Báo cáo toán học: "Boundedness of Multilinear Littlewood-Paley Operators for the Extreme Cases"

Chia sẻ: Nguyễn Phương Hà Linh Nguyễn Phương Hà Linh | Ngày: | Loại File: PDF | Số trang:12

Thêm vào BST

Báo xấu

43
lượt xem 3
download

Download Vui lòng tải xuống để xem tài liệu đầy đủ

Mục đích của bài viết này là để nghiên cứu tính chất boundedness multilinear Littlewood-Paley các nhà khai thác cho các trường hợp cực kỳ.

Chủ đề:

Bình luận(0) Đăng nhập để gửi bình luận!

Lưu

Nội dung Text: Báo cáo toán học: "Boundedness of Multilinear Littlewood-Paley Operators for the Extreme Cases"

9LHWQDP -RXUQDO Vietnam Journal of Mathematics 33:2 (2005) 123–134 RI 0$7+(0$7,&6 9$67 Boundedness of Multilinear Littlewood-Paley Operators for the Extreme Cases* Liu Lanzhe College of Mathematics, Changsha University of Science and Technology Changsha 410077, China Received July 7, 2003 Revised December 4, 2004 Abstract. The purpose of this paper is to study the boundedness properties of multilinear Littlewood-Paley operators for the extreme cases. 1. Introduction and Results Fix δ > 0. Let ψ be a ﬁxed function which satisﬁes the following properties: (1) ψ (x)dx = 0, Rn (2) |ψ (x)| ≤ C (1 + |x|)−(n+1−δ) , (3) |ψ (x + y ) − ψ (x)| ≤ C |y |(1 + |x|)−(n+2−δ) when 2|y | < |x|. We denote Γ(x) = {(y, t) ∈ Rn+1 : |x − y | < t} and the characteristic + function of Γ(x) by χΓ(x) . Let m be a positive integer and A be a function on Rn . The multilinear Littlewood-Paley operator is deﬁned by 1/2 dydt A |FtA (f )(x, y )|2 Sδ (f )(x) = , tn+1 Γ(x) where ∗ This work was supported by the NNSF (Grant: 10271071).
124 Liu Lanzhe Rm+1 (A; x, z ) FtA (f )(x, y ) = f (z )ψt (y − z )dz, |x − z |m Rn 1α D A(y )(x − y )α Rm+1 (A; x, y ) = A(x) − α! |α|≤m and ψt (x) = t−n+δ ψ (x/t) for t > 0. Set Ft (f )(y ) = f ∗ ψt (y ). We also deﬁne dydt 1/2 |Ft (f )(y )|2 n+1 Sδ (f )(x) = , t Γ(x) which is the Littlewood-Paley operator (see [14]). 1/2 |h(t)|2 dydt/tn+1 Let H be the Hilbert space H = h : h = < Rn+1 + ∞ . Then for each ﬁxed x ∈ Rn , FtA (f )(x, y ) may be viewed as a mapping from (0, +∞) to H , and it is clear that Sδ (f )(x) = χΓ(x) FtA (f )(x, y ) , Sδ (f )(x) = χΓ(x) Ft (f )(y ) . A A We also consider the variant of Sδ , which is deﬁned by 1/2 dt ˜A |FtA (f )(x)|2 n+1 ˜ Sδ (f )(x) = , t Γ(x) where Qm+1 (A; x, y ) FtA (f )(x) = ˜ ψt (x − y )f (y )dy |x − y |m Rn and 1α D A(x)(x − y )α . Qm+1 (A; x, y ) = Rm (A; x, y ) − α! |α|=m A Note that when m = 0, Sδ is just the commutator of Littlewood-Paley operator (see [1, 11, 12]). It is well known that multilinear operators, as the extension of Commutators, are of great interest