part (b).Clearly we have Also by Hölder's inequality, we obtain Since q/p > 1 , using Hölder's inequality, we have Thus the Lemma 2.2 is proved completely.
We now proceed with proofs of our theorems.
Proof of the Theorem 1.1: Fix a cube Q = Q(x Q , s), centered at x Q ,with the side length s, and x ∈ Q.For f ∈ L p , decompose f as We estimate these terms respectively.
To the term I , for x ∈ Q , by Hölder's inequality and the part (a) in Lemma 2.2, we have To get II, choosing 1 < r < q such that rq/(q − r) > 1, we firstly estimate (b Take r such that 1/r+1/r = 1 .By the boundedness of Calderon-Zygmund Singular integral operator and the estimate of (b Now we estimate III .Before we start the proof, we recall the smoothness condition of Calderon-Zygmund kernel, that is, Then we have It is easy to see that Thus by Hölder's inequality, (2.1) and (2.7), the last inequality in (2.6) is dominated by where we use the assumption 0 < β − n/p < 1 .Thus we have Putting these estimates (2.3), (2.5) and (2.8) together, we obtain Since 0 < β − n/p < 1, d > q/(q − 1) = q , we can choose 1 < r and closely to 1 such that d > rq/(q − r) > q/(q − 1).Taking the supremum over all Q such that x ∈ Q , and L d norm on both sides, and using Lemma 2.1, we conclude that Thus, the proof of Theorem 1.1 is completed.
The proof of Theorem 1.2 is similar to that of Theorem 1.1.
The last inequality follows from the following lemma, with β − n/p in the place of γ.
Suppose that for each cube Q we have a function h Q , defined on this cube.Then, for γ ≥ 0 , where the constant C depends only on d, r, α and n.Lemma 2.3, as in [8], specifies that the function h Q have the form h−h Q , but the proof carries over immediately to the more general case above.Thus, Using Lemma 2.2 (a), we can estimate I as follows Since r > q , we have We now estimate II.Choose 1 < h < q, and h such that then Since d > q/(q − 1), we can choose 1 < h such that d > qh/(q − h) > q/(q − 1), and obtain Now we estimate III.Set y ∈ Q .By (2.7) and Lemma 2.2, we have where we first assume that α + β − n/p < 1.Thus, Putting (2.9), (2.10) and (2.11) together, Theorem 1.3 is proved under the assumption α + β − n/p < 1 .Following the facts 1.9 and 1.10 in [6] on pages 10 and 11, we can also get the result when 0 < α < n.
The proof of the Theorem 1.4 is similar to that of Theorem 1.3, and we omit the details.