Weighted sharp maximal function inequalities and boundedness of multilinear singular integral operator with variable Calderón-Zygmund kernel

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1 Fuctioal Aalysis, Approximatio ad omputatio 7 3) 205), Published by Faculty of Scieces ad Mathematics, Uiversity of Niš, Serbia Available at: Weighted sharp maximal fuctio iequalities ad boudedess of multiliear sigular itegral operator with variable alderó-zygmud kerel iog he a a Shuyag Rudog Seior High School, Shuyag, Jiagsu Provice, , P. R. of hia Abstract. I this paper, we establish the weighted sharp maximal fuctio iequalities for the multiliear operator associated to the sigular itegral operator with variable alderó-zygmud kerel. As a applicatio, we obtai the boudedess of the operator o weighted Lebesgue spaces.. Itroductio As the developmet of sigular itegral operatorssee [9][20][2]), their commutators ad multiliear operators have bee well studied. I [6][8][9], the authors proved that the commutators geerated by the sigular itegral operators ad BMO fuctios are bouded o L p R ) for < p <. haillo see [3]) proved a similar result whe sigular itegral operators are replaced by the fractioal itegral operators. I [2][7], the boudedess for the commutators geerated by the sigular itegral operators ad Lipschitz fuctios o Triebel-Lizorki ad L p R ) < p < ) spaces are obtaied. I [][], the boudedess for the commutators geerated by the sigular itegral operators ad the weighted BMO ad Lipschitz fuctios o L p R ) < p < ) spaces are obtaied also see [0]). I [4][5], the authors studied some multiliear sigular itegral operators as followig also see [7]): T b R m+b; x, y) f )x) = R x y m Kx, y) f y)dy, ad obtaied some variat sharp fuctio estimates ad boudedess of the multiliear operators if D α b BMOR ) for all α with α = m. I [2], alderó ad Zygmud itroduced some sigular itegral operators with variable kerel ad discussed their boudedess. I [3-5][22], the authors obtaied the boudedess for the commutators ad multiliear operators geerated by the sigular itegral operators with variable kerel ad BMO fuctios. I [6], the authors prove the boudedess for the multiliear oscillatory sigular itegral operators geerated by the operators ad BMO fuctios. Motivated by these, i this paper, we will study the multiliear operator geerated by the sigular itegral operator with 200 Mathematics Subject lassificatio. 42B20, 42B25. Keywords. Multiliear operator; Sigular itegral operator; Variable alderó-zygmud Kerel; Sharp maximal fuctio; Weighted BMO; Weighted Lipschitz fuctio. Received: 26 February 205; Accepted: 9 April 205. ommuicated by Draga S. Djordjević address: @qq.com iog he)

