Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces

In this article, we consider the Marcinkiewicz integrals with variable kernels defined by μΩ(f)(x) = (∫ ∞ 0 ∣∣∣∣ ∫ |x−y|≤t Ω(x, x− y) |x− y|n−1 f(y)dy ∣∣∣∣ 2 dt t3 )1/2 , where Ω(x, z) ∈ L∞(Rn)×Lq(Sn−1) for q > 1. We prove that the operator μΩ is bounded from Hardy space, H(R) , to L(R) space; and is bounded from weak Hardy space, Hp,∞(Rn) , to weak L(R) space for max{ 2n 2n+1 , n n+α } < p < 1, if Ω satisfies the L -Dini condition with any 0 < α ≤ 1.


Introduction
Let R n (n ≥ 2) be the n-dimensional Euclidean space and S n−1 denote the unit sphere in R n equipped with induced Lebesgue measure dσ , and let x = x |x| for any x = 0.
We remark that the Marcinkiewicz integral is essentially a Littlewood-Paley g -function.If let φ(x) = ω(x)|x| −n+1 χ B (x) and φ t (x) = t −n φ(x/t), where B denotes the unit ball of R n and χ B denotes the characteristic function of B , then In order to study non-smoothness partial differential equations with variable coefficients, mathematicians pay more attention to the singular integral with variable kernels, see [2], [4], [5] and [6] among others.Specially, in 1955 Calderón and Zygmund [4] considered the singular integral with variable kernel defined by |x − y| n f (y)dy.
In this paper, we study the Marcinkiewicz integral with variable kernel defined by We point out that μ Ω can be interpreted as a Hilbert-valued function.In fact, denote the Hilbert space H by where Then we obtain that μ Ω (f Before stating our theorems, we first introduce some definitions about the variable kernel Ω(x, z).
z) satisfies the following three conditions: (1) Ω(x, λz) = Ω(x, z), for any x, z ∈ R n and any λ > 0; ( (3) In [4], Calderón and Zygmund proved that if Ω satisfies (1), ( 3) and (2 ) s u p /n.They also found that for no n can we replace the exponent 2(n − 1)/n by a smaller one.Since the condition (2) implies (2'), so the ) with q ≥ 2(n−1)/n.Recently in [7], the authors proved that if Ω ∈ L ∞ (R n ) × L q (S n−1 ) with q ≥ 2(n − 1)/n, then μ Ω is bounded on L 2 (R n ); and they also showed the and O is a rotation in R n with |O| = O − I , where I is the identity operator.For the special case α = 0 , it reduces to the L 1 -Dini condition Our first aim is to show that the Marcinkiewicz integral μ Ω with variable kernel is bounded on Hardy spaces H p (R n ) with some p < 1.
The smallest constant C satisfying the above inequality is called the H p,∞ norm of f , and is denoted by ) with q > 2(n − 1)/n, and let Ω(x, z ) satisfy the L 1,α -Dini condition with 0 < α ≤ 1 .Then, if max{ 2n 2n+1 , n n+α } < p < 1 , there exists a constant C independent of f and β such that Thus, the conclusions of Corollary 1.2 and 1.5 may be regarded as an improvement and extension of Stein's results about the Marcinkiewicz integrals with convolution kernels in [12] and [8].
Remark 1.7.It is worthy noting that the H 1 − L 1 boundedness of μ Ω may be regarded as the limit case of Theorem 1.1 by choosing p = 1 and letting α → 0 .Hence, Theorem 1.3 and Corollary 1.7 in [7] are the special cases of above Theorem 1.1.
Throughout the paper, C always denotes a positive constant not necessarily the same at each occurrence.We use a ∼ b to mean the equivalence of a and b ; that is, there exists a positive constant C independent of a, b such that C −1 a ≤ b ≤ Ca.

