Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator

Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator

In this paper, a sharp inequality for the vector-valued multilinear commutator of strongly singular integral operator are obtained. By using this inequality, we prove the weighted Lp-norm inequality for the multilinear commutator.

Gespeichert in:
Bibliographische Detailangaben
Datum:2008
Hauptverfasser: Zhang, H., Cai, H.
Format: Artikel
Sprache:English
Veröffentlicht: Інститут прикладної математики і механіки НАН України 2008
Schriftenreihe:Український математичний вісник
Online Zugang:http://dspace.nbuv.gov.ua/handle/123456789/124350
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Zitieren:Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator / H. Zhang, H. Cai // Український математичний вісник. — 2008. — Т. 5, № 3. — С. 406-419. — Бібліогр.: 13 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-124350
record_format dspace
spelling irk-123456789-1243502017-09-24T03:03:52Z Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator Zhang, H. Cai, H. In this paper, a sharp inequality for the vector-valued multilinear commutator of strongly singular integral operator are obtained. By using this inequality, we prove the weighted Lp-norm inequality for the multilinear commutator. 2008 Article Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator / H. Zhang, H. Cai // Український математичний вісник. — 2008. — Т. 5, № 3. — С. 406-419. — Бібліогр.: 13 назв. — англ. 1810-3200 2000 MSC. 42B20, 42B25. http://dspace.nbuv.gov.ua/handle/123456789/124350 en Український математичний вісник Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description In this paper, a sharp inequality for the vector-valued multilinear commutator of strongly singular integral operator are obtained. By using this inequality, we prove the weighted Lp-norm inequality for the multilinear commutator.
format Article
author Zhang, H.
Cai, H.
spellingShingle Zhang, H.
Cai, H.
Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator
Український математичний вісник
author_facet Zhang, H.
Cai, H.
author_sort Zhang, H.
title Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator
title_short Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator
title_full Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator
title_fullStr Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator
title_full_unstemmed Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator
title_sort weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator
publisher Інститут прикладної математики і механіки НАН України
publishDate 2008
url http://dspace.nbuv.gov.ua/handle/123456789/124350
