Weighted Boundedness of Toeplitz Operator Related to Singular Integral Operator

Let b be a locally integrable function on Rⁿ and T be an integral operator. For a suitable function f , the commutator generated by b and T is defined by [b,T] f = bT ( f ) −T (bf ). The investigation of the commutator begins with Coifman-Rochberg- Weiss pioneering study and classical result (see [3]). There are two major reasons for considering the problem of commutators. The first one is that the boundedness of commutator can produces some characterizations of function spaces (see [3,9,23]). The other one is that the theory of commutator plays an important role in the study of the regularity of solutions to elliptic and parabolic partial differential equations (PDEs) of the second order (see [5,22]). The well-posedness problem of solutions to many PDEs can be attributed to the corresponding boundedness of commutators of integral operators. Now, with the development of singular integral operators (see [7,27]), their commutators have been well studied. In [3,25,26], the authors prove that the commutators generated by the singular integral operators and BMO functions are bounded on L^p (Rⁿ ) for 1< p < ∞. In [9,23], the boundedness for the commutators generated by the singular integral operators and Lipschitz functions on L^p (Rⁿ )(1< p < ∞) and Triebel-Lizorkin spaces are obtained. In [1,8], the boundedness for the commutators generated by the singular integral operators and the weighted BMO and Lipschitz functions on L^p (Rⁿ )(1< p < ∞) spaces are obtained. In [11,12,17,18], some Toeplitz type operators related to the singular integral operators and strongly singular integral operators are introduced, and the boundedness for the operators generated by BMO and Lipschitz functions are obtained. In [2], Calder´on and Zygmund introduce certain singular integral operator with variable kernel and discuss its boundedness. In [14- 16,28], the authors obtain the boundedness for the commutator generated by the singular integral operator with variable kernel and BMO function. In [19], the authors prove the boundedness for the multilinear oscillatory singular integral operator generated by the operator and BMO function.

On the other hand, the classical Morrey space was introduced by Morrey in [21] to investigate the local behaviour of solutions to second order elliptic partial differential equations (also see [22]). As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of operator on the Morrey spaces. The boundedness of the maximal operator, the singular integral operator, the fractional integral operator and their commutators on Morrey spaces have been studied by many authors (see [4,5,13,20]). In [10], Komori and Shirai studied the boundedness of these operators on weighted Morrey spaces.

Motivated by these, in this paper, we will study the Toeplitz type operator generated by the singular integral operator with variable kernel and the weighted Lipschitz and BMO functions on weighted Morrey spaces.

First, let us introduce some notations. Throughout this paper, Q = Q(x, r ) will denote a cube of Rn with sides parallel to the axes and centered at x and edge length is r, and 2^kQ denote a cube with same center as Q and edge length is 2^kr for k ≥ 0. For any locally integrable function f , the sharp maximal function of f is defined by

where, and in what follows, It is well-known that (see [7,27]).

The Ap weight is defined by (see [7]), for 1< p < ∞,

For 0 < β <1 and the non-negative weight function w, the weighted Lipschitz space Lipβ (w) is the space of functions b such that

and the weighted BMO space BMO(w) is the space of functions b such that

Remark. (1) It has been known that (see [6]), for b∈Lipβ(w), w∈ A₁ and x∈Q,

(2) It has been known that (see [6]), for b∈BMO(w), w∈ A₁ and x∈Q,

(3) Let b∈Lipβ(w)or b∈BMO(w) and w∈ A₁. By [6,7], we know that spaces Lipβ (w) or BMO(w) coincide and the norms are equivalent with respect to different values 1≤ p < ∞.

Definition 1. Let 1≤ p < ∞, 0 < k <1, u and v be two non-negative weight functions on Rⁿ and f be a locally integrable function on .Rⁿ Set

The generalized weighted Morrey space , is defined by

Remark. (4) We write L^p,k (u,v) = L^p,k (u) if u = v. As the Morrey space may be considered as an extension of the Lebesgue space, it is natural and important to study the boundedness of the operator on the Morrey spaces (see [3,24]). In this paper, we will study certain singular integral operator with variable Calder´on- Zygmund kernel as following (see [2]).

Definition 2. Denote a multi-indices by α = (α₁,....,α_n) with α_j is a nonnegative integer for 1≤ j ≤ n and Let is said to be a Calder´on-Zygmund kernel if
a) ( {0}); Ω∈C^∞ Rⁿ b) Ω is homogeneous of degree zero; c) for all multi-indices with α = N, where Σ = {x∈Rⁿ : x =1} is the unit sphere of Rn .

Definition 3. Let is said to be a variable Calder´on-Zygmund kernel if

(d) K (x,.) is a Calder´on-Zygmund kernel for a.e. x∈Rⁿ ;

Moreover, let b be a locally integrable function on Rⁿ and T be the singular integral operator with variable Calder´on-Zygmund kernel as

variable Calder´on-Zygmund kernel.