in harmonic analysis and have been widely studied by many authors (see [3 - 6, 8]). In [2, 7], the Lp (p > 1) boundedness of commutators generated by the Calder´n-Zygmund operator or o fractional integral operator and BMO functions are obtained, and in [11], the endpoint boundedness of commutators generated by the Calder´n-Zygmund o operator and BMO functions are obtained. The main purpose of this paper is to discuss the boundedness properties of the multilinear Littlewood-Paley operators for the extreme cases of p. Throughout this paper, the letter C s will denote the positive constants which may have diﬀerent values in each line; B will denote a ball of Rn . For a ball B , set fB = |B |−1 f (x)dx and f # (x) = B sup |B |−1 |f (y ) − fB |dy . x ∈B B
Boundedness of Multilinear Littlewood-Paley Operators for the Extreme Cases 125 We shall prove the following theorems in Sec. 3. Theorem 1. Let 0 ≤ δ < n and Dα A ∈ BM O(Rn ) for |α| = m. Then Sδ is A n/δ n n bounded from L (R ) to BM O(R ). Theorem 2. Let 0 ≤ δ < n and Dα A ∈ BM O(Rn ) for |α| = m. Then Sδ is ˜A 1 n n/(n−δ ) n bounded from H (R ) to L (R ). Theorem 3. Let 0 ≤ δ < n and Dα A ∈ BM O(Rn ) for |α| = m. Then Sδ is A bounded from H 1 (Rn ) to weak Ln/(n−δ) (Rn ). Theorem 4. Let 0 ≤ δ < n and Dα A ∈ BM O(Rn ) for |α| = m. (i) If for any H 1 -atom a supported on certain cube Q and u ∈ 3Q \ 2Q, there is 1 (x − u)α n/(n−δ ) Dα A(z )a(z )dz ψt (y − u) dx ≤ C, χΓ(x) α! |x − u|m |α|=m Q (4Q)c then Sδ is bounded from H 1 (Rn ) to Ln/(n−δ) (Rn ); A (ii) If for any cube Q and u ∈ 3Q \ 2Q, there is 1 1 (Dα A(x) − (Dα A)Q ) χΓ(x) |Q| α! |α|=m Q (u − z )α ψt (u − z )f (z )dz dx ≤ C ||f ||Ln/δ , |u − z |m (4Q)c then Sδ is bounded from Ln/δ (Rn ) to BM O(Rn ). ˜A 2. Proofs of Theorems We begin with some preliminary lemmas. Lemma 1. (see [6]) Let A be a function on Rn and Dα A ∈ Lq (Rn ) for |α| = m and some q > n. Then 1/q 1 |Rm (A; x, y )| ≤ C |x − y |m |Dα A(z )|q dz , |B ˜ (x, y )| |α|=m ˜ B (x,y ) √ where B (x, y ) is the ball centered at x and having radius 5 n|x − y |. ˜ Lemma 2. Let 0 ≤ δ < n, 1 < p < n/δ and Dα A ∈ BM O(Rn ) for |α| = m, 1 < r ≤ ∞, 1/q = 1/p + 1/r − δ/n. Then Sδ is bounded from Lp (Rn ) to A q n L (R ), that is
126 Liu Lanzhe A ||Dα A||BMO ||f ||Lp . ||Sδ (f )||Lq ≤ C |α|=m Proof. By Minkowski inequality and by the condition of ψ , we have |f (z )| |Rm+1 (A; x, z )| 1/2 dydt A |ψt (y − z )|2 Sδ (f )(x) ≤ dz |x − z |m t1+n Rn Γ(x) ∞ t−2n+2δ |f (z )||Rm+1 (A; x, z )| 1/2 dydt ≤C dz |x − z |m (1 + |y − z |/t)2n+2−2δ t1+n 0 |x−y |≤t Rn ∞ 22n+2−2δ · t1−n |f (z )||Rm+1 (A; x, z )| 1/2 ≤C dydt dz. |x − z |m (2t + |y − z |)2n+2−2δ 0 |x−y |≤t Rn Noting that 2t + |y − z | ≥ 2t + |x − z | − |x − y | ≥ t + |x − z | when |x − y | ≤ t and ∞ tdt = C |x − z |−2n+2δ , (t + |x − z |)2n+2−2δ 0 we obtain ⎛ ⎞1/2 ∞ |f (z )||Rm+1 (A; x, z )| ⎝ tdt ⎠ dz A Sδ (f )(x) ≤ C |x − z |m (t + |x − z |)2n+2−2δ 0 Rn |f (z )||Rm+1 (A; x, z )| =C dz. |x − z |m+n−δ Rn Thus, the lemma follows from [8]. Proof of Theorem 1. It suﬃces to prove that there exists a constant C depending on B such that 1 A |Sδ (f )(x) − CB |dx ≤ CB ||f ||Ln/δ |B | B √ ˜ ˜ holds for any ball B . Fix a ball B = B (x0 , l). Let B = 5 nB and A(x) = 1 (Dα A)B xα , then Rm (A; x, y ) = Rm (A; x, y ) and Dα A = Dα A − A(x) − ˜ ˜ ˜ α! |α|=m (Dα A)B for |α| = m. We write, for f1 = f χB and f2 = f χRn \B , FtA (f )(x) = ˜ ˜ ˜ FtA (f1 )(x) + FtA (f2 )(x), then
Boundedness of Multilinear Littlewood-Paley Operators for the Extreme Cases 127 1 A A |Sδ (f )(x) − Sδ (f2 )(x0 )|dx |B | B 1 ||χΓ(x) FtA (f )(x, y )|| − ||χΓ(x) FtA (f2 )(x0 , y )|| dx = |B | B 1 1 A ||χΓ(x) FtA (f2 )(x, y ) − χΓ(x) FtA (f2 )(x0 , y )||dx ≤ Sδ (f1 )(x)dx + |B | |B | B B := I + II. Now, let us estimate I and II . First, taking p > 1 and q > 1 such that 1/q = 1/p − δ/n, by the (Lp , Lq ) boundedness of Sδ (Lemma 2), we gain A 1/q 1 (Sδ (f1 )(x))q dx A ≤ C |B |−1/q ||f1 ||Lp = C ||f ||Ln/δ . I≤ |B | B To estimate II , we write χΓ(x) FtA (f2 )(x, y ) − χΓ(x) FtA (f2 )(x0 , y ) 1 1 − χΓ(x) ψt (y − z )Rm (A; x, z )f2 (z )dz = |x − z |m |x0 − z |m χΓ(x) ψt (y − z )f2 (z ) [Rm (A; x, z ) − Rm (A; x0 , z )]dz + |x0 − z |m ψt (y − z )Rm (A; x0 , z )f2 (z ) + (χΓ(x) − χΓ(x0 ) ) dz |x0 − z |m χΓ(x) (x − z )α χΓ(x0 ) (x0 − z )α 1 ψt (y − z )Dα A(z )f2 (z )dz − − ˜ m |x0 − z |m |x − z | α! |α|=m t t t t := II1 (x) + II2 (x) + II3 (x) + II4 (x). We choose r > 1 such that 1/r + δ/n = 1. Note that |x − z | ∼ |x0 − z | for x ∈ B ˜ and z ∈ Rn \ B , similar to the proof of Lemmas 2 and 1, we have ˜ 1 t ||II1 (x)||dx |B | B |x − x0 ||f (z )| C ≤ |Rm (A; x, z )|dz dx ˜ |x − z |n+m+1−δ |B | ˜ B R n \B ∞ |x − x0 ||f (z )| C ≤ |Rm (A; x, z )|dz dx ˜ |x − z |n+m+1−δ |B | k=0 k+1 ˜ k ˜ B 2 B \2 B
128 Liu Lanzhe ∞ l(2k l)m ||Dα A||BMO ≤C |f (z )|dz k (2k l)n+m+1−δ k=0 |α|=m ˜ 2k B ∞ ||Dα A||BMO ||f ||Ln/δ k 2 −k ≤C k=0 |α|=m α ≤C ||D A||BMO ||f ||Ln/δ . |α|=m t For II2 (x), by the formula (see [6]) Rm (A; x, z ) − Rm (A; x0 , z ) ˜ ˜ 1 Rm−|β | (Dβ A; x0 , z )(x − x0 )β ˜ ˜ = Rm (A; x, x0 ) + β! 0
