Boundedness of a Class of Oscillatory Singular Integral Operators and Their Commutators with Rough Kernel on Weighted Central Morrey Spaces

: In this paper, we establish the boundedness of a class of oscillatory singular integral operators with rough kernel on central Morrey spaces. Moreover, the boundedness for each of their commutators on weighted central Morrey spaces was also obtained. We generalized some existing results.


Introduction
Morrey spaces play an significant role in harmonic analysis and partial differential equations. After Morrey [1] introduced that to investigate the local behavior of solutions of the second-order elliptic partial differential equations in 1938, Morrey space was studied by quite a few authors. For example, in the study of the boundedness of Riesz potential on the Morrey space, the first remarkable result was due to Spanne [2] and the second milestone result was due to Adams [3]. Later, the boundedness of Hardy-Littlwood maximal function and singular integral operators on Morrey spaces was obtained by Chiarenza and Frasca [4] in 1987. In addition, for more boundedness of other operators on Morrey space, see [5][6][7] et al. Now we review some definitions of Morrey type spaces. Given 1 ≤ p < ∞ and 0 ≤ λ < n, the Morrey space M p,λ (R n ) is defined as the set of real-valued measurable function f on R n enjoying where B(x 0 , r) is the Euclidean ball with center x 0 ∈ R n and radius r ∈ (0, ∞), |B(x 0 , r)| ≈ r n . By slightly modifying the definition, we can get some generalizations of Morrey space, such as is a positive measurable function on R n × (0, ∞), we obtain the generalized Morrey space M p,ϕ and refer to [8,9] for the known results of M p,ϕ for some suitable ϕ.
In this paper, we focus on local versions of Morrey spaces. More precisely, denoting the ball with center at origin and radius r by B = B(0, r) = {y ∈ R n : |y| < r}, the central Morrey space M p,λ (R n ) is defined as the set of function f with finite norm f M p,λ (R n ) , where f M p,λ (R n ) := sup Before giving the main results of this paper, let us state some existing results. Denote the unit sphere in R n by S n−1 . Suppose that Ω is a homogeneous function with degree zero, has mean value zero on S n−1 and belongs to L q (S n−1 )(1 < q ≤ ∞); then the Calderón-Zygmund singular integral with a rough kernel Ω is defined by It is worth pointing out that the kernel in the above formula is a convolution kernel. However, there are many kinds of operators which have non-convolution kernels, such as the Fourier transform and Radon transform [11], which belong to oscillatory integrals. Due to F. Ricci and E. M. Stein, the following form of oscillatory singular integrals was studied in [12]: where P(x, y) is a real valued polynomial defined as R n × R n , and K(x) is a standard Calderón-Zygmund kernel. That means K satisfies: (a) K(x) is a C 1 -continuous away from the origin; |x| n with Ω homogeneous of degree 0 on S n−1 ; (c) S n−1 Ω(x )dσ(x ) = 0. The following theorem is the main result in [12], which was the first result concerning the boundedness of oscillatory sigular integral operator with polynomial phase. Theorem 1. If K(x) satisfies (a)-(c), then the operator T K can be extended to be a bounded operator on L p (R n ) to itself, with 1 < p < +∞, and the norm of this operator depends only on the total degree of P(x, y), but not on the coefficients of P(x, y).
By examining Condition (a), Lu and Zhang [13] introduced new conditions: (d) Ω(x ) ∈ L q (S n−1 ), 1 < q ≤ ∞ for some q. With those mild conditions, they considered a more general oscillatory singular integral operator with rough kernel T which is defined by where P(x, y) is a real valued polynomial defined as R n × R n and Ω is a rough kernel to be specified later, and promoted Theorem 1 to the following theorem.

Theorem 2.
If Ω satisfies (b)-(d), then the operator T can be extended to be a bounded operator on L p (R n ) to itself, with 1 < p < +∞, and the norm of this operator depends only on the total degree of P(x, y), but not on the coefficients of P(x, y).
In addition, Jiang and Lu [14] gave the weighted form of Theorem 2.
It should be mentioned that Yu et al. [15] studied the boundedness of singular integral operators and their commutators with rough kernel on weighted central Morrey spaces. For more about the boundedness of operators with rough kernels, readers can refer to [16][17][18][19][20][21].
For a function b ∈ L loc (R n ), let A be a linear operator on some measurable function space. The commutator between A and b is defined by where T is as in (1). Commutators of oscillatory singular operators have been considered by many authors. The boundedness and weighted boundedness for commutators of oscillatory singular operators were studied by Chen and Zhu [22] and Ding and Lu [23] respectively, with Ω being a rough kernel satisfying some conditions.
On the basis of these works, we studied the boundedness of oscillatory singular integral operators with rough kernel and their commutator on center Morrey spaces and weighted center Morrey spaces, respectively. Now we are in a position to state our main results.
Here and in what follows, the operator T and its commutators [b, T] are always defined by (1) and (2), respectively.