2 iog he / FAA 7 3) 205), variable alderó-zygmud kerel ad the weighted Lipschitz ad BMO fuctios, that is D α b BMOw) or D α b Lip β w) for all α with α = m. 2. Prelimiaries I this paper, we will study some sigular itegral operators as followig see [2]). Defiitio. Let Kx) = Ωx)/ x : R \ {0} R. K is said to be a alderó-zygmud kerel if a) Ω R \ {0}); b) Ω is homogeeous of degree zero; c) Σ Ωx)xα dσx) = 0 for all multi-idices α N {0}) with α = N, where Σ = {x R : x = } is the uit sphere of R. Defiitio 2. Let Kx, y) = Ωx, y)/ y : R R \{0}) R. K is said to be a variable alderó-zygmud kerel if d) Kx, ) is a alderó-zygmud kerel for a.e. x R ; e) max γ γ 2 γ y Ωx, y) L = L <. R Σ) Moreover, let m be the positive iteger ad b be the fuctio o R. Set R m+ b; x, y) = bx) α m α! Dα by)x y) α. Let T be the sigular itegral operator with variable alderó-zygmud kerel as T f )x) = Kx, x y) f y)dy, R where Kx, x y) = Ωx,x y) x y ad that Ωx, y)/ y is a variable alderó-zygmud kerel. The multiliear operator related to the operator T is defied by T b R m+b; x, y) f )x) = R x y m Kx, x y) f y)dy. Note that the commutator [b, T] f ) = bt f ) Tb f ) is a particular operator of the multiliear operator T b if m = 0. The multiliear operator T b are the o-trivial geeralizatios of the commutator. It is well kow that commutators ad multiliear operators are of great iterest i harmoic aalysis ad have bee widely studied by may authors see [4][5][7]). The mai purpose of this paper is to prove the sharp maximal iequalities for the multiliear operator T b. As the applicatio, we obtai the weighted L p -boudedess for the multiliear operator T b. Now, let us itroduce some otatios. Throughout this paper, will deote a cube of R with sides parallel to the axes. For a oegative itegrable fuctio ω, let ω) = ωx)dx ad ω = ωx)dx. For ay locally itegrable fuctio f, the sharp maximal fuctio of f is defied by It is well-kow that see [9]) Let M # f )x) = sup x f y) f dy. M # f )x) sup if f y) c dy. x c M f )x) = sup x f y) dy. For η > 0, let M # η f )x) = M # f η ) /η x) ad M η f )x) = M f η ) /η x).

3 ad ad iog he / FAA 7 3) 205), For 0 < η <, p < ad the o-egative weight fuctio ω, set M η,p,ω f )x) = sup x The A p weight is defied by see [9]) A p = M ω f )x) = sup x ω) pη/ ) 0 < ω L loc R ) : sup ωx)dx f y) p ωy)dy f y) ωy)dy). ω) A = {0 < ω L p loc R ) : Mω)x) wx), a.e.}. ) /p p ωx) dx) /p ) <, < p <, Give a o-egative weight fuctio ω. For p <, the weighted Lebesgue space L p R, ω) is the space of fuctios f such that ) /p f L p ω) = f x) p ωx)dx <. R Give the o-egative weight fuctio ω. The weighted BMO space BMOω) is the space of fuctios b such that b BMOω) = sup by) b dy <. ω) For 0 < β <, the weighted Lipschitz space Lip β ω) is the space of fuctios b such that b Lipβ ω) = sup ω) β/ ω) /p by) b p ωx) dy) p <. Remark.). It has bee kow thatsee [8]), for b Lip β ω), ω A ad x, b b 2 k k b Lipβ ω)ωx)ω2 k ) β/. 2). Let b Lip β ω) ad ω A. By [8], we kow that spaces Lip β ω) coicide ad the orms b Lipβ ω) are equivalet with respect to differet values p <. We give some Prelimiary lemmas. Lemma.[9, p.485]) Let 0 < p < q < ad for ay fuctio f 0. We defie that, for /r = /p /q, f WL q = sup λ {x R : f x) > λ} /q, λ>0 N p,q f ) = sup f χ L p/ χ L r, where the sup is take for all measurable sets with 0 < <. The f WL q N p,q f ) q/q p)) /p f WL q. Lemma 2.see [2]) Let T be the sigular itegral operator as Defiitio 2. The T is bouded o L p R, ω) for ω A p with < p <, ad weak L, L ) bouded. Lemma 3.see []) Let b BMOω). The where ω j = max ij 2 i ωx)dx. 2 i b b 2 j j b BMOω) ω j,

