Boundedness for Parametrized Littlewood-Paley Operators with Rough Kernels on Weighted Weak Hardy Spaces

and Applied Analysis 3 We say that w ∈ A1 if 1 |B| ∫ B w (x) dx ⩽ C ⋅ ess inf x∈B w (x) for every ball B ⊂ R. (17) A weight function w ∈ A∞ if it satisfies the Ap condition for some 1 < p < ∞. It is well known that ifw ∈ Ap, 1 < p < ∞, then w ∈ Ar for all r > p, and w ∈ Aq for some 1 < q < p. We thus write qw ≡ inf{q > 1 : w ∈ Aq} to denote the critical index of w. Lemma 4 (see [21]). Let 1 ⩽ p < ∞, w ∈ Ap. Then, for any ball B, there exists an absolute constant C > 0, such that w (2B) ⩽ Cw (B) . (18) In general, for any λ > 0, we have w (λB) ⩽ Cλ np w (B) , (19) where C does not depend on B nor on λ. Lemma 5 (see [19]). Let 0 < p ⩽ 1, w ∈ A∞. For every f(x) belongs toWH w (R), there exists a sequence of bounded measurable functions {fk(x)} ∞ k=−∞ such that (i) f(x) = ∑ k=−∞ fk(x), in S, (ii) each fk can be further decomposed into fk = ∑i b k i , where b i satisfies the following conditions. (a) supp(b i ) ⊂ Q k i , where Q i denotes the ball with center x i and radius r i . Moreover, ∑ i w(Q k i ) ⩽ C12 −kp , ∑ i χ Q k i ⩽ C1, (20) where χE denotes the characteristic function of the set E and C1 ⩽ ‖f‖ p WH p w . (b) ‖b i ‖ L∞ ⩽ C2 , where C > 0 is independent of i, k. (c) ∫ Rn b k i x dx = 0 for every multi-index α with |α| ⩽ [n(qw/p − 1)]. Conversely, if f ∈ S(R) have a decomposition satisfying (i) and (ii), then f ∈ WH w (R). Moreover, we have ‖f‖p WH p w ∼ C. In the end of this section, we need the following lemmas used in the next section. Lemma 6 (see [22]). Suppose that Ω ∈ L(S) satisfies (1) and the following condition

Before stating our main results, let us recall some definitions.Firstly, let Ω(  ) ∈   ( −1 ),  ⩾ 1.Then, the integral modulus   () of continuity of order  of Ω is defined by Ω (  ) − Ω (  )       d (  )) where,  denotes a rotation on  −1 and ‖‖ = sup   ∈ −1 |  −   |.The function Ω is said to satisfy the   -Dini condition, if Secondly, given a weight function  on R  , for 1 ⩽  < ∞, the weighted Lebesgue spaces is defined by ()      ()d) And also, the weighted weak Lebesgue spaces is defined by Let us now turn to recall the definition of the weighted weak Hardy spaces.The weak Hardy spaces were first introduced in [16].The atomic decomposition theory of weak  1 spaces on R  was given by Fefferman and Soria in [17].Later, Liu established the weak   spaces on homogeneous groups in [18].In 2000, Quek and Yang introduced the weighted weak Hardy spaces    (R  ) in [19] and established their atomic decompositions.Moreover, by using the atomic decomposition theory of    (R  ), Quek and Yang also obtained the boundedness of  −  operators on these weighted spaces in [19].Let  ∈  ∞ , 0 <  ⩽ 1, and  = [(  / − 1)].Define where For  ∈ S  (R  ), the grand maximal function of  is defined by Then, weighted weak Hardy space is defined by    (R  ) = { ∈ S  (R  ) :    ∈    (R  )}.Moreover, we set ‖‖    = ‖  ‖    .Our main results are stated as follows.
Theorem 1.Let Ω ∈  2 ( −1 ) satisfying (1) and the following condition Then, for  > /2,  ∈ The relationship between condition (11) and Lip  ( −1 ) condition is not clear up to now.We point that the conclusion of Theorem 1 still holds if we replace the condition (11) by the Lip  ( −1 ) (0 <  ⩽ 1) condition.In other words, we have the following result.