Proof of theorem 1.1
In order to show the H p − L p boundedness of μ Ω , we will use the atomic decomposition theory of the real Hardy space H p (R n ) for n n+1 < p ≤ 1, see for instance [13].A function a(x) is said to be (p, 2, 0) atom if it satisfies the following three conditions: and in the sense of distributions; where ( and all a k (x) are (p, 2, 0) atoms.
Start with f in a nice dense class of function, say , where Now following [13] (Page 115), we write the distribution kernel K = K 0 + K ∞ , where K 0 has compact support, and thus the distribution F Ω,t,ε is well defined for every fixed ε and t, and We claim that, for almost every To see the claim, we use the cancellation condition of Ω and the fact Ω On the other hand, Hölder inequality gives for any 1 < r < n/(n − 1).Therefore uniformly on ε .So by the Lebesgue dominated convergence theorem we get that μ Ω (f Thus, by similar approximation arguments as in [9] (Theorem 7.3) and in [13] (Page 115), we can obtain Therefore, to derive the inequality (1.3) for any f ∈ H p (R n ) and prove Theorem 1.1, it suffices to show that for any (p, 2, 0) atom a(x), there exists a constant C > 0 independent of a(x) such that Without loss of generality, we let the support of the atom a(x) is B = B(0, r), and denote B * = B(0, 8r).Using the L 2 boundedness of μ Ω , we have Note that (u + v) s ≤ u s + v s for any u, v ≥ 0 and 0 ≤ s ≤ 1 .It is left to give the estimate for the integral We note that, for y ∈ B and x ∈ (B * ) c , |x − y| ∼ |x| ∼ |x| + 2r .Thus by the mean value theorem we have Applying this inequality and the Minkowski's inequality, we obtain that , so we can choose ε satisfying 0 < ε < n + 1 2 − n p .Using Hölder inequality for integrals, we have where  From this and Minkowski's inequality for integrals, we obtain where and Applying Minkowski inequality for integrals, Hölder inequality for integrals and Fubini theorem successively, we can obtain To estimate the inner integral above, we note |y| < r and |x| > 2r , which implies, And thus This and a direct computation give

Now using the condition
1 0 (δ) δ 1+α dδ < ∞, and 1 > p > n n+α , we can get At last, we give the estimate of I 22 .Obviously, mean value theorem gives Applying this inequality, Minkowski's inequality for integrals, and the fact n n+1 < p < 1 , we deduce that Combining (2.3), and the estimates of I 1 , I 21 and I 22 , we get (2.2) and then complete the proof of Theorem 1.1.

Proof of theorem 1.4
In order to prove theorem 1.4, we need the following decomposition theorem for distributions in H p,∞ (R n ).Lemma 3.1.[10] Given a distribution f ∈ H p,∞ (R n ), there exits a bounded function sequence {f k } +∞ k=−∞ which has the following properties: , where {h k i } satisfies the following three conditions: , where B(x, r) denotes the ball in R n with the center at x and radius r .Moreover, We now give the proof of Theorem 1.4.For any f ∈ H p,∞ (R n ) and β > 0, we choose k 0 satisfying 2 k0 ≤ β < 2 k0+1 .Applying Lemma 3.1, we can write where h k i satisfies the properties (i), (ii) and (iii) of Lemma 3.1.
By the L 2 -boundedness of μ Ω , it follows On the other hand, we denote and let where the last inequality holds owing to p > n n+ 1   2   .Therefore, in order to prove Theorem 1.4, it suffices to show Firstly, a similar argument as the one used in (2.1) and Minkowski's inequality for series give where and It is obvious that when y ∈ B k i , x ∈ B c k0 , we will have |x − Combining the above inequality and Minkowski's inequality for integrals, the size condition of h k i and Hölder inequality for integrals, we deduce that ) Decompose J 2,1 as following Using a similar argument as in inequalities (2.4) and (2.5), we can deduce that , and then This follows .
To deal with J 2,2 , we use the mean value theorem, Hölder inequality for integrals, and the condition n n+ 1 2 < p < 1 to give Combining the estimates of J 2,1 and J 2,1 , we obtain that Obviously, the inequalities (3.5), (3.6) and (3.4) yield the inequality (3.3) and the result follows.
we have used that |x − y| ∼ |x|.As to the estimate of I 2 .Noting that if t ≥ |x| + 2r , then B ⊂ {y : |x − y| < t} .So by the cancellation condition (iii) of a, we have