A note on maximal singular integrals with rough kernels

In this note we study the maximal singular integral operators associated with a homogeneous mapping with rough kernels as well as the corresponding maximal operators. The boundedness and continuity on the Lebesgue spaces, Triebel–Lizorkin spaces, and Besov spaces are established for the above operators with rough kernels in H1(Sn–1), which complement some recent developments related to rough maximal singular integrals.


Introduction
During the last several years, a considerable amount of attention has been given to investigate the boundedness for various kinds of integral operators on Triebel-Lizorkin spaces. For examples, see [1,4,5] for singular integrals, [15,17,26,27] for Marcinkiewicz integrals, [16,27] for Littlewood-Paley functions, [14,18,20] for maximal functions, and [21,22] for maximal singular integrals. The main purpose of this paper is to prove the boundedness and continuity of the maximal singular integral and maximal operators related to homogeneous mappings on Triebel-Lizorkin spaces when their kernels are given by function in the Hardy space H 1 (S n-1 ). Let n, d ≥ 2 and m = (m 1 , . . . , m d ) ∈ R d . We say that : R n → R d is a (nonisotropic) homogeneous mapping of degree m if (ty) = δ t (y) holds for all t > 0 and y ∈ R d . Here, {δ t } t>0 is a family of dilations on R d defined by sure dσ . Let be integrable over S n-1 and satisfy S n-1 (u) dσ (u) = 0. (1.1) For a suitable mapping : R n → R d , we define the singular integral operator T , by R n f x -(y) (y/|y|) |y| n dy, (1.2) where f ∈ S(R d ) (the Schwartz class on R d ). When n = d and (y) = y, the operator T , reduces to the classical Calderón-Zygmund singular integral operator, which is denoted by T . In their fundamental work on singular integrals, Calderón and Zygmund [3] first proved that T is bounded on L p (R n ) for 1 < p < ∞ if ∈ L log L(S n-1 ). The same conclusion was obtained independently by Coifman and Weiss [7] and Connett [8] under the less restrictive condition that ∈ H 1 (S n-1 ). Here, H 1 (S n-1 ) denotes the Hardy space on the unit sphere and contains L log L(S n-1 ) as a proper subspace. The above results were later extended to singular Radon transforms by many authors (see [2,6,9,10]). In particular, Cheng [6] proved the following result.
Theorem A ( [6]) Let = ( 1 , . . . , d ) be a homogeneous mapping of degree m = (m 1 , . . . , m d ) with each m i = 0. Assume that ∈ H 1 (S n-1 ) satisfies (1.1) and | S n-1 is real-analytic. Then, for 1 < p < ∞, there exists a positive constant C p such that In this paper, we study the maximal singular integrals and maximal operators related to homogeneous mappings. Let , be given as in (1.2). The maximal singular integral operator T * , and the maximal operator M , are defined by For the sake of simplification, we denote T * , = T and M , = M when n = d and (y) = y. Particularly, when = P is a real-valued polynomial mapping from R n to R d , we denote T * , = T * ,P and M , = M ,P . In 1997, Fan and Pan [10] proved that T * ,P is bounded on L p (R d ) for 1 < p < ∞, provided that ∈ H 1 (S n-1 ). It follows from a theorem of Stein and Wainger that M ,P is bounded on L p (R d ) for 1 < p ≤ ∞ if ∈ L 1 (S n-1 ). Other relevant results on the L p bounds for M ,P can be found in [2,24].
Based on the above, a natural question is the following: Here, the above constant C > 0 is independent of . (ii) Theorem 1.1 is new. In a very recent paper [19], the authors have established the L p (1 < p < ∞) bounds for the singular integral operator and maximal singular integral operator related to homogeneous mappings when their integral kernels are given by the unit sphere kernel in H 1 (S n-1 ) and the weak size radial kernel, which contradicts the main result of [13].
On the other hand, the boundedness properties of maximal singular integral operator and maximal operator in Triebel-Lizorkin spaces have also received some attention of many authors. Let S (R d ) be the tempered distribution class on R d . For α ∈ R and 0 < p, q ≤ ∞(p = ∞), the homogeneous Triebel-Lizorkin spacesḞ . The inhomogeneous version of Triebel-Lizorkin spaces, which is denoted by F p,q α (R d ), is obtained by adding the term * f L p (R d ) to the right-hand side of (1.3) with i∈Z replaced by i≥1 , where ∈ S(R d ), supp(ˆ ) ⊂ {ξ : |ξ | ≤ 2},ˆ (x) > c > 0 if |x| ≤ 5/3. It is well known that the following are valid (see [11,25] for more details): Recently, Liu et al. [21,22] have established the bounds for T * ,P and M ,P on Triebel-Lizorkin spaces when ∈ L log L(S n-1 ) or ∈ F β (S n-1 ) (the Grafakos-Stefanov function class (see [12])). It should be pointed out that the following relationships are valid: L(log L) α 1 S n-1 L(log L) α 2 S n-1 for 0 < α 2 < α 1 ; L(log L) α S n-1 H 1 S n-1 L 1 S n-1 for α ≥ 1; As far as I know the bounds for T * ,P and M ,P on the Triebel-Lizorkin spaces and Besov spaces are unknown under the condition that ∈ H 1 (S n-1 ), even in the special case n = d and P(y) = y.
A natural question, which arises from the above, is the following: Question 1.1 is the main motivation for this work. In this paper we give an affirmative answer to the above question by treating more general operators. Our main result can be stated as follows. (i) For any α ∈ (0, 1) and 1 < p, q < ∞, there exists a constant C > 0 such that for all α ∈ (0, 1) and 1 < p, q < ∞. The same conclusions hold for M , .
The rest of this section is to present the bounds and continuity for T * , and M , on Besov spaces. For α ∈ R and 0 < p, q ≤ ∞(p = ∞), the homogeneous Besov spacesḂ where i is given as in (1.3). The inhomogeneous version of Besov spaces B p,q α (R d ) is obtained by adding the term * f L p (R d ) to the right-hand side of (1.5) with i∈Z replaced by i≥1 , where is given as in the definition of F p,q α (R d ). The following property is well known (see [11,25] for more details): Recently, Liu and Wu [20] established a criterion on the boundedness and continuity of a class of operators on Besov spaces, which is listed as follows.