Notations and Preliminaries
In this section, we will introduce some notations and preliminary lemmas used in the proofs of our main theorems in the next section.The classical   weighted theory was first introduced by Muckenhoupt in the study of weighted   boundedness of Hardy-Littlewood maximal functions in [20].A weight  is a locally integrable function on R  which takes values in (0, ∞) at almost everywhere.Given a ball  and  > 0,  denotes the ball with the same center as  whose radius is  times that of .We also denote the weighted measure of  by (); that is, () = ∫  ()d.We say that  ∈   with 1 <  < ∞ if there exists a constant  > 0, such that for every ball  ⊂ R  , Abstract and Applied Analysis 3 We say that  ∈  1 if A weight function  ∈  ∞ if it satisfies the   condition for some 1 <  < ∞.It is well known that if  ∈   , 1 <  < ∞, then  ∈   for all  > , and  ∈   for some 1 <  < .
We thus write   ≡ inf{ > 1 :  ∈   } to denote the critical index of .

∼ 𝐶.
In the end of this section, we need the following lemmas used in the next section.Lemma 6 (see [22]).Suppose that Ω ∈  2 ( −1 ) satisfies (1) and the following condition Lemma 7 (see [23]).Suppose that  > 0, Ω is homogeneous of degree zero and satisfies the  2 -Dini condition.If there exists a constant 0 <  < 1/2 such that || < , then we have where the constant  > 0 is independent of , .

Proof of Main Results
Proof of Theorem 1.In order to prove Theorem 1, it suffices to show that there exists a constant  > 0, for any  ∈ Take  0 ∈ Z such that 2  0 ⩽  < 2  0 + 1; then by Lemma 5 we can write where (26) First, we claim that the following inequality holds: In fact, since supp(   ) ⊂    = (   ,    ), ‖   ‖  ∞ ⩽ 2  , then it follows from Minkowski's integral inequality that Now we turn our attention to the estimate of  2 .If we set where Q  = (   , 8 (− 0 )/    ) and  is a fixed positive number such that 1 <  < 2, therefore, Since  ∈  1 , then by Lemma 4 we can get An application of Chebyshev's inequality and Minkowski integral inequality gives us that Firstly, let us estimate By Lemma 4, we obtain that Noticing that  > /2, 1 <  < 2, we have Using the Minkowski inequality, we get that ) Abstract and Applied Analysis 7 By Lemma 5, we have ( (− 0 )/ ) − It is easy to see that Using the same method as what used to deal with the inequality (39), we can obtain that For  2 , we have Hence, by the inequalities (44) and (45), we have (48) Repeating this process which is similar to the one of estimating   42 (from (42) to ( 46)), we may have Thus by ( 36) and (40), we get This completes the proof of Theorem 1.
Proof of Theorem 2. Combining the idea of proving Theorem 1 with the similar steps as in [12] it is not difficult to get the proof of Theorem 2. We omit the details here.
Proof of Theorem 3. We follow the strategy of the proof of Theorem 1.It suffices to show that there exists a constant  > 0, such that, for any  ∈ where the notations  1 ,  2 are the same as in the proof of Theorem 1.Using the same method of the proof of Theorem 1, we can get Below, we will give the estimate of  2 .If we set where Q  = (   , 8 (− 0 )/    ),  is a fixed positive number such that 1 <  < 2; thus, Noting that  ∈  2 , then by Lemmas 4 and 5, we have where .Now, we are in a position to give the estimates of  3 ,  41 ,  51 ,  52 ,  6 , respectively.First, we take 0 <  < min{1/2,  − /2, , (−2)/2} in the whole proof of Theorem 2. Obviously, By the proof of Theorem 1, we have We conclude the proof of Theorem 3.
each   can be further decomposed into   = ∑     , where    satisfies the following conditions.