Boundedness of Multilinear Littlewood-Paley Operators for the Extreme Cases 129 t ||II3 (x)|| |ψt (y − z )||f2 (z )||Rm (A; x0 , z )| ˜ ≤C m |x0 − z | Rn Rn+1 + 2 dydt 1/2 × |χΓ(x) (y, t) − χΓ(x0 ) (y, t)| dz tn+1 |f2 (z )|Rm (A; x0 , z )| ˜ ≤C |x0 − z |m Rn t1−n dydt t1−n dydt 1/2 × − dz (t + |y − z |)2n+2−2δ (t + |y − z |)2n+2−2δ Γ(x) Γ(x0 ) |f2 (z )|Rm (A; x0 , z )| ˜ ≤C |x0 − z |m Rn 1/2 1 1 dydt × − dz (t + |x + y − z |)2n+2−2δ (t + |x0 + y − z |)2n+2−2δ tn−1 |y |≤t |f2 (z )|Rm (A; x0 , z )| ˜ ≤C |x0 − z |m Rn |x − x0 |t1−n dydt 1/2 × dz (t + |x + y − z |)2n+3−2δ |y |≤t |f2 (z )||x − x0 |1/2 |Rm (A; x0 , z )| ˜ ≤C dz m+n+1/2−δ |x0 − z | Rn ∞ kl1/2 (2k l)m ||Dα A||BMO ≤C ||f ||Ln/δ (2k l)n+m+1/2−δ k=0 |α|=m ∞ ||Dα A||BMO ||f ||Ln/δ k 2−k/2 ≤ C ||Dα A||BMO ||f ||Ln/δ . ≤C k=0 |α|=m |α|=m t t For II4 (x), similar to the estimate of II3 (x), we have |x − x0 |1/2 |x − x0 | t |Dα A(z )||f (z )|dz I I4 (x) ≤ C ˜ + |x − z |n+1−δ |x − z |n+1/2−δ |α|=m ˜ R n \B ∞ ||Dα A||BMO ||f ||Ln/δ k (2−k + 2−k/2 ) ≤C k=0 |α|=m ||Dα A||BMO ||f ||Ln/δ . ≤C |α|=m Combining these estimates, we complete the proof of Theorem 1.
130 Liu Lanzhe Proof of Theorem 2. It suﬃces to show that there exists a constant C > 0 such that for every H 1 -atom a (that is: supp a ⊂ B = B (x0 , r), ||a||L∞ ≤ |B |−1 and a(y )dy = 0 (see[9, 13])), we have Rn ˜A ||Sδ (a)||Ln/(n−δ) ≤ C. We write [Sδ (a)(x)]n/(n−δ) dx = ˜A [Sδ (a)(x)]n/(n−δ) dx := J + JJ. ˜A + Rn |x−x0 |≤2r |x−x0 |>2r For J , by the following equality 1 (x − y )α (Dα A(x) − Dα A(y )), Qm+1 (A; x, y ) = Rm+1 (A; x, y ) − α! |α|=m we have, similar to the proof of Lemma 2, |Dα A(x) − Dα A(y )| ˜A A Sδ (a)(x) ≤ Sδ (a)(x) + C |a(y )|dy, |x − y |n−δ |α|=mRn thus, Sδ is (Lp , Lq )-bounded by Lemma 2 and [1, 2], where 1/q = 1/p − δ/n. ˜A We see that n/((n−δ )q) n/(n−δ ) ˜A |2B |1−n/((n−δ)q) ≤ C ||a||Lp |B |1−n/((n−δ)q) ≤ C. J ≤ C ||Sδ (a)||Lq 1 To obtain the estimate of JJ , set A(x) = A(x) − |α|=m α! (Dα A)2B xα . Then ˜ ˜ Qm (A; x, y ) = Qm (A; x, y ). We write, by the vanishing moment of a and Qm+1 1 (A; x, y ) = Rm (A; x, y ) − |α|=m α! (x − y )α Dα A(x), for x ∈ (2B )c , FtA (a)(x, y ) ˜ ψt (y − z )Rm (A; x, z ) ˜ = a(z )dz m |x − z | Rn ψt (y − z )Dα A(z )(x − z )α ˜ 1 − a(z )dz |x − z |m α! |α|=m Rn ψt (y − z )Rm (A; x, z ) ψt (y − x0 )Rm (A; x, x0 ) ˜ ˜ − = a(z )dz m m |x − z | |x − x0 | Rn ψt (y − z )(x − z )α ψt (y − x0 )(x − x0 )α 1 Dα A(x)a(z )dz, − − ˜ |x − z |m |x − x0 |m α! |α|=m Rn
Boundedness of Multilinear Littlewood-Paley Operators for the Extreme Cases 131 thus, similar to the proof of II in Theorem 1, we obtain ||FtA (a)(x, y )|| ˜ ||Dα A||BMO |B |1/n |x–x0 |–n–1+δ +|B |1/n |x–x0 |–n–1+δ |Dα A(x)| , ≤C ˜ |α|=m so that, n/(n−δ ) ∞ ||Dα A||BMO k 2−kn/(n−δ) ≤ C, JJ ≤ C k=1 |α|=m which together with the estimate for J yields the desired result. This ﬁnishes the proof of Theorem 2. Proof of Theorem 3. By