citation_txt Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator / H. Zhang, H. Cai // Український математичний вісник. — 2008. — Т. 5, № 3. — С. 406-419. — Бібліогр.: 13 назв. — англ.
series Український математичний вісник
work_keys_str_mv AT zhangh weightedsharpinequalityforvectorvaluedmultilinearcommutatorofstronglysingularintegraloperator
AT caih weightedsharpinequalityforvectorvaluedmultilinearcommutatorofstronglysingularintegraloperator
first_indexed 2025-07-09T01:18:20Z
last_indexed 2025-07-09T01:18:20Z
_version_ 1837130214341083136
fulltext Український математичний вiсник Том 5 (2008), № 3, 406 – 419 Weighted sharp inequality for vector-valued multilinear commutator of strongly singular integral operator Hongwei Zhang, Haitao Cai (Presented by M. M. Malamud) Abstract. In this paper, a sharp inequality for the vector-valued mul- tilinear commutator of strongly singular integral operator are obtained. By using this inequality, we prove the weighted L p-norm inequality for the multilinear commutator. 2000 MSC. 42B20, 42B25. Key words and phrases. Strongly singular integral operator, Multi- linear Commutator; Sharp inequality, BMO, Ap-weight. 1. Introduction and Theorems As the development of Calderón–Zygmund singular integral opera- tors, the commutators of the singular integral operators have been well studied (see [7–11]). In this paper, we study the vector-valued multilin- ear commutator of strongly singular integral operator which is defined as following. Let 0 < b < 1 and θ(ξ) be a smooth radial cut-off function on R n such that θ(ξ) = 1 if |ξ| ≥ 1 and θ(ξ) = 0 if |ξ| ≤ 1/2. The strongly singular integral operator is a multiplier operator which is defined by (T (f))̂(ξ) = θ(ξ) ei|ξ|b |ξ|nb/2 f̂(ξ). The kernel K for T is very singular. Roughly speaking, it looks like K(x) = ϑ(x)ei|x|−b′ /|x|n, where b′ = b/(1 − b) and supp ϑ ⊂ {x ∈ R n : |x| ≤ 2}. In fact, we know that Received 22.06.2008 ISSN 1810 – 3200. c© Iнститут математики НАН України H. Zhang, H. Cai 407 T (f)(x) = p.v. ∫ Rn K(x − y)f(y) dy and for |x| ≥ 2|y| (see [1]) |K(x − y) − K(x)| ≤ C|y||x|−n−b′−1. Let bj(j = 1, . . . , m) be the fixed locally integrable functions on R n. For 1 < s < ∞, the vector-valued multilinear commutator related to T is defined by |Tb̃(f)(x)|s = ( ∞ ∑ i=1 |Tb̃(fi)(x)|s )1/s , where Tb̃(fi)(x) = p.v. ∫ Rn ( m ∏ j=1 (bj(x) − bj(y)) ) K(x − y)fi(y) dy. We also denote |T (f)(x)|s = ( ∞ ∑ i=1 |T (fi)(x)|s )1/s and |f(x)|s = ( ∞ ∑ i=1 |fi(x)|s )1/s . The strongly singular integral operator has been studied by several authors (see [1, 4, 5, 13]). In [2], the weighted norm inequality for the commutator of strongly singular integral operator is