Boundedness of Higher-order Marcinkiewicz-type Integrals

Let A be a function with derivatives of order m and D γ A ∈ ˙ Λ β (0 < β < 1, |γ| = m). The authors in the paper proved that if Ω ∈ L s (S n−1) (s ≥ n/(n − β)) is homogeneous of degree zero and satisfies a vanishing condition, then both the higher-order Marcinkiewicz-type integral μ A Ω and its variation μ A Ω are bounded from L p (R n) to L q (R n) and from L 1 (R n) to L n/(n−β),∞ (R n), where 1 < p < n/β and 1/q = 1/ p − β/n. Furthermore, if Ω satisfies some kind of L s-Dini condition, then both μ A Ω and μ A Ω are bounded on Hardy spaces, and μ A Ω is also bounded from L p (R n) to certain Triebel-Lizorkin space.


Introduction
Suppose that S n−1 is the unit sphere of R n (n ≥ 2) equipped with the normalized Lebesgue measure dσ = dσ(x ).Let Ω ∈ L 1 (S n−1 ) be homogeneous of degree zero and satisfy where x = x/|x| for any x = 0. Then the Marcinkiewicz integral operator of higher dimension is defined by where F Ω,t ( f )(x) = (1/t) |x−y|≤t (Ω(x − y)/|x − y| n−1 ) f (y)dy.
And if we denote H as the Hilbert space H = {h : h H = ( ∞ 0 |h(t)| 2 (dt/t)) 1/2 < ∞}, then μ Ω ( f ) can be looked as the vector-valued function in H, that is 2 Boundedness of higher-order Marcinkiewicz-type integrals It is well known that the operator μ Ω was defined first by Stein in [13], where the author proved that if Ω is continuous and satisfies a Lip α (0 < α ≤ 1) condition on S n−1 , then μ Ω is an operator of type (p, p) for 1 < p ≤ 2 and of weak type (1,1).Benedek et al. in [1] showed that if Ω ∈ C 1 (S n−1 ), then μ Ω is an operator of type (p, p) for 1 < p < ∞.Recently, Ding et al. in [4] improved the results mentioned above.They gave the L p (R n ) (1 < p < ∞) boundedness of μ Ω for Ω ∈ H 1 (S n−1 ), where H 1 denotes the Hardy space on S n−1 (see [3] for the definition of H 1 ).
On the other hand, let b ∈ L loc (R n ), then the commutator of Marcinkiewicz integral is defined by where In 1990, Torchinsky and Wang [14] proved that if Ω is continuous and satisfies a Lip α (0 < α ≤ 1) condition, then for b ∈ BMO, μ b Ω is bounded on L p (ω), here ω ∈ A p (1 < p < ∞).
Moreover, let m ∈ N and let A be a function on R n .We denote (1.7) Then the higher-order Marcinkiewicz-type integral and its variation are defined, respectively, by S. Lu and H. Mo 3 where (1.9) When Ω ∈ Lip α (S n−1 ) and D γ A ∈ Λβ (0 < β < min{1/2,α}), Liu [8] considered the boundedness of μ A Ω and got the following results.Theorem 1.1 [8].Let 1 < p < ∞, let 0 < α ≤ 1, let Ω be homogeneous of degree zero on R n and satisfy (1.1) It is well known that any weakness or removal of smoothness assumed on kernels is very interesting to the boundedness of singular integrals.Inspired by [9,15,11], we want to know whether the conditions assumed on Ω in Theorem 1.1 can be weakened or removed.In fact, the answer is affirmative.And we will also study the boundedness of μ A Ω and μ A Ω on Hardy spaces.Let us now give a definition and formulate our results.Definition 1.2.For Ω ∈ L s (S n−1 ) (s ≥ 1), the integral modulus ω s (δ) of continuity of order s of Ω is defined by where ρ is a rotation on it is said that Ω(x ) satisfies the L s -Dini condition.
. If there exists some s ≥ n/(n − β) such that Ω ∈ L s (S n−1 ) satisfying (1.1) and (1.11), then both μ A Ω and μ A Ω are bounded from is the commutator of Marcinkiewicz integral.So, our results in this paper are extensions of those in [9,15,11].
Remark 1.9.It is easy to see that if Ω ∈ Lip α (S n−1 ) (0 < α ≤ 1), then Ω ∈ L s (S n−1 ) for any s ≥ 1 and satisfies the L s -Dini condition (1.12).In addition, (1.11) is weaker than (1.12) (see [6]).So, Theorems 1.3, 1.4, and 1.5 in the paper are substantial improvements of Theorem A. It should be pointed out that any smooth condition assumed on Ω is not needed in Theorems 1.3 and 1.4.

Some basic notations and lemmas
Lemma 2.1 [7].Let A be a function with derivatives of order m in Λβ (0 < β < 1), then there exists a constant C > 0 such that (2.1) (2.4) S. Lu and H. Mo 5 , then the fractional integral operator T Ω,α defined by ) , then for any λ > 0 and . (2.7) It is easy to see that TΩ,α satisfies Lemmas 2.2 and 2.3.
Lemma 2.5 [12]. where Lemma 2.6 [6].Suppose that 0 < λ < n and Ω is homogeneous of degree zero and satisfies the L s -Dini condition (1.11) for s > 1.If there exists a constant where the constant C > 0 is independent of R and x.

