Rough singular integrals associated to polynomial curves

In this paper, the authors establish the boundedness of singular integral operators associated to polynomial curves as well as the related maximal operators with rough kernels ∈ H1(Sn–1) and h ∈ γ (R+) for some γ > 1 on the Triebel–Lizorkin spaces. It should be pointed out that the bounds are independent of the coefficients of the polynomials in the definition of the operators. The main results of this paper not only improve and generalize essentially some known results but also complement some recent boundedness results.


Introduction
It is well known that the Triebel-Lizorkin spaces contain many important function spaces such as Lebesgue spaces, Hardy spaces, Sobolev spaces, and Lipschitz spaces. Over the last several years, a considerable amount of attention has been given to investigate the boundedness for singular integral operators with various rough kernels on the Triebel-Lizorkin spaces. Particularly, many scholars devoted to studying the bounds for singular integral operators with singularity along various sets under the rough kernels ∈ H 1 (S n-1 ) and h ∈ γ (R + ) for some γ > 1. For example, see [10] for the polynomial mappings, [29] for the homogeneous mappings, [27] for the surfaces to revolution. It is unknown whether the singular integral operators associated to polynomial curves under the rough kernels are bounded on the Triebel-Lizorkin spaces. The main purpose of this paper is to address the question. In addition, we establish the bounds for the related maximal singular integral operators on the Lebesgue and Triebel-Lizorkin spaces.
Before stating our main results, let us recall some pertinent definitions, notations, and backgrounds. Let n ≥ 2 be an integer and let S n-1 denote the unit sphere in R n equipped with the normalized Lebesgue measure dσ . Let ∈ L 1 (S n-1 ) be a homogeneous function of degree zero on R n and satisfy S n-1 (u) dσ (u) = 0. (1.1)

The singular integral operator T h, is defined as
T h, f (x) := p.v.
The theory of singular integral originated in Calderón and Zygmund's work [4] in which they used the rotation method to establish the L p (R n )(1 < p < ∞) of T if ∈ L log L(S n-1 ). Since then, more and more scholars have been devoted to studying the boundedness of singular integrals with various rough kernels. Particularly, Coifman and Weiss [12] proved that T is of type (p, p) for 1 < p < ∞ if ∈ H 1 (S n-1 ) (see also [15]). It was remarkable that ∈ H 1 (S n-1 ) turned out to be the weakest size condition for the L p boundedness of T up to now. Later on, an active extension to the theory was due to Fefferman [23] who discovered that the Calderón-Zygmund rotation method is no longer available if T h, is also rough in the radial direction, for instance h ∈ L ∞ (R + ), so that new methods must be addressed. More precisely, Fefferman [23] showed that T h, is of type (p, p) for 1 < p < ∞ if ∈ Lip α (S n-1 ) for some α > 0 and h ∈ L ∞ (R + ). Fefferman's result was later improved by Namazi [32] by assuming ∈ L q (S n-1 ) for some q > 1 instead of ∈ Lip α (S n-1 ). Meanwhile, Duoandikoetxea and Rubio de Francia [16] used the Littlewood-Paley theory to improve the results to the case ∈ L q (S n-1 ) for any q > 1 and h ∈ 2 (R + ). The boundedness for rough singular integral operators on Tribel-Lizorkin spaces has also been studied extensively by many authors. In 2002, Chen, Fan, and Ying [5] first showed that T is bounded onḞ p,q α (R n ) if ∈ L r (S n-1 ) for some r > 1. Later on, the result was extended and improved by many authors. For example, see [2,6] for the case ∈ F β (S n-1 ) (the Grafakos-Stefanov function class in [25]), [9,10] for the case ∈ H 1 (S n-1 ).
For the operators T and T h, , the singularities are along the diagonal {x = y}. However, many problems in analysis have led one to consider singular integral operators with singularity along more general sets. One of the principal motivations for the study of such operators is the requirements of several complex variables and large classes of "subelliptic" equations (see [37,39]). So more and more scholars are devoted to studying the L p bounds for rough singular integral operators with singularity along various sets. For example, see [3,22,34] for polynomial mappings, [17,19] for real-analytic submanifolds, [11,28] for homogeneous mappings, [1,18,20,26] for polynomial curves. Other interesting works can be found in [7,8,35,36,42], among others.
In this paper we focus on the singular integrals associated to polynomial curves with rough kernels. Let h, be given as in (1.2) and P be a real polynomial on R satisfying P(0) = 0. For a function ϕ : R + → R, we define the singular integral operator associated to polynomial compound curves {P(ϕ(|y|))y/|y|; y ∈ R n } by where f ∈ S(R n ). When ϕ(t) ≡ t, we denote T h, ,P,ϕ = T h, ,P . Particularly, T h, ,P = T h, when P(t) ≡ t. In 1997, Fan and Pan [20] first established the L 2 boundedness for T h, ,P if h ∈ L ∞ (R + ) and ∈ H 1 (S n-1 ). Subsequently, Al-Hasan and Pan [1] improved the result by establishing the following.
and ∈ H 1 (S n-1 ) satisfy (1.1). Then, for 1 < p < ∞, there exists a constant C > 0 independent of h, and the coefficients of P such that Later on, the L p mapping properties for T h, ,P have been investigated by many authors. For example, see [18] for the case h ≡ 1 and ∈ F β (S n-1 ), [26] for the case ∈ L log L(S n-1 ).
Based on (2.4) and Theorem A, a natural question is the following.
Our investigation will not only address this question, but also deal with a more general class of operators. More specifically, we have the following result. Theorem 1.1 Let P be a real polynomial on R satisfying P(0) = 0 and ϕ ∈ F 1 or F 2 . Here, F 1 (resp., F 2 ) is the set of all functions φ : R + → R satisfying the following condition (a) (resp., (b)): (a) φ is an increasing Suppose that ∈ H 1 (S n-1 ) satisfies (1.1) and h ∈ γ (R + ) for some γ ∈ (1, ∞]. Then (i) For α ∈ R and (1/p, 1/q) ∈ R γ , there exists a constant C > 0 independent of h, γ , and the coefficients of P such that Here, R γ is the interior of the convex hull of three squares ( 1 For α > 0 and (1/p, 1/q) ∈ R γ , there exists a constant C > 0 independent of h, γ , and the coefficients of P such that Remark 1.1 There are some model examples in the class F 1 such as t α (α > 0), t α (ln(1 + t)) β (α, β > 0), t ln ln(e + t), real-valued polynomials P on R with positive coefficients and P(0) = 0, and so on. We now give examples in the class F 2 such as t δ (δ < 0) and t -1 ln(1 + 1/t). It was pointed out in [26] that for ϕ ∈ F 1 (or F 2 ) there exists a constant B ϕ > 1 such that ϕ(2t) ≥ B ϕ ϕ(t) (or ϕ(t) ≥ B ϕ ϕ(2t)). Remark 1.2 (i) It is clear that R γ 1 R γ 2 for γ 1 < γ 2 and R ∞ = (0, 1) × (0, 1). In view of (2.4), we see that Theorem 1.1 essentially improved and generalized Theorem A.
(ii) Our methods used to deal with Fourier transform estimates of some measures are different from those in the proof of Theorem A. In fact, the authors in [1] used the TT * method to prove Theorem A. However, the TT * method is not needed in the proof of Theorem 1.1.
(iii) Part (i) of Theorem 1.1 improved and generalized Theorem 1 in [9], in which the authors showed that T h, is bounded onḞ p,q α (R n ) for α ∈ R and 1 < p, q < ∞, provided that h ∈ L ∞ (R + ) and ∈ H 1 (S n-1 ).
(iv) Theorem 1.1 is new, even in the special case h ≡ 1 or The second motivation of this paper is concerned with the L p boundedness of maximal truncated singular integrals associated to polynomial curves. Let h, , P, ϕ be given as in (1.3). The maximal truncated singular integral operator T * h, ,P,ϕ is defined by where f ∈ S(R n ). The type of operator T * h, ,P,ϕ was first studied by Fan, Guo, and Pan [18] and ∈ F β (S n-1 ) for some β > 3/2. Recently, Liu [26] proved that T * h, ,P,ϕ is of type (p, p) for 1 < p < ∞, provided that ϕ ∈ F 1 or F 2 , ∈ L log L(S n-1 ) and h satisfies certain radial condition.
The third motivation of this paper is concerned with the boundedness of maximal truncated singular integrals associated to polynomial curves on Triebel-Lizorkin spaces. The first work related to the boundedness for maximal singular integral operator on Triebel-Lizorkin spaces was due to Zhang and Chen [43], who showed that the maximal singular integral operator is bounded onḞ p,q α (R n ) and F p,q α (R n ) for 0 < α < 1 and 1 < p, q < ∞ by assuming that ∈ H 1 (S n-1 ). Recently, Liu, Xue, and Yabuta [30] established the boundedness for the maximal singular integral operators associated to polynomial mappings on Triebel-Lizorkin spaces under the conditions h ∈ γ (R + ) with some γ > 1 and ∈ L log L(S n-1 ). Very recently, the authors [31] obtained the boundedness for T * h, ,P,ϕ on Triebel-Lizorkin spaces, provided that h ≡ 1, ∈ F β (S n-1 ) with some β > 3/2 and ϕ ∈ F 3 , where F 3 is the set of all functions φ satisfying the following conditions: (a) φ is a positive increasing function on (0, ∞) such that t δ φ (t) is monotonic on (0, ∞) for some δ ∈ R; (b) There exist positive constants C φ and c φ such that It is clear that F 3 F 1 . There are some model examples for the class F 3 such as t α (α > 0), t β ln(1 + t)(β ≥ 1), t ln ln(e + t), real-valued polynomials P on R with positive coefficients and P(0) = 0 and so on.
Based on the above, it is natural to ask the following question. Our next result will give a positive answer to Question 1.3.