The Toeplitz type operator associated to T is defined by

where T^j,1 are the singular integral operator with variable Calder´on-Zygmund kernel T or ±I (the identity operator), T^j,2 are the linear operators,

Remark. (5) Note that the commutator [b,T ]( f ) = bT ( f ) −T (bf ) is a particular operator of the

Toeplitz type operator Tb. The Toeplitz type operator Tb is the non-trivial generalizations of the commutator. It is well known that commutators are of great interest in harmonic analysis and have been widely studied by many authors (see [25,26]). The main purpose of this paper is to prove the sharp maximal inequalities for the Toeplitz type operator Tb. As the application, we obtain the weighted boundedness on Morrey space for the Toeplitz type operator Tb.

We shall prove the following theorems.

Theorem 1. Let T be the singular integral operator as Definition 3, w∈ A₁, 0 <η <1, 1< s < ∞, 0 < β <1 and b∈Lipβ (w). If T¹ (g ) = 0 for any g ∈L^r (Rⁿ )(1< r < ∞), then there exists a constant C > 0 such that, for any f ∈C^∞₀(Rⁿ) and x∈Rⁿ,

Theorem 2. Let T be the singular integral operator as Definition 3, w∈ A₁ , 0 <η <1, 1< s < ∞ and b∈BMO(w). If T¹(g ) = 0 for any g ∈L^r (Rⁿ)(1< r < ∞), then there exists a constant C > 0 such that, for any f ∈C^∞₀(Rⁿ) and x∈Rⁿ,

Theorem 3. Let T be the singular integral operator as Definition 3, w∈ A₁, 0 < β <1, b∈Lipβ (w), 1< p < n β , 1 q =1 p −β n and 0 < k < p q. If T¹ (g ) = 0 for any g ∈L^r (Rⁿ )(1< r < ∞) and T^j,2 are bounded on Lp,k (w) for 1≤ j ≤ m, then T^b is bounded from

Theorem 4. Let T be the singular integral operator as Definition 3, w∈ A₁, 1< p < ∞, b∈BMO(w) and 0 < k <1. If T¹ (g ) = 0 for any g ∈L^r (Rⁿ )(1< r < ∞) and T^j,2 are bounded on Lp,k (w) for 1≤ j ≤ m, then T^b is bounded from L^p,k (w) to Lp

Corollary. Let [b,T ]( f ) = bT ( f ) −T (bf ) be the commutator generated by the singular integral operator T as Definition 3 and b. Then Theorems 1-4 hold for [b, T].

To prove the theorems, we need the following lemmas.

Lemma 1. (see [7]) Let 0 < p < q < ∞ and for any function f ≥ 0. We define that, for 1 r =1 p −1 q,

where the sup is taken for all measurable sets Q with 0 < Q < ∞. Then

Lemma 2. (see [2,19]) Let T be the singular integral operator as Definition 3 and 1 < p < ∞. Then T is bounded on L^p (Rⁿ ,w) for w∈ A _p with 1 < p < ∞, and weak (L¹, L² ) bounded.

Lemma 3. Let 1 , 0 , 0 1 p k η < < ∞ < < ∞ < < and , . u v A∞ ∈ Then, for any smooth function f for which the left-hand side is finite,

Proof. By [7,27], we have the following weighted version of the local good inequality, for all cube Q and λ,δ > 0,

Thus, the conclusion is obtained by using the standard argument of Whitney decomposition theorem (see [7,27]). This finishes the proof.

Lemma 4. Let 0 < k <1, 1≤ s < p < ∞ and w A . ∞ ∈ Then

Proof. By [10], we know

This completes the proof.

Proof. By using a similar argument as in the proof of [10, Theorem 3.5], we get

This completes the proof.

Proof of Theorem 1. It suffices to prove for f ∈C^∞₀(Rⁿ) and some constant C₀, the following inequality holds:

Then, for C₀= I₂(x₀) ,

For I₁ , by the weak (L¹, L² ) boundedness of T^j,1 (see Lemma 2) and Kolmogoro’s inequality (see Lemma 1), we obtain

For I₂, by [2,19], we know that

thus

These complete the proof of Theorem 1.

Proof of Theorem 2. It suffices to prove for f ∈C^∞₀(Rⁿ) and some constant C₀, the following inequality holds:

Without loss of generality, we may assume T^j,1 are T ( j =1,...,m). Fix a cube Q = Q (x₀,d) and x∈Q. Similar to the proof of Theorem 1, we have

By using the same argument as in the proof of Theorem 1, we get

This completes the proof of Theorem 2.

Proof of Theorem 3. Choose 1 < s < p in Theorem 1, we have, by Lemmas 3 and 5,

This completes the proof.

Proof of Theorem 4. Choose 1 < s < p in Theorem 2, we have, by Lemmas 3 and 4,

This completes the proof.

Acknowledgment

The authors would like to express their deep gratitude to the referee for his/her valuable comments and suggestions.