Proofs of Theorems 1.3, 1.4, and 1.5
We first prove Theorems 1.3 and 1.4.By Lemmas 2.2, 2.3 and Remark 2.4, we need only to show that there exists a constant C > 0 such that for any x ∈ R n .
6 Boundedness of higher-order Marcinkiewicz-type integrals In fact, for any fixed x ∈ R n , by the Minkowski inequality and (2.1), we have Similarly, by the Minkowski inequality and (2.2), So, we complete the proofs of Theorems 1.3 and 1.4.Let us now turn to prove Theorem 1.5.
Fix a cube Q(x Q ,l) x with its center at x Q and denote the half side length of Q by l.Let Since by the definition of μ A Ω ( f ), we have (3.5) S. Lu and H. Mo 7 Choose 1 < p 1 < n/β and 1/q 1 = 1/ p 1 − β/n such that 1 < p 1 < p. Then by Hölder's inequality and the (L p1 ,L q1 ) boundedness of μ A Ω (see Theorem 1.3), we have . By the Minkowski inequality, (2.1), and Hölder's inequality, we have 8 Boundedness of higher-order Marcinkiewicz-type integrals Since Ω ∈ L s (S n−1 ), it is easy to see that Therefore, by 0 < β < 1/2, we have (3.10) In the same way, we have Let us now estimate W.
S. Lu and H. Mo 9 Since, (3.12) By the Minkowski inequality and |y − z| For W 1 , using (2.1) and Hölder's inequality, 10 Boundedness of higher-order Marcinkiewicz-type integrals However, by Lemma 2.6 and (1.12), we obtain Hence, , it is similar to the estimate of V , and we have Lu and H. Mo 11 (3.17) Let us now estimate W 3 .By (2.4), Hölder's inequality, and (3.9), (3.18) 12 Boundedness of higher-order Marcinkiewicz-type integrals Thus, Combining the estimates of U, V with W, we have Combining the estimates of J 1 with J 2 , we obtain 1 where 1 < p 1 , s < p. So, by Lemma 2.5 and the L p (R n ) boundedness of M p1 and M s , we conclude that We complete the proof of Theorem 1.5.

Proofs of Theorems 1.6 and 1.7
First, let us introduce some notations related to Hardy spaces.
Definition 4.1 [10].Let 0 < p ≤ 1 ≤ q ≤ ∞, let p < q, and let s ≥ s 0 , where A function a is said to be a (p, q,s) atom, if a ∈ L q (R n ) and satisfies the following conditions:

S. Lu and H. Mo 13
Definition 4.2 [10].Let 0 < p ≤ 1 ≤ q and let p < q, then the atomic Hardy space H p,q,s a (R n ) is defined by here a j is a (p, q,s) atom and j λ j p < ∞ . (4.1) Then, , for all decompositions of f = j λ j a j .(4.2) Lemma 4.3 [10].Let 0 < p ≤ 1 ≤ q and let p < q, then H p,q,s a Let us now turn to prove Theorem 1.6.First, we estimate μ A Ω ( f ).Notice that, when n/(n + β) < p ≤ 1 and 0 By Lemma 4.3 and a standard argument, it is sufficient for us to show that there is a constant C > 0 such that for any (p,∞,0) atom a, μ A Ω (a) L r ≤ C. Take a (p,∞,0) atom a with suppa ⊂ B(x 0 ,l).Then, Choose p 1 and q 1 satisfying 1 < p 1 < n/β and 1/q 1 = 1/ p 1 − β/n.It is obvious that r < q 1 .So, by Hölder's inequality and the (L p1 ,L q1 ) boundedness of μ A Ω (see Theorem 1.3), 14 Boundedness of higher-order Marcinkiewicz-type integrals we have Since for any . By the Minkowski inequality, Hölder's inequality, (2.1), and (3.9), S. Lu and H. Mo 15 Notice that for any y ∈ B, we have dy . ( On the other hand, it is similar to (3.12), and we have So, 16 Boundedness of higher-order Marcinkiewicz-type integrals For III 1 , by the Minkowski inequality, (2.1), Hölder's inequality, and (3.15), we have , by the Minkowski inequality, (2.1), Hölder's inequality, and (3.9), × a(y) dy S. Lu and H. Mo 17 Moreover, by the Minkowski inequality, (2.3), and (3.9), 18 Boundedness of higher-order Marcinkiewicz-type integrals So, Combining the estimates of I, II with III, we get

.14)
Replacing μ A Ω ( f ) by μ A Ω ( f ) and using (2.2) and (2.5) instead of (2.1) and (2.3) in the above estimates, we can show that μ A Ω is also bounded from In fact, we need only to check III 3 , where R m+1 is replaced by Q m+1 : Thus we complete the proof of Theorem 1.6.
Let us now prove Theorem 1.7.The main idea is the same as that of proving Theorem 1.6.
Let a be a (1,∞,0) atom with suppa ⊂ B(x 0 ,l) and r = n/(n − β), then (4.17) As in the estimate of (4.9), we have (4.20)Thus, we get the estimate of μ A Ω ( f ) for f ∈ H 1 (R n ).It is analogous to the argument for μ A Ω in the proof of Theorem 1.6, and we can get the desired result for μ A Ω by repeating the above estimates and using (4.15), when f ∈ H 1 (R n ).So, we complete the proofs of Theorem 1.7.
μ A Ω (a) L r ≤ I + II + III, (4.16)where I, II, and III are the same as in the proof of Theorem 1.6.In the same way as in the estimates of (4.5) and (4.6), when r = n/(n − β), we haveI ≤ C |γ|=m D γ A Λβ , II ≤ C |γ|=m D γ A Λβ Ω L s (S n−1 ) .