Theorem 1.3
Let P be a real polynomial on R satisfying P(0) = 0 and ϕ ∈ F 3 . Suppose that h ≡ 1 and ∈ H 1 (S n-1 ) satisfies (1.1). Then, for 0 < α < 1 and 1 < p, q < ∞, there exists a constant C > 0 independent of and the coefficients of P such that It should be pointed out that Theorem 1.3 is new, even in the special case ϕ(t) ≡ t.
The paper is organized as follows. In Sect. 2 we present some preliminary definitions and lemmas, which are the main ingredients of proving Theorems 1.1-1.3. The proofs of Theorems 1.1-1.3 will be given in Sect. 3. It should be pointed out that the main methods and ideas employed in this paper are a combination of ideas and arguments from [1,21,22,27,30,41]. However, some new techniques are needed in the main proofs. The new ideas invented in our proofs are to define suitable measures and to estimate them suitably.
Throughout the paper, for any p ∈ [1, ∞], we denote p by the conjugate index of p, which satisfies 1/p + 1/p = 1. Here, we set 1 = ∞ and ∞ = 1. The letter C or c, sometimes with certain parameters, will stand for positive constants not necessarily the same one at each occurrence, but are independent of the essential variables. In what follows, we set R n = {ζ ∈ R n ; 1/2 < |ζ | ≤ 1}. Let ζ (f ) be the difference of f for an arbitrary function f defined on R n and ζ ∈ R n , i.e., For any t ∈ R, we set exp(t) = e -2π it . We also use the conventions i∈∅ a i = 0 and i∈∅ a i = 1.

Preliminary definitions and lemmas 2.1 Preliminary definitions
In this subsection we give the definitions of several rough kernels and their relationships.
The class L(log L) α (S n-1 ) for α > 0 denotes the class of all measurable functions on S n-1 which satisfy

Definition 2.3 (Grafakos-Stefanov class)
The Grafakos-Stefanov class F β (S n-1 ) for β > 0 denotes the set of all integrable functions over S n-1 which satisfy the condition We remark that F β (S n-1 ) was introduced by Grafakos and Stefanov [25] in the study of the L p boundedness of singular integral operator with rough kernels.
The following inclusion relations are known: Let us present the definitions of Triebel-Lizorkin spaces.

Definition 2.4 (Triebel-Lizorkin spaces)
Let S (R n ) be the tempered distribution class on R n . For α ∈ R and 0 < p, q ≤ ∞(p = ∞), we define the homogeneous Triebel-Lizorkin spacesḞ The inhomogeneous versions of Triebel-Lizorkin spaces are denoted by F p,q α (R n ) and are obtained by adding the term * f L p (R n ) to the right-hand side of (2.3) with i∈Z replaced by i≥1 , where ∈ S(R n ), The following properties are well known (see [24,40]): Our next definition is concerned with the H 1 (S n-1 ) atom.

Preliminary lemmas
We start now the following atomic decomposition of H 1 (S n-1 ).

Lemma 2.1 ([13, 14])
Let ∈ H 1 (S n-1 ) satisfy (1.1). Then there exist a sequence of complex numbers {c j } j≥1 and a sequence of (1, ∞) atoms { j } j≥1 such that In order to deal with certain estimates for Fourier transforms of some measures, we need the following properties for (1, ∞) atom.
Then there exists a positive constant C, independent of b, such that The following oscillatory estimates are useful for our proofs.
Then, for any r > 0 and λ = 0, , but may depend on ϕ and δ.
We end this section by presenting a well-known result.
Here, C p > 0 is independent of the coefficients of {P i } d i=1 and f .

Proofs of Theorems 1.1-1.3
In this section we prove Theorems 1.1-1.3. In Sect. 3.1 we present some notation and lemmas, which are the main ingredients of proving Theorems 1.1-1.3. The proofs of Theorems 1.1-1.3 will be given in Sect. 3.2.

Proofs of Theorems 1.1-1.3
Proof of Theorem 1.1 Let h, , P, ϕ be given as in Theorem 1.1. Invoking Lemma 2.1, there exist a sequence of complex numbers {c j } ∞ j=1 and a sequence of (1, ∞) atoms { j } ∞ j=1 such that = ∞ j=1 c j j and H 1 (S n-1 ) ≈ ∞ j=1 |c j |. By the definition of T h, ,P,ϕ , one has In view of (3.52) and the definition ofḞ p,q α (R n ), we have that, for 1 < p, q < ∞ and α ∈ R, Therefore, to prove Theorem 1.1, it suffices to prove that there exists C > 0 is independent of h, γ , and the coefficients of P such that holds for any (1, ∞) atom and α ∈ R and (p, q) ∈ R γ .