Boundedness of singular integral operators with variable kernels on weighted weak Hardy spaces

Let $T_\Omega$ be the singular integral operator with variable kernel $\Omega(x,z)$. In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of $T_\Omega$ on these spaces, under some Dini type conditions imposed on the variable kernel $\Omega(x,z)$.


Introduction
Let S n−1 be the unit sphere in R n (n ≥ 2) equipped with the normalized Lebesgue measure dσ. A function Ω(x, z) defined on R n × R n is said to belong to L ∞ (R n ) × L r (S n−1 ), r ≥ 1, if it satisfies the following conditions: (1) for all λ > 0 and x, z ∈ R n , Ω(x, λz) = Ω(x, z); (2) for all x ∈ R n , S n−1 Ω(x, z ′ ) dσ(z ′ ) = 0; (3) Ω L ∞ (R n )×L r (S n−1 ) := sup x∈R n S n−1 |Ω(x, z ′ )| r dσ(z ′ ) 1/r < ∞, where z ′ = z/|z| for any z ∈ R n \{0}. Set K(x, z) = Ω(x,z ′ ) |z| n . In this paper, we consider the singular integral operator with variable kernel defined by In [2] and [3], Calderón and Zygmund investigated the L p boundedness of singular integral operators with variable kernels. They found that these operators are closely related to the problem about second order elliptic partial differential equations with variable coefficients. We will denote the conjugate exponent of p > 1 by p ′ = p/(p − 1). In [4], Calderón and Zygmund proved the following theorem.
In 2008, Lee et al. [13] considered the weighted case and showed that if the kernel K(x, y) satisfies the L r -Hörmander condition with respect to x and y variables respectively, then T Ω is bounded on L p w (R n ). More precisely, they proved hold for all R > 0. If r ′ ≤ p < ∞ and w ∈ A p/r ′ , then T Ω is bounded on L p w (R n ). In [5,6], Ding et al. introduced some definitions about the variable kernel Ω(x, z) when they studied the H 1 -L 1 boundedness of Marcinkiewicz integral. Replacing the above condition (3), they strengthened it to the condition For r ≥ 1, a function Ω(x, z) is said to satisfy the L r -Dini condition if the conditions (1), (2), (3 ′ ) hold and where ω r (δ) := sup x∈R n ρ≥0 In order to obtain the H p w -L p w boundedness of T Ω , Lee et al. [13] generalized the L r -Dini condition by replacing (1.3) to the following condition(see also [14]) (1.4) If Ω satisfies (1.4) for some r ≥ 1 and 0 ≤ α ≤ 1, we call it satisfies the L r,α -Dini condition. For the special case α = 0, it reduces to the L r -Dini condition. For 0 < β < α ≤ 1, if Ω satisfies the L r,α -Dini condition, then it also satisfies the L r,β -Dini condition. We thus denote by Din r α the class of all functions which satisfy the L r,β -Dini condition for all 0 < β < α.
Theorem C ( [13]). Let 0 < α ≤ 1 and n/(n + α) < p < 1. Suppose Ω ∈ Din r α such that (1.1) and (1.2) hold for a certain large number r. If w r ′ ∈ A (p+ pα n − 1 r )r ′ , then there exists a constant C > 0 independent of f such that The main purpose of this article is to study the weak type estimates of T Ω on the weighted Hardy spaces H p w (R n ) at the endpoint case of p = n/(n + α). We now present our main result as follows.
In particular, if we take w to be a constant function, then we can get Corollary 1.2. Let 0 < α < 1, p = n/(n + α) and r > 1. Suppose Ω ∈ Din r α such that (1.1) and (1.2) hold. Then there exists a constant C > 0 independent of f such that T