Remark 1.
In the above theorems, the well-definedness of the operators or commutators on weighted Morrey spaces must be checked before the proof. Fortunately, this can proceed in a very similar manner as Remark 7.2 in [24], so we omit the details.

Remark 2.
In fact, our theorems says that the (weighted) L p boundedness of T and its commutators [b, T] imply the boundedness on (weighted) Morrey spaces. As a consequence, we can deduce the (weighted) boundedness of T on Morrey spaces if the conditions of Theorem 1, Theorem 2 or Theorem 3 are satisfied.  [15,25] in some sense.

Remark 4.
In [26], V. S. Guliyev outlines a very similar results, which are the boundedness of homogeneous singular integrals with rough kernel on the local generalized Morrey spaces LM {x 0 } p,ϕ for s ≤ p or p < s and the commutator operators formed by a local BMO function b and these rough operators on the local generalized Morrey spaces LM {x 0 } p,ϕ . Essentially, Theorems 4 and 6 of our paper are consistent with the main results in [26] in terms of the line of proof, while our arguments are more direct.
Throughout this paper, for a real number p > 1, we denote p by 1/p + 1/p = 1. The letter C appearing in this paper is a constant which is independent of the main variables, but may vary from line to line. Denote tB by B(0, tr) for a real number t > 0. Use symbol T b for the commutator [b, T].

Preliminaries
Here are some definitions and lemmas needed in the proof of our main results. In the study of weighted inequalities, an interesting type of weight function is A p weight, which characterizes the weighted L p boundedness of many operators in harmonic analysis, such as the Riesz transforms. Below are the definitions of A p (1 < p < ∞) weight and A ∞ weight, and the relationship between them.

Definition 1. We say a non-negative function ω(x) belongs to the Muckenhoupt class
[ω] A p denotes the infimum of C.

Definition 2.
We say ω ∈ A ∞ if there exist two constant C and δ > 0 such that for any measurable set On the one hand, the study of BMO function space has its own value. For example, its duality with Hardy space H 1 has crucial theoretical worth; on the other hand, as a symbolic function for forming commutators, the BMO function also plays an important role in the study of the commutator boundedness. The following is the definition and some properties of the (weighted) BMO function.
. Then for any ball B ⊂ R n ,

Definition 4. A locally integrable function b is said to be in BMO(ω) if for any ball
. Then there exists a constant C such that For the sake of conciseness, we will only prove Theorems 6 and 7, since Theorems 4 and 5 are simpler.

Proofs of Main Results
Compared with the argument in [26], where the author uses the relevant conclusions of the weighted Hardy operator, our method is quite straightforward. With the help of discrete ring decomposition, Holder's inequalities, the relevant properties of the BMO function, some properties of A p weight, etc., we can proof the desired result.
Proof of Theorem 6. As usual, we decompose f as Then By the L p boundedness of T b , we have Note that when x ∈ B and y ∈ (2B) c , We have the following estimates We consider T b f 2 (x) in two different cases, i.e. Ω ∈ L ∞ (S n−1 ) and Ω ∈ L q (S n−1 )(1 < q < ∞). Case I. Ω ∈ L ∞ (S n−1 ). Using Holder's inequalities, we have For the term T b f 22 (x), we have the following estimates Case II. Ω ∈ L q (S n−1 )(1 < q < ∞). By applying Holder's inequalities, we have which yields We obtain that from which it turns out that For both cases, we arrive at the same estimates We proceed to estimate the L p norm of T b f 21 and T b f 22 by using (3) and (4). From the definition of BMO, Lemmas 2 and 3, we deduce that and By the estimates (5) and (6), we have the following estimates Obviously, there holds which yields Combining the estimate of T b f 1 L p (B) and (8), we get So we complete the proof.
Proof of Theorem 7. The proof of Theorem 7 is a little similar to that of Theorem 6, but there are some differences because of the appearance of weights.
We also decompose f by Then By the weighted L p boundedness of T b , we have Through the similar discussion as in Theorem 6, we obtain We now divide our discussion into two cases. Case I. Ω ∈ L ∞ (S n−1 ).
For the term T b f 23 (x), we have the following pointwise estimates: To get a precise pointwise estimate for T b f 24 (x), we need the following observation by using Lemma 4 where c = (s p ) , s > 1 such that c ≥ p/q . By using (10), we now estimate T b f 24 (x) as By using (11), we can get a pointwise estimates for T b f 23 (x), From the definition of A p weights and Lemma 4, we have where m is any number greater than 1. From this Taking (13) into consideration, we obtain (2 j r) n Ω L q (S n−1 ) (2 j+2 r) n q 2 j+1 B |b(y) − b 2 j+1 B,ω | q | f (y)| q dy For both cases, we arrive at the same estimates In the following, we will proceed to estimate the L p norm of T b f 23 and T b f 24 by using (14) and (15).