4 iog he / FAA 7 3) 205), Lemma 4.see []) Let ω A p, < p <. The there exists ε > 0 such that ω r/p A r for ay p r p + ε. Lemma 5.see []) Let b BMOω), ω = µν ) /p, µ, ν A p ad p >. The there exists ε > 0 such that for p r p + ε, bx) b r µx) r/p dx b r BMOω) νx) r/p dx. Lemma 6.see []) Let ω A p, < p <. The there exists 0 < δ < such that ω r /p A p/r dµ) for ay p < r < p + δ), where dµ = ω r /p dx. Lemma 7.see []) Let µ, ν A p, ω = µν ) /p, < p <. The there exists < q < p such that ) /q ω ν ) /q ωx) q νx) q /q dx. Lemma 8.see [3][9]) Let 0 η <, s < p < /η, /q = /p η/ ad ω A. The M η,s,ω f ) L q ω) f L p ω). Lemma 9.see [9]). Let 0 < p, η < ad ω r< A r. The, for ay smooth fuctio f for which the left-had side is fiite, M η f )x) p ωx)dx R M # η f )x) p ωx)dx. R Lemma 0.see [5]) Let b be a fuctio o R ad D α A L s R ) for all α with α = m ad ay s >. The R m b; x, y) x y m x, y) where is the cube cetered at x ad havig side legth 5 x y. 3. Theorems ad Proofs /s D α bz) dz) s, x,y) We shall prove the followig theorems. Theorem. Let T be the sigular itegral operator as Defiitio 2, < p <, µ, ν A p, ω = µν ) /p, 0 < η < ad D α b BMOω) for all α with α = m. The there exists a costat > 0, ε > 0, 0 < δ <, < q < p ad p < r < mip + ε, p + δ)) such that, for ay f 0 R ) ad x R, M # ηt b f )) x) D α b BMOω) [M ν ωt f ) q ) x)] /q + [M ν r /p ω f r ) x)] /r + [M ν ω f q ) x)] /q). Theorem 2. Let T be the sigular itegral operator as Defiitio 2, ω A, 0 < η <, < r <, 0 < β < ad D α b Lip β ω) for all α with α = m. The there exists a costat > 0 such that, for ay f 0 R ) ad x R, M # ηt b f )) x) D α b Lipβ ω)ω x)m β,r,ω f ) x). Theorem 3. Let T be the sigular itegral operator as Defiitio 2, < p <, µ, ν A p, ω = µν ) /p ad D α b BMOω) for all α with α = m. The T b is bouded from L p R, µ) to L p R, ν). Theorem 4. Let T be the sigular itegral operator as Defiitio 2, ω A, 0 < β <, < p < /β, /q = /p β/ ad D α b Lip β ω) for all α with α = m. The T b is bouded from L p R, ω) to L q R, ω q ). orollary. Let [b, T] f ) = bt f ) Tb f ) be the commutator geerated by the sigular itegral operator T as Defiitio 2 ad b. The Theorems -4 hold for [b, T].

5 iog he / FAA 7 3) 205), Proof of Theorem. It suffices to prove for f 0 R ) ad some costat 0, the followig iequality holds: ) /η T b η f )x) 0 dx D α b BMOω) [M ν ωt f ) q ) x)] /q + [M ν r /p ω f r ) x)] /r + [M ν ω f q ) x)] /q). Fix a cube = x 0, d) ad x. Let = 5 ad bx) = bx) R m b; x, y) ad D α b = D α b D α b) for α = m. We write, for f = f χ ad f 2 = f χ R \, α! Dα b) xα, the R m b; x, y) = T b f )x) = + = T R m b; x, y) R x y m Kx, x y) f y)dy x y)α Dα by) α! R x y m Kx, x y) f y)dy R m+ b; x, y) R x y m Kx, x y) f 2 y)dy Rm b; x, ) x m f ) T x ) α D α b α! x m f + T b f 2 )x), the T b ) /η f )x) T b f 2 )x 0 ) η dx = I + I 2 + I 3. T Rm b; x, ) x m f ) η dx )/η ) /η T b f 2 )x) T b f 2 )x 0 ) η dx T x ) α D α b η x m f dx /η For I, otig that ω A, w satisfies the reverse of Hölder s iequality: ) /p0 ωx) p 0 dx ωx)dx for all cube ad some < p 0 < see [9]). We take s = rp 0 /r + p 0 ) i Lemma 0 ad have < s < r