the equality 1 (x − y )α (Dα A(x) − Dα A(y )) Rm+1 (A; x, y ) = Qm+1 (A; x, y ) + α! |α|=m and similar to the proof of Lemma 2, we get |Dα A(x) − Dα A(y )| A ˜A Sδ (f )(x) ≤ Sδ (f )(x) + C |f (y )|dy. |x − y |n−δ |α|=mRn By Theorems 1 and 2 with [1, 2], we obtain |{x ∈ Rn : Sδ (f )(x) > λ}| A ≤ |{x ∈ Rn : S A (f )(x) > λ/2}| ˜ δ |Dα A(x) − Dα A(y )| n x∈R : |f (y )|dy > Cλ + |x − y |n−δ |α|=mRn n/(n−δ ) ≤ C (||f ||H 1 /λ) . This completes the proof of Theorem 3. Proof of Theorem 4 (i). It suﬃces to show that there exists a constant C > 0 such that for every H 1 (w)-atom a with suppa ⊂ Q = Q(x0 , d), there is A ||Sδ (a)||Ln/(n−δ) ≤ C. 1 α α Let A(x) = A(x) − ˜ ˜ α! (D A)Q x , then Rm (A; x, y ) = Rm (A; x, y ) and |α|=m Dα A = Dα A − (Dα A)Q for all α with |α| = m. We write, by the vanishing ˜ moment of a and for u ∈ 3Q \ 2Q,
132 Liu Lanzhe FtA (a)(x, y ) = χ4Q (x)FtA (a)(x, y ) Rm (A; x, z )ψt (y − z ) Rm (A; x, u)ψt (y − u) ˜ ˜ − + χ(4Q)c (x) a(z )dz m |x − u|m |x − y | Rn ψt (y –z )(x–z )α ψt (y –u)(x–u)α α ˜ 1 − χ(4Q)c (x) – D A(z )a(z )dz |x–z |m |x–u|m α! |α|=m Rn (x − u)α 1 ψt (y − u)Dα A(z )a(z )dz, − χ(4Q)c (x) ˜ |x − u|m α! |α|=m Rn then Sδ (a)(x) = χΓ(x) (y, t)FtA (a)(x, y ) A ≤ i4Q (x) χΓ(x) (y, t)FtA (a)(x, y ) + χ(4Q)c (x) Rm (A; x, z )ψt (y − z ) Rm (A; x, u)ψt (y − u) ˜ ˜ × χΓ(x) (y, t) − a(z )dz m |x − u|m |x − z | Rn ψt (y − z )(x − z )α 1 + χ(4Q)c (x) χΓ(x) (y, t) |x − z |m α! |α|=m Rn α ψt (y − u)(x − u) Dα A(z )a(z )dz − ˜ |x − u|m (x–u)α 1 ψt (y –u)Dα A(z )a(z )dz ˜ + χ(4Q)c (x) χΓ(x) (y, t) |x–u|m α! |α|=m Rn = L1 (x) + L2 (x, u) + L3 (x, u) + L4 (x, u). By the (Lp , Lq )-boundedness of Sδ for n/(n − δ ) < q and 1/q = 1/p − δ/n (see A Lemma 2), we get A (n−δ )/n−1/q 1−1/p L1 (·) ≤ Sδ (a) Lq |4Q| ≤C a Lp |Q| ≤ C. Ln/(n−δ) Similar to the proof of Theorem 1, we obtain ≤ C and L3 (·, u) ≤ C. L2 Ln/(n−δ) Ln/(n−δ) Thus, using the condition of L4 (x, u), we obtain A ≤ C. Sδ (a) Ln/(n−δ) (ii). We write, for f = f χ4Q + f χ(4Q)c = f1 + f2 and u ∈ 3Q \ 2Q, ˜ Rm (A; x, z ) FtA (f )(x, y ) = FtA (f1 )(x, y ) + ˜ ˜ ψt (y − z )f2 (z )dz |x − z |m Rn ψt (y –z )(x–z )α ψt (u − z )(u − z )α 1 (Dα A(x)–(Dα A)Q ) − – f2 (z )dz |x–z |m |u − z |m α! |α|=m Rn (u − z )α 1 (Dα A(x) − (Dα A)Q ) − ψt (u − z )f2 (z )dz, |u − z |m α! |α|=m Rn