obtained by using a sharp estimate. Note that when b1 = · · · = bm, |Tb̃|s is just the m order vector-valued commutator (see, for example, [6, 10] and [11]). In [11], Perez and Trujillo–Gonzalez prove a sharp estimate for the vector-valued commutator of Calderon–Zygmund singular integral operator. The main purpose of this paper is to prove a sharp inequality for the vector-valued multilinear commutator of strongly singular integral operator. As an ap- plication, we obtain the weighted Lp-norm inequality for the multilinear commutator. First, let us introduce some notations. Throughout this paper, Q = Q(x, r) will denote a cube of Rn centered at x and sidelength r with sides parallel to the axes. For any locally integrable function f , the sharp function of f is defined by 408 Weighted sharp inequality... f#(x) = sup Q∋x 1 |Q| ∫ Q |f(y) − fQ| dy, where, and in what follows, fQ = |Q|−1 ∫ Q f(x) dx. It is well-known that (see, for example, [3] and [12]) f#(x) ∼= sup Q∋x inf c∈C 1 |Q| ∫ Q |f(y) − c| dy. We say that f belongs to BMO(Rn) if f# belongs to L∞(Rn) and ‖f‖BMO = ‖f#‖L∞ . Given the functions bj(j = 1, . . . , m) and 0 < q < ∞, denote that Cq b̃ (f)(x) = sup Q∋x 1 |Q| ∫ Q m ∏ j=1 |bj(x) − bj(y)|q|f(y)| dy. We write Cq b̃ (f) = Cb̃(f) if q = 1. Let M be the Hardy–Littlewood maximal operator defined by M(f)(x) = sup Q∋x |Q|−1 ∫ Q |f(y)| dy. We write Mp(f) = (M(fp))1/p for 0 < p < ∞. Given a positive in- teger m and 1 ≤ j ≤ m, we denote by Cm j the family of all finite subsets σ = {σ(1), . . . , σ(j)} of {1, . . . , m} of j different elements. For σ ∈ Cm j , denote that σc = {1, . . . , m} \ σ; For b̃ = (b1, . . . , bm) and σ = {σ(1), . . . , σ(j)} ∈ Cm j , denote b̃σ = (bσ(1), . . . , bσ(j)), bσ = bσ(1) . . . bσ(j) ‖b̃‖BMO = ‖b1‖BMO . . . ‖bm‖BMO and ‖b̃σ‖BMO = ‖bσ(1)‖BMO · · · × ‖bσ(j)‖BMO. We denote the Muckenhoupt weights by Ap for 1 ≤ p < ∞ (see [3]), that is Ap = { w : sup Q ( 1 |Q| ∫ Q w(x) dx )( 1 |Q| ∫ Q w(x)−1/(p−1) dx )p−1 < ∞ } , 1 < p < ∞, and H. Zhang, H. Cai 409 A1 = {w : M(w)(x) ≤ Cw(x), a.e.}. We shall prove the following theorems. Theorem 1.1. Let 1 < s < ∞, bj ∈ BMO(Rn) for j = 1, . . . , m. Then there exists a constant C > 0 such that for any f ∈ C∞ 0 (Rn), 1 < r < ∞ and x̃ ∈ R n, (|Tb̃(f)|s) #(x̃) ≤ C ( ‖b̃‖BMOMr(|f |s)(x̃) + M((Cs b̃ (|f |rs)) 1/r)(x̃) + m ∑ j=1 ∑ σ∈Cm j ‖b̃σ‖BMOMr(|Tb̃σc (f)|s)(x̃) ) . Theorem 1.2. Let 1 < s < ∞, bj ∈ BMO(Rn) for j = 1, . . . , m and w ∈ Ap, 1 < p < ∞. Then |Tb̃|s is bounded on Lp(w), that is ‖ |Tb̃(f)|s‖Lp(w) ≤ C‖ |f |s‖Lp(w). 2. Proof of the main results To prove the theorems, we need the following lemmas. Lemma 2.1 ([1]). Let T be the strongly singular integral operator and 1 < s < ∞. Then |T |s is bounded on Lp(w) for w ∈ Ap and 1 < p < ∞. Lemma 2.2 ([1]). Let 0 < b < 1, b′ = b/(1−b), 1 < p < ∞, 1/p+1/p′ = 1, (2 + b′)/p ≤ 1, 1 < s < ∞ and K̃(x) = ei|x|−b′ |x|n(b′+2)/p . Then ‖K̃ ∗ |f |s‖Lp ≤ C‖ |f |s‖Lp′ . Lemma 2.3 ( [3]). Suppose that 1 < s < ∞, 1 ≤ r < p < ∞ and w ∈ Ap. Then ‖Mr(|f |s)‖Lp(w) ≤ C‖ |f |s‖Lp(w). Lemma 2.4 ([2]). Suppose that bj ∈ BMO(Rn)(j = 1, . . . , m), 1 < s < ∞, 1 ≤ q < ∞, 1 < p < ∞ and w ∈ Ap. Then ‖Cq b̃ (|f |s)‖Lp(w) ≤ C‖ |f |s‖Lp(w). 410 Weighted sharp inequality... The Proofs of Lemma 2.1, 2.3 and 2.4 is similar to the proofs in [1–3], we omit the detail. Proof of Theorem 1.1. It suffices to prove for f ∈ C∞ 0 (Rn) and some constant C0, the following inequality holds: 1 |Q| ∫ Q | |Tb̃(f)(x)|s − C0| dx ≤ C ( ‖b̃‖BMOMr(|f |s)(x̃) + M((Cr b̃ (|f |rs)) 1/r)(x̃) + m ∑ j=1 ∑ σ∈Cm j ‖b̃σ‖BMOMr(|Tb̃σc (f)|s)(x̃) ) . Fix a cube Q = Q(x0, d) such that x̃ ∈ Q. Let d0 be a positive number satisfying 4d0 = d 1/(1+b′) 0 and Q̃ = Q(x0, d 1/(1+b′)). Let us first consider the case m = 1. In this time, we will prove the following inequality 1 |Q| ∫ Q | |Tb̃(f)(x)|s − C0| dx ≤ C ( Mr(|f |s)(x̃) + M((Cr b̃ (|f |rs)) 1/r)(x̃) + Mr(|T (f)|s)(x̃) ) . Consider the following two cases: Case 1. d < d0. For each i ∈ N , we split fi = f1 i + f2 i + f3 i , where f1 i = fiχ4Q, f2 i = fiχQ̃\4Q and f3 i = fiχRn\Q̃ and we define f (j) = {f j i } for j = 1, 2, 3. By Minkowski’ inequality and taking C0 = |Tb̃(f (3))(x0)|s, we have 1 |Q| ∫ Q | |Tb̃(f)(x)|s − |Tb̃(f (3))(x0)|s| dx ≤ 1 |Q| ∫ Q ( ∞ ∑ i=1 |Tb̃(fi)(x) − Tb̃(f 3 i )(x0)| s )1/s dx ≤ 1 |Q| ∫ Q |b(x) − b2Q| |T (f)(x)|s dx + 1 |Q| ∫ Q |T ((b − b2Q)f (1))(x)|s dx H. Zhang, H. Cai 411 + 1 |Q| ∫ Q |T ((b − b2Q)f (2))(x)|s dx + 1 |Q| ∫ Q |T ((b − b2Q)f (3))(x) − T ((b − b2Q)f (3))(x0)|s dx = I1 + I2 + I3 + I4. Let us now estimate I1, I2, I3 and I4, respectively. By applying Hölder’s inequality and Lemma 2.1, we get I1 ≤ C ( 1 |Q| ∫ Q |T (f)(x)|rs dx )1/r( 1 |2Q| ∫ 2Q |b(x) − b2Q| r′ dx )1/r′ ≤ C‖b‖BMOMr(|T (f)|s)(x̃). Now we proceed to estimate I2. If 1 < p < r, from Lemma 2.1, we obtain I2 ≤ C ( 1 |Q| ∫ Rn |T ((b − b2Q)f (1)(x))(x)|ps dx )1/p ≤ ( 1 |Q| ∫ Rn |b(x) − b2Q| p|f (1)(x)|ps dx )1/p ≤ C ( 1 |2Q| ∫ 2Q |f(x)|rs dx )1/r( 1 |2Q| ∫ 2Q |b(x) − b2Q| pr/(r−p) dx )(r−p)/pr ≤ C‖b‖BMOMr(|f |s)(x̃). To estimate I3, we follow Chanillo’s argument (see [1]). Then we choose r > 1 such that (2 + b′)/r′ < 1 to get T ((b − b2Q)f2 i (x) := ∫ Rn ϑ(x − y)ei|x−y|−b′ |x − y|n(2+b′)/r′ ( 1 |x − y|n(1−(2+b′)/r′) − 1 |x0 − y|n(1−(2+b)/r′) ) (b(y) − b2Q)f2 i (y) dy + ∫ Rn ϑ(x − y)ei|x−y|−b′ |x − y|n(2+b′)/r′ · 1 |x0 − y|n(1−(2+b′)/r′) (b(y) − b2Q)f2 i (y) dy 412 Weighted sharp inequality... = I (1) 3 (x) + I (2) 3 (x). Note that |b2Q − b2k+1Q| ≤ k‖b‖BMO, then, by Minkowski’ inequality, ( ∞ ∑ i=1 ∣ ∣I (1) 3 (x) ∣ ∣ s )1/s ≤ C ∞ ∑ k=1 2−k 1 |2k+1Q| ∫ 2k+1Q |b(y) − b2Q‖f(y)|s dy ≤ C ∞ ∑ k=1 2−k ( 1 |2k+1Q| ∫ 2k+1Q |f(x)|rs dx )1/r × ( 1 |2k+1Q| ∫ 2k+1Q |b(x) − b2Q| r′ dx )1/r′ ≤ C‖b‖BMO ∞ ∑ k=1 k2−k ( 1 |2k+1Q| ∫ 2k+1Q |f(x)|rs dx )1/r ≤ C‖b‖BMOMr(|f |s)(x̃). Taking k0 such that 2k0d < d1/(1+b′) ≤ 2k0+1d, by Lemma 2.2 and Minkowski’ inequality, we get 1 Q ∫ Q ( ∞ ∑ i=1 ∣ ∣I (2) 3 (x) ∣ ∣ s )1/s dx ≤ C|Q|−1/r′ ( ∫ Rn |b(y) − b2Q| r|f (2)(y)|rs |x0 − y|nr(1−(2+b′)/r′) dy )1/r ≤ C|Q|−1/r′ ( k0 ∑ k=1 (2kd)n(r−1)(1+b′) 1 |2k+1Q| × ∫ 2k+1Q |f(y)|rs|b(y) − b2Q| r dy )1/r ≤ C|Q|−1/r′ ( k0 ∑ k=1 (2kd)n(r−1)(1+b′) [ 1 |2Q| ∫ 2Q ( 1 |2k+1Q| H. Zhang, H. Cai 413 × ∫ 2k+1Q |f(y)|rs|b(y) − b(z)|r dy )1/r dz ]r)1/r ≤ CM((Cr b̃ (|f |rs)) 1/r)(x̃). Thus I3 ≤ C ( ‖b‖BMOMr(|f |s)(x̃) + M((Cr b (|f |rs)) 1/r)(x̃) ) . For I4, note that |x− y| ≈ |x0 − y| for x ∈ Q and y ∈ Rn \ Q̃, we obtain, by the condition on K, I4 ≤ C 1 |Q| ∫ Q ∫ Rn |x − x0| |x0 − y|n+b′+1 |f (3)(y)|s|b(y) − b2Q| dy dx ≤ C 1 |Q| ∫ Q ∞ ∑ k=0 ∫ 2k+1Q̃\2kQ̃ |x − x0| |x0 − y|n+b′+1 |f(y)|s|b(y) − b2Q| dy dx ≤ C ∞ ∑ k=0 2−k 1 |2kQ̃| ∫ 2k+1Q̃\2kQ̃ |f(y)|s|b(y) − b2Q| dy ≤ C ∞ ∑ k=1 2−k 1 |2Q| ∫ 2Q ( 1 |2kQ̃| ∫ 2kQ̃ |f(y)|s|b(y) − b(z)| dy ) dz ≤ CM(Cb̃(|f |s))(x̃). Case 2. d ≥ d0. We do not subtract the constant C0. Let l = 4d−1 0 and f (4) i = fiχlQ, by the location of the support of K, we have Tb̃(fi) = Tb̃(f (4) i ), thus 1 |Q| ∫ Q |Tb̃(f)(x)|s dx ≤ 1 |Q| ∫ Q |b(x) − blQ| |T (f (4))(x)|s dx + 1 |Q| ∫ Q |T ((b − blQ)f (4))(x)|s dx = J1 + J2. Similar to the proof of I1 and I2 for case 1, we get J1 ≤ C ( 1 |Q| ∫ Q |T (f (4))(x)|rs dx )1/r( 1 |lQ| ∫ Q |b(x) − blQ| r′ dx )1/r′ 414 Weighted sharp inequality... ≤ C ( 1 |lQ| ∫ lQ |f(x)|rs dx )1/r( 1 |lQ| ∫ Q |b(x) − blQ| r′ dx )1/r′ ≤ C‖b‖BMOMr(|f |s)(x̃); J2 ≤ C ( 1 |Q| ∫ Rn |T ((b − blQ)f (4))(x)|ps dx )1/p ≤ C ( 1 |Q| ∫ Rn |b(x) − blQ| p|f (4)(x)|ps dx )1/p ≤ C ( 1 |lQ| ∫ lQ |f(x)|rs dx )1/r( 1 |lQ| ∫ lQ |b(x) − blQ| pr/(r−p) dx )(r−p)/pr ≤ C‖b‖BMOMr(|f |s)(x̃), which proves the case 1. Now we turn to the case m ≥ 2. Also consider the following two cases: Case 1. d < d0. Following [10], we write Tb̃(fi)(x) = ∫ Rn ( m ∏ j=1 (bj(x) − bj(y)) ) K(x − y)fi(y) dy = (b1(x) − (b1)2Q) · · · (bm(x) − (bm)2Q)T (fi)(x) + (−1)mT ((b1 − (b1)2Q) · · · (bm − (bm)2Q)fi)(x) + m−1 ∑ j=1 ∑ σ∈Cm j (−1)m−j(b(x) − (b)2Q)σ ∫ Rn (b(y) − b(x))σcK(x − y)fi(y) dy = (b1(x) − (b1)2Q) · · · (bm(x) − (bm)2Q)T (fi)(x) + (−1)mT ((b1 − (b1)2Q) · · · (bm − (bm)2Q)fi)(x) + m−1 ∑ j=1 ∑ σ∈Cm j cm,j(b(x) − (b)2Q)σTb̃σc (fi)(x), thus, for f (j) = {f j i } for j = 1, 2, 3 with f1 i = fiχ4Q, f2 i = fiχQ̃\4Q and f3 i = fiχRn\Q̃, H. Zhang, H. Cai 415 1 |Q| ∫ Q ||Tb̃(f)(x)|s − |T ((b1 − (b1)2Q) · · · (bm − (bm)2Q))f (3))(x0)|s dx ≤ 1 |Q| ∫ Q |(b1(x) − (b1)2Q) · · · (bm(x) − (bm)2Q)T (f)(x)|s dx + 1 |Q| ∫ Q m−1 ∑ j=1 ∑ σ∈Cm j |(b(x) − (b)2Q)σTb̃σc (f)(x)|s dx + 1 |Q| ∫ Q |T ((b1 − (b1)2Q) · · · (bm − (bm)2Q)f (1))(x)|s dx + 1 |Q| ∫ Q |T ((b1 − (b1)2Q) · · · (bm − (bm)2Q)f (2))(x)|s dx + 1 |Q| ∫ Q |T ((b1 − (b1)2Q) · · · (bm − (bm)2Q)f (3))(x) − T ((b1 − (b1)2Q) · · · (bm − (bm)2Q)f (3))(x0)|s dx = L1 + L2 + L3 + L4 + L5. Similar to the proof of m = 1, we get, for 1 < p1, . . . , pm < ∞, 1 < p < r, 1 < q1, . . . , qm < ∞, 1/r + 1/p1 + · · ·+ 1/pm = 1 and p/r + 1/q1 + · · ·+ 1/qm = 1, L1 ≤ C ( 1 |Q| ∫ Q |T (f)(x)|rs dx )1/r( 1 |Q| ∫ Q |b1(x) − (b1)2Q| p1 dx )1/p1 × · · · × ( 1 |2Q| ∫ 2Q |bm(x) − (bm)2Q| pm dx )1/pm ≤ C m ∏ j=1 ‖bj‖BMOMr(|T (f)|s)(x̃); L2 ≤ C m−1 ∑ j=1 ∑ σ∈Cm j ‖b̃σ‖BMOMr(|Tb̃σc (f)|s)(x̃); L3 ≤ C ( 1 |Q| ∫ Rn |T ((b1 − (b1)2Q) · · · (bm − (bm)2Q)f (1))(x)|ps dx )1/p 416 Weighted sharp inequality... ≤ C ( 1 |Q| ∫ Rn (|b1(x) − (b1)2Q| · · · |bm(x) − (bm)2Q| |f (1)(x)|)p dx )1/p ≤ C ( 1 |2Q| ∫ 2Q |f(x)|rs dx )1/r( 1 |2Q| ∫ 2Q |b1(x) − (b1)2Q| pq1 dx )1/pq1 × · · · × ( 1 |2Q| ∫ 2Q |bm(x) − (bm)2Q| pqm dx )1/pqm ≤ C m ∏ j=1 ‖bj‖BMOMr(|f |s)(x̃). Similarly, for L4, we get, for 1 < p1, . . . , pm < ∞ and 1/r + 1/p1 + · · · + 1/pm = 1, L4 ≤ 1 Q ∫ Q ∫ Rn |ϑ(x − y)|ei|x−y|−b′ |x − y|n(2+b′)/r′ × ∣ ∣ ∣ 1 |x − y|n(1−(2+b′)/r′) − 1 |x0 − y|n(1−(2+b)/r′) ∣ ∣ ∣ × m ∏ j=1 |bj(y) − (bj)2Q| |f (2)(y)|s dy dx + 1 Q ∫ Q ∫ Rn |ϑ(x − y)|ei|x−y|−b′ |x − y|n(2+b′)/r′ · 1 |x0 − y|n(1−(2+b′)/r′) × m ∏ j=1 |bj(y) − (bj)2Q| |f (2)(y)|s dy dx ≤ C ∞ ∑ k=1 2−k 1 |2k+1Q| ∫ 2k+1Q m ∏ j=1 |bj(y) − (bj)2Q||f(y)|s dy + C|Q|−1/r′ ( ∫ Rn ∏m j=1 |bj(y) − (bj)2Q| r|f (2)(y)|rs |x0 − y|nr(1−(2+b′)/r′) dy )1/r ≤ C ∞ ∑ k=1 2−k ( 1 |2k+1Q| ∫ 2k+1Q |f(x)|rs dx )1/r H. Zhang, H. Cai 417 × m ∏ j=1 ( 1 |2k+1Q| ∫ 2k+1Q |bj(x) − (bj)2Q| p′j dx )1/p′j + C|Q|−1/r′ ( k0 ∑ k=1 (2kd)n(r−1)(1+b′) 1 |2k+1Q| × ∫ 2k+1Q |f(y)|rs m ∏ j=1 |bj(y) − (bj)2Q| r dy )1/r ≤ C m ∏ j=1 ‖bj‖BMO ∞ ∑ k=1 km2−k ( 1 |2k+1Q| ∫ 2k+1Q |f(x)|rs dx )1/r + C|Q|−1/r′ ( k0 ∑ k=1 (2kd)n(r−1)(1+b′) [ 1 |2Q| ∫ 2Q ( 1 |2k+1Q| × ∫ 2k+1Q |f(y)|rs m ∏ j=1 |bj(y) − (bj)(z)|r dy )1/r dz ]r)1/r ≤ C ( m ∏ j=1 ‖bj‖BMOMr(|f |s)(x̃) + M((Cr b̃ (|f |rs)) 1/r)(x̃) ) . For L5, we get L5 ≤ C 1 |Q| ∫ Q ∫ Rn |x − x0| |x0 − y|n+b′+1 |f (3)(y)|s m ∏ j=1 |bj(y) − (bj)2Q| dy dx ≤ C 1 |Q| ∫ Q ∞ ∑ k=0 ∫ 2k+1Q̃\2kQ̃ |x − x0| |x0 − y|n+b′+1 |f(y)|s m ∏ j=1 |bj(y) − (bj)2Q| dy ≤ C ∞ ∑ k=0 2−k 1 |2kQ̃| ∫ 2k+1Q̃\2kQ̃ |f(y)|s m ∏ j=1 |bj(y) − (bj)2Q| dy ≤ C ∞ ∑ k=1 2−k 1 |2Q| ∫ 2Q ( 1 |2kQ̃| ∫ 2kQ̃ |f(y)|s m ∏ j=1 |bj(y) − bj(z)| dy ) dz ≤ CM(Cb̃(|f |s))(x̃). 