6 iog he / FAA 7 3) 205), ad p 0 = sr )/r s), the by the Lemma 0 ad Hölder s iequality, we gai ) /s R m b; x, y) x y m D x, α bz) s dz y) x,y) ) /s x y m /s D α bz) s ωz) s r)/r ωz) sr )/r dz x,y) ) /r x y m /s D α bz) r ωz) r dz ωz) sr )/r s) dz x,y) x,y) x y m /s D α b BMOω) ω ) /r r s)/rs ωz) x, p 0 dz y) x,y) x y m D α b BMOω) /q ω ) /r /s /r x, y) x y m D α b BMOω) /q ω ) /r /s /r ω ) /r /r x y m D α ω ) b BMOω), thus, by Lemma 7, we obtai ) I T Rm b; x, ) x m f dx D α ω ) b BMOω) D α b BMOω) ω ωz)dz x,y) T f )y) ωy)νy) /q ωy) νy) /q dy ωy)t f )y) q νy)dy ) /q ) /q ωy) q νy) q /q dy D α b BMOω) ω ν ) /q ωy)t f )y) q νy)dy ν) ) /q ωy) q νy) q /q dy D α b BMOω) [M ν ωt f ) q ) x)] /q ω ν ) /q ωy) q νy) q /q dy D α b BMOω) [M ν ωt f ) q ) x)] /q. For I 2, we kow ν r/p A r by Lemma 4, thus /r ) /r νx) dx) r/p νx) r /p dx, ) /q ) /q ) r s)/rs ) r s)/rs ) r )/r

7 iog he / FAA 7 3) 205), the, by the weak L, L ) boudedess of Tsee Lemma 2) ad Kolmogoro s iequalitysee Lemma ), we obtai, by Lemma 5, I 2 = ) /η TD α b f )x) η dx /η /η TD α b f )χ Lη χ L η/ η) TDα b f ) WL D α bx) f x) dx R D α bx) D α b) µx) /p f x) ωx)νx) /p dx ) /r D α bx) D α b) ) r µx) r/p dx D α b BMOω) D α b BMOω) D α b BMOω) ) /r νx) r/p dx ) /r νx) r /p dx ν ) r /p D α b BMOω) [M ν r /p ω f r ) x)] /r. f x)ωx) r νx) r /p dx For I 3, ote that x y x 0 y for x ad y R \, we write T b f 2 )x) T b f 2 )x 0 ) f x) r ωx) r νx) r /p dx f x)ωx) r νx) r /p dx f x)ωx) r νx) r /p dx ) /r Rm b; x, y) R m b; x 0, y) Ωx, x y) R x y m+ f 2y) dy Ωx, x y) R x y +m Ωx 0, x 0 y) x 0 y +m R m b; x 0, y) f 2 y) dy Ωx, x y) α! R x y +m Ωx 0, x 0 y) x 0 y +m x y)α D α by) f2 y) dy x y) α α! x y m+ x 0 y) α x 0 y m+ Ωx 0, x 0 y) D α by) f2 y) dy = I ) 3 R x) + I2) x) + I3) x) + I4) x). For I ), by the formula see [5]): 3 R m b; x, y) R m b; x 0, y) = γ <m ) /r γ! R m γ D γ b; x, x0 )x y) γ ) /r ) /r