Boundedness of Multilinear Littlewood-Paley Operators for the Extreme Cases 133 then Rm (A; x0 , ·) ˜ ˜A Sδ (f )(x) − Sδ f2 (x0 ) |x0 − ·|m Rm (A; x0 , ·) ˜ χΓ(x) FtA (f )(x, y ) − χΓ(x0 ) Ft ˜ = f2 (y ) |x0 − ·|m Rm (A; x0 , ·) ˜ ≤ χΓ(x) (y, t)FtA (f )(x, y ) − χΓ(x0 ) (y, t)Ft ˜ f2 (y ) |x0 − ·|m ≤ χΓ(x) (y, t)FtA (f1 )(x, y ) ˜ ˜ Rm (A; x, z ) ψt (y − z ) + χΓ(x) (y, t) |x − z |m Rn ˜ Rm (A; x0 , z ) − χΓ(x0 ) (y, t) ψt (y − z ) f2 (z )dz |x0 − z |m Rn 1 (Dα A(x) − (Dα A)Q ) + χΓ(x) (y, t) α! |α|=m ψt (y − z )(x − z )α ψt (u − z )(u − z )α × − f2 (z )dz m |u − z |m |x − z | Rn (u − z )α 1 (Dα A(x) − (Dα A)Q ) ψt (u − z )f2 (z )dz + χΓ(x) (y, t) |u − z |m α! |α|=m Rn = M1 (x) + M2 (x) + M3 (x, u) + M4 (x, u). By the (Lp , Lq )-boundedness of Sδ for 1 < p < n/δ and 1/q = 1/p − δ/n, we ˜A get 1 M1 (x)dx ≤ |Q|−1/q ||Sδ (f1 )||Lq ≤ C |Q|−1/q ||f1 ||Lp ≤ C ||f ||Ln/δ . ˜A |Q| Q Similar to the proof of Theorem 1, we obtain 1 1 M2 (x)dx ≤ C ||f ||Ln/δ and M3 (x, u)dx ≤ C ||f ||Ln/δ . |Q| |Q| Q Q Thus, by using the condition of M4 (x, u), we obtain Rm (A; x0 , ·) ˜ 1 ˜A Sδ (f )(x) − Sδ f2 (x0 ) dx ≤ C ||f ||Ln/δ . |x0 − ·|m |Q| Q This completes the proof of Theorem 4. Acknowledgement. The author would like to express his gratitude to the referee for his comments and suggestions.
134 Liu Lanzhe References 1. J. Alvarez, R. J. Babgy, D. S. Kurtz, and C. Perez, Weighted estimates for com- mutators of linear operators, Studia Math. 104 (1993) 195–209. 2. S. Chanillo, A note on commutators, Indiana Univ. Math. J. 31 (1982) 7–16. 3. W. Chen and G. Hu, Weak type (H 1 , L1 ) estimate for multilinear singular integral operator, Adv. in Math. 30 (2001) 63–69 (Chinese). 4. J. Cohen, A sharp estimate for a multilinear singular integral on Rn , Indiana Univ. Math. J. 30 (1981) 693–702. 5. J. Cohen and J. Gosselin, On multilinear singular integral operators on Rn , Studia Math. 72 (1982) 199–223. 6. J. Cohen and J. Gosselin, A BMO estimate for multilinear singular integral op- erators, Illinois J. Math. 30 (1986) 445–465. 7. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976) 611–635. 8. Y. Ding and S. Z. Lu,Weighted boundedness for a class rough multilinear opera- tors, Acta Math. Sinica 17 (2001) 517–526. 9. J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. 16, Amsterdam, 1985. 10. E. Harboure, C. Segovia, and J. L.Torrea, Boundedness of commutators of frac- tional and singular integrals for the extreme values of p, Illinois J. Math. 41 (1997) 676–700. 11. L. Z. Liu , Weighted weak type estimates for commutators of Littlewood-Paley operator, Japanese J. Math. 29 (2003) 1–13. 12. L. Z. Liu, Weighted weak type (H 1 , L1 ) estimates for commutators of Littlewood- Paley operator, Indian J. Math. 45 71–78. 13. E. M. Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Os- cillatory Integrals, Princeton Univ. Press, Princeton NJ, 1993. 14. A. Torchinsky, The Real Variable Methods in Harmonic Analysis, Pure and Ap- plied Math. 123, Academic Press, New York, 1986.