418 Weighted sharp inequality... Similarly, for case 2 d ≥ d0, we get 1 |Q| ∫ Q |Tb̃(f)(x)|s dx ≤ C m ∏ j=1 ‖bj‖BMOMr(|f |s)(x̃) + C m−1 ∑ j=1 ∑ σ∈Cm j ‖b̃σ‖BMOMr(|Tb̃σc (f)|s)(x̃). These complete the proof of Theorem 1.1. Proof of Theorem 1.2. We choose 1 < r < p in Theorem 1.1, and by using Lemma 2.3, 2.4 and induction on m, we get ‖|Tb̃(f)|s‖Lp(w) ≤ C‖(|Tb̃(f)|s) #‖Lp(w) ≤ C‖|f |s‖Lp(w). This finishes the proof. Remark 2.1. The Theorem 1.1 and 1.2 also hold for bj ∈Oscexp Lrj (Rn), where Oscexp Lrj (Rn) is defined in [10]. It is obvious that Oscexp Lr(Rn) ⊂ BMO(Rn) and Oscexp Lr(Rn) coincides with the BMO(Rn) space if r=1. References [1] S. Chanillo, Weighted norm inequalities for strongly singular convolution opera- tors // Trans. Ammer. Math. Soc., 281 (1984), 77–107. [2] J. Garcia-Cuerva, E. Harboure, C. Segovia and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J., 40 (1991), 1397–1420. [3] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. 16, Amsterdam, 1985. [4] C. Feffeman, Inequalities for strongly singular convolution operators, Acta Math., 124 (1970), 9–36. [5] C. Feffeman and E. M. Stein, H p spaces of several variables // Acta Math., 129 (1972), 137–193. [6] O. Gorosito, G. Pradolini and O. Salinas, Weighted weak type estimates for mul- tilinear commutators of fractional integral on spaces of homogeneous type, Acta Math. Sinica, 15 (2005), 1–11. [7] C. Perez, Endpoint estimate for commutators of singular integral operators // J. Func. Anal., 128 (1995), 163–185. [8] C. Perez, Sharp estimates for commutators of singular integrals via iterations of the Hardy–Littlewood maximal function, J. Fourier Anal. Appl., 3 (1997), 743– 756. [9] C. Perez and G. Pradolini, Sharp weighted endpoint estimates for commutators of singular integral operators // Michigan Math. J., 49 (2001), 23–37. H. Zhang, H. Cai 419 [10] C. Perez and R. Trujillo-Gonzalez, Sharp weighted estimates for multilinear com- mutators // J. London Math. Soc., 65 (2002), 672–692. [11] C. Perez and R. Trujillo-Gonzalez, Sharp weighted estimates for vector-valued singular integral operators and commutators // Tohoku Math. J., 55 (2003), 109– 129. [12] E. M. Stein, Harmonic Analysis: real variable methods, orthogonality and oscil- latory integrals, Princeton Univ. Press, Princeton NJ, 1993. [13] S. Wainger, Special trigonometric series in k-dimensions // Mem. Amer. Math. Soc., 59 (1965). Contact information Hongwei Zhang, Haitao Cai School of Mathematical Sciences and Computing Technology Central South University Changsha University of Science and Technology Changsha, P.R. of China China E-Mail: zhouxiaosha57@126.com

Ähnliche Einträge