8 ad Lemma 0, we have, similar to the proof of I ad for k 0, R m b; x, y) R m b; x 0, y) iog he / FAA 7 3) 205), γ <m x x 0 m γ x y γ D α ω2 k ) b BMOω) 2 k, thus I ) 3 x) 2k+ \2 k R m b; x, y) R m b; x 0, y) Ωx, x y) D α ω2 k+ ) b BMOω) 2k+ D α d b BMOω) ω 2 k 2 k d) + k= D α b BMOω) 2 k ω 2 k 2 k= k ) /q ωy) q νy) q /q dy 2 k 2 k 2k+ \2 k f y) dy x y m+ D α b BMOω) 2 k ω 2 k ν 2 k )/q ν2 k= k ) ) /q 2 k ωy) q νy) q /q dy 2 k D α b BMOω) [M ν ω f q ) x)] /q k2 k k= ω 2 ν k 2 ) /q k 2 k ωy) q νy) q /q dy 2 k D α b BMOω) [M ν ω f q ) x)] /q k2 k D α b BMOω) [M ν ω f q ) x)] /q. k= x x 0 f y) dy x 0 y + f y) ωy)νy) /q ωy) νy) /q dy 2 k ωy) f y) q νy)dy 2 k ) /q ) /q ωy) f y) q νy)dy 2 k ) /q For I 2), by [2], we kow that 3 Ωx, x y) x y m+ u u= v= where u u 2, a uv L u 2, Y uv x y) u /2 ad a uv x) Y uvx y) x y, Y uv x y) x y Y uvx 0 y) x 0 y u/2 x x 0 / x 0 y +

9 for x y > 2 x 0 x > 0. Thus, we get u I 2) 3 x) R m b; x 0, y) 2k+ \2 k u= v= D α b BMOω) u 2 u /2 iog he / FAA 7 3) 205), u= Y uv x y) a uv x) x y +m Y uvx 0 y) x 0 y +m f y) dy ω2 k+ ) x x 0 f y) dy 2k+ x 0 y + D α b BMOω) [M ν ω f q ) x)] /q k= k2 k ω 2 k ν 2 k )/q 2 k ωy) q νy) q /q dy 2 k D α b BMOω) [M ν ω f q ) x)] /q. ) /q 2k+ \2 k For I 3) ad I 4), by usig the same argumets as I2) ad I , we have I 3) x) + I4) 3 3 x) u 2 u /2 Thus k= k= 2k+ \2 k d 2 k d) + u= 2k+ \2 k x x 0 f y) Dα by) dy x 0 y + D α by) D α b) 2 k 2 k f y) dy d 2 k d) + Dα b) 2 k Dα b) D α b BMOω) 2 k 2 k= k ) /r 2 k f y)ωy) r νy) r /p dy 2 k D α b BMOω) [M ν ω f q ) x)] /q f y) dy 2 k νy) r/p dy 2 k k= k2 k ) /r ) /q ω 2 k ν 2 k )/q 2 k ωy) q νy) q /q dy 2 k D α b BMOω) [Mν r /p ω f r ) x)] /r + [M ν ω f q ) x)] /q). x x 0 f y) Dα by) dy x 0 y + I 3 D α b BMOω) [Mν r /p ω f r ) x)] /r + [M ν ω f q ) x)] /q). These complete the proof of Theorem. Proof of Theorem 2. It suffices to prove for f 0 R ) ad some costat 0, the followig iequality holds: ) /η T b η f )x) 0 dx D α b Lipβ ω)w x)m β,r,ω f ) x).

10 iog he / FAA 7 3) 205), Fix a cube = x 0, d) ad x. Similar to the proof of Theorem, we have, for f = f χ ad f 2 = f χ R \, T b ) /η f )x) T b f 2 )x 0 ) η dx T = J + J 2 + J 3. Rm b; x, ) x m f ) η dx )/η ) /η T b f 2 )x) T b f 2 )x 0 ) η dx T x ) α D α b η x m f dx /η For J, by usig the same argumet as i the proof of Theorem, we get R m b; x, y) x y m /s x y m /s D α bz) r ωz) r dz x,y) x y m /s D α b Lipβ ω)ω ) β/+/r r s)/rs ) /s D α bz) s ωz) s r)/r ωz) sr )/r dz x,y) ) /r ωz) sr )/r s) dz x,y) ωz) x, p 0 dz y) x,y) x y m D α b Lipβ ω) /s ω ) β/+/r /s /r x, y) x y m D α b Lipβ ω) /s ω ) β/+/r /s /r ω ) /r /r x y m D α ω ) β/+ b Lipβ ω) x y m D α b Lipβ ω)ω ) β/ ω x), ωz)dz x,y) ) r s)/rs ) r s)/rs ) r )/r