Notations and preliminaries
The definition of A p class was first used by Muckenhoupt [16], Hunt, Muckenhoupt and Wheeden [9], and Coifman and Fefferman [1] in the study of weighted L p boundedness of Hardy-Littlewood maximal functions and singular integrals. Let w be a nonnegative, locally integrable function defined on R n ; all cubes are assumed to have their sides parallel to the coordinate axes. We say that w ∈ A p , where C is a positive constant which is independent of the choice of Q.
A weight function w is said to belong to the reverse Hölder class RH s if there exist two constants s > 1 and C > 0 such that the following reverse Hölder inequality holds It is well known that if w ∈ A p with 1 < p < ∞, then w ∈ A r for all r > p, and w ∈ A q for some 1 < q < p. We thus write q w ≡ inf{q > 1 : w ∈ A q } to denote the critical index of w. Given a cube Q and λ > 0, λQ denotes the cube with the same center as Q whose side length is λ times that of Q. Q = Q(x 0 , l) denotes the cube centered at x 0 with side length l. For a weight function w and a measurable set E, we denote the Lebesgue measure of E by |E| and set the weighted measure w(E) = E w(x) dx.
We give the following results that will be used in the sequel.
. Let w ∈ A 1 . Then, for any cube Q and any λ > 1, there exists an absolute constant C > 0 such that where C does not depend on Q nor on λ.
for any measurable subset E of a cube Q.
Given a weight function w on R n , for 0 < p < ∞, we denote by L p w (R n ) the space of all functions satisfying We also denote by W L p w (R n ) the weighted weak L p space which is formed by all functions satisfying We write S (R n ) to denote the Schwartz space of all rapidly decreasing infinitely differentiable functions and S ′ (R n ) to denote the space of all tempered distributions, i.e., the topological dual of S (R n ). For any 0 < p ≤ 1, the weighted Hardy spaces H p w (R n ) can be defined in terms of maximal functions. Let ϕ be a function in S (R n ) satisfying R n ϕ(x) dx = 1. Set We will define the maximal function M ϕ f (x) by The real-variable theory of weighted Hardy spaces has been investigated by many authors. For example, Garcia-Cuerva [7] studied the atomic decomposition and the dual spaces of H p w for 0 < p ≤ 1. The molecular characterization of H p w for 0 < p ≤ 1 was given by Lee and Lin [12]. We refer the readers to [7,12,18] and the references therein for further details.
In this article, we will use Garcia-Cuerva's atomic decomposition theory for weighted Hardy spaces in [7,18]. We characterize weighted Hardy spaces in terms of atoms in the following way.
Let 0 < p ≤ 1 ≤ q ≤ ∞ and p = q such that w ∈ A q with critical index q w . Set [ · ] the greatest integer function. For s ∈ Z + satisfying s ≥ N = [n(q w /p − 1)], a real-valued function a(x) is called a (p, q, s)-atom centered at x 0 with respect to w(or a w-(p, q, s)-atom centered at x 0 ) if the following conditions are satisfied: (a) a ∈ L q w (R n ) and is supported in a cube Q centered at x 0 ; (b) a L q w ≤ w(Q) 1/q−1/p ; (c) R n a(x)x α dx = 0 for every multi-index α with |α| ≤ s. Theorem 2.4. Let 0 < p ≤ 1 ≤ q ≤ ∞ and p = q such that w ∈ A q with critical index q w . For each f ∈ H p w (R n ), there exist a sequence {a j } of w-(p, q, N )-atoms and a sequence {λ j } of real numbers with j |λ j | p ≤ C f p H p w such that f = j λ j a j both in the sense of distributions and in the H p w norm. In particular, for w equals to a constant function, we shall denote W L p w (R n ) and H p w (R n ) simply by W L p (R n ) and H p (R n ). Throughout this article C denotes a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence.

Proof of Theorem 1.1
In order to prove our main theorem, we shall need the following superposition principle on weighted weak type estimates.
Lemma 3.1. Let w ∈ A 1 and 0 < p < 1. If a sequence of measurable functions {f j } satisfy w x ∈ R n : |f j (x)| > α ≤ α −p for all j ∈ Z and j∈Z |λ j | p ≤ 1, then we have that j λ j f j (x) is absolutely convergent almost everywhere and Proof. The proof of this lemma is similar to the corresponding result for the unweighted case which can be found in [17]. See also [15, p. 123].
Following the same arguments as in the proof of Lemma 5 in [11], we can also establish the following lemma on the variable kernel Ω. See [6] and [13].
satisfies the L r -Dini condition in Section 1. If there exists a constant 0 < γ ≤ 1 2 such that |y| < γR, then for any x 0 ∈ R n , we have where the constant C > 0 is independent of R and y.
We are now in a position to give the proof of Theorem 1.1.
Proof. Since w r ′ ∈ A 1 , then we have w ∈ A 1 by Lemma 2.2. We now observe that for w ∈ A 1 and p = n/(n + α), then [n(q w /p − 1)] = [α] = 0. By Theorem 2.4 and Lemma 3.1, it suffices to verify that for any w-(p, r ′ , 0)-atom a(x), there exists a constant C > 0 independent of a such that T Ω (a) W L p w ≤ C. Let a(x) be a w-(p, r ′ , 0)-atom centered at x 0 with supp a ⊆ Q = Q(x 0 , l), and let Q * = 2 √ nQ. For any fixed λ > 0, we write |T Ω a(x)| > λ = I 1 + I 2 .
Since w ∈ A 1 , then by using Theorem B, we can see that T Ω is bounded on L r ′ w . Applying Chebyshev's inequality, Hölder's inequality, Lemma 2.1 and the size condition of atom a, we thus obtain We turn our attention to the estimate of I 2 . First we note that if x ∈ (Q * ) c : |T Ω a(x)| > λ = Ø, then the inequality Integrating over Q * k+1 \Q * k (k ∈ Z + ) on both sides of the inequality λ < |T Ω a(x)|, then we get |T Ω a(x)|dx, for all k = 1, 2, . . . .
By the cancellation condition of atom a, we have Using Hölder's inequality and Lemma 3.2, we can see that for any y ∈ Q, the inner integral of the above expression is dominated by On the other hand, it follows directly from Hölder's inequality, the A r condition and the size condition of atom a that Q |a(y)| dy ≤ Q |a(y)| r ′ w(y) dy In addition, since Q ⊆ Q * k for any k = 1, 2, . . ., then by Lemma 2.3, we can get Summarizing the estimates (3.2) and (3.3) derived above and using the fact that p = n/(n + α), we then obtain It is obvious that for any k = 1, 2, . . ., we have |Q * k+1 \Q * k | ≥ |Q * k |. Hence w(Q * k ) 1/p . Furthermore, for p = n/(n + α), it is easy to check that lim k→∞ 1 w(Q * k ) 1/p = 0. Then for any given λ > 0, we are able to find a maximal positive integer K such that λ < C · 1 w(Q * K ) 1/p . Therefore Combining the above inequality (3.4) with (3.1) and taking the supremum over all λ > 0, we conclude the proof of Theorem 1.1.