Proposition 1.1 ([20]) Let T be a sublinear operator and be bounded on L p
for any α ∈ (0, 1) and q ∈ (1, ∞). Specially, if T satisfies the following: It is easy to check that both the operators T * , and M , satisfy conditions (1.7) and (1.8). This together with Theorem 1.1 and Proposition 1.1 implies the following theorem. (i) For any α ∈ (0, 1) and 1 < p, q < ∞, there exists a constant C > 0 such that The paper is organized as follows. Sect. 2 contains the atomic decomposition of Hardy space and some estimates of oscillatory integrals, which play key roles in the estimates of Fourier transforms on some measures. In Sect. 3, we prove Theorem 1.1 after presenting a general criterion on the L p bounds of the convolution operators (see Lemma 3.1). The proof of Theorem 1.2 is given in Sect. 4. It should be pointed out that the proof of the boundedness (resp., continuity) part in Theorem 1.2 is greatly motivated by the idea in [21] (resp., [22]).
Throughout the paper, the letter C or c, sometimes with certain parameters, stands for positive constants not necessarily the same one at each occurrence, but are independent of the essential variables. For notational convenience, we set exp(it) = e it for any t ∈ R.

Preliminary definitions and lemmas
We start with the definition of Hardy space on S n-1 and its atomic decomposition. Recall that the Poisson kernel on S n-1 is defined by P rw (θ ) = 1-r 2 |rw-θ| n for 0 ≤ r < 1 and θ , w ∈ S n-1 .
The Hardy space H 1 (S n-1 ) is the set of all L 1 (S n-1 ) functions which satisfy The following atomic decomposition of Hardy space can be obtained by the idea in [7,8].
In our proofs of the main results we shall encounter oscillatory integrals with generalized polynomials as their phase functions. Thus the following lemma of van der Corput type is needed.
Applying Lemma 2.2 and the arguments similar to those used in deriving [6, Lemma 2.5], we can get the following lemma.
Then, by applying an inequality proved by Ricci where C > 0 is independent of v 1 , η, r. On the other hand, by Hölder's inequality and a change of variable, we have where C > 0 is independent of r. (2.5) together with (2.2) and (2.4) yields (2.1).

Lemma 2.4 ([6]) For j ∈ {1, 2}, let U j be a domain in R n j and K j be a compact subset of U j .
Let h(·, ·) be a real-analytic function on U 1 × U 2 such that h(·, z) is a nonzero function for every z ∈ U 2 . Then there exists a positive number δ = δ(h, K 1 , K 2 ) such that

Proof of Theorem 1.1
In order to prove Theorem 1.1, we need the following lemma, which is the main tool of proving Theorem 1.1.
(iv) for any 1 ≤ s ≤ , {a k,s } satisfies one of the following conditions: Then, for any 1 < p < ∞, there exists C p > 0 such that Here, the constant C p is independent of {L s } λ=1 and {v s } s=1 , but depends on , p, {δ s } s=1 , {β s } s=1 , and {η s } s=1 .
We now proceed with the proof of Theorem 1.1. For notational convenience, we denote by V n-1 the set of polynomials in n -1 variables with real coefficients and set [x] := max{k ∈ Z : k ≤ x} for any x ∈ R. For s ∈ N, let V n-1,s denote the subset of V n-1 which contains homogeneous polynomials of degree s.
Proof of Theorem 1.1 By Lemma 2.1 and Minkowski's inequality, it suffices to show that where is an H 1 atom on S n-1 satisfying conditions (i)-(iii) of Definition 2.1.