11 iog he / FAA 7 3) 205), thus, by the L s -boudedess of T for < s < r ad w A A r/s, we obtai ) J T Rm b; x, ) x m f dx ) /s D α b Lipβ ω)ω ) β/ ω x) T f )x) s dx R D α b Lipβ ω)ω ) β/ ω x) R /s f x) s dx D α b Lipβ ω)ω ) β/ ω x) /s ) /s f x) r ωx)dx ) /r D α b Lipβ ω)ω x) /s ω ) /r f x) ω ) r ωx)dx rβ/ ) r s)/rs ) /r ωx) s/r s) dx ωx)dx /s ω ) /r D α b Lipβ ω)ω x)m β,r,ω f ) x). ωx) s/r s) dx For J 2, by the weak L, L ) boudedess of T ad Kolmogoro s iequality, we obtai ) /η J 2 TD α b f )x) η dx /η /η TD α b f )χ Lη χ L η/ η) TDα b f ) WL D α bx) f x) dx R D α bx) D α b) ωx) /r f x) ωx) /r dx ) /r ) /r ) /r D α bx) D α b) ωx) r r dx f x) r ωx)dx Dα b Lipβ ω)ω ) β/+/r ω ) /r β/ D α ω ) b Lipβ ω) M β,r,ω f ) x) D α b Lipβ ω)ω x)m β,r,ω f ) x). γ <m ω ) rβ/ f x) r ωx)dx ) /r ) r s)/rs For J 3, we have R m b; x, y) R m b; x 0, y) x x 0 m γ x y γ D α b Lipβ ω)ω x)ω2 k ) β/,

12 iog he / FAA 7 3) 205), thus T b f 2 )x) T b f 2 )x 0 ) R m b; x, y) R m b; x 0, y) Ωx, x y) f y) dy 2k+ \2 k x y m+ Ωx, x y) + 2k+ \2 k x y +m Ωx 0, x 0 y) x 0 y +m R m b; x 0, y) f y) dy Ωx, x y) x y +m Ωx 0, x 0 y) x 0 y +m x y)α D α by) f y) dy 2k+ \2 k 2k+ \2 k x y) α x y m+ x 0 y) α x 0 y m+ Ωx 0, x 0 y) D α by) f y) dy D α b Lipβ ω)ω x) ω2 k+ ) β/ k= k= 2k+ \2 k x x 0 f y) dy x 0 y + d D α by) D α b) 2 k d) + 2 k 2 k ωy) /r f y) ωy) /r dy d 2 k d) + D α b Lipβ ω)ω x) D α b) 2 k 2 k Dα b) f y) ωy)/r ωy) /r dy d 2 k d) + ω2k ) β/ f y) r ωy)dx k= 2 k ) r )/r ) /r 2 k ωy) /r ) dy 2 k 2 k ωy)dy 2 k ω2 k ) /r 2 k ) /r d D α by) D α b) 2 k d) + 2 k k= 2 k ωy) ) r r dy ) /r D α b Lipβ ω)ω x) kω2 k ) β/ d f y) r ωy)dx 2 k d) + k= 2 k ) r )/r ) /r 2 k ωy) /r ) dy 2 k 2 k ωy)dy 2 k ω2 k ) /r 2 k ) /r D α b Lipβ ω)ω x) k2 k f y) ω2k ) r ωy)dx rβ/ k= 2 k ) /r D α b Lipβ ω) 2 k ω2k ) 2 k f y) ω2k ) r ωy)dx rβ/ 2 k k= D α b Lipβ ω)ω x)m β,r,ω f ) x). ) /r f y) r ωy)dy 2 k ) /r This completes the proof of Theorem 2. Proof of Theorem 3. Notice ν r /p A r + r /p A p ad νx)dx A p/r νx) r /p dx) by Lemma 6, thus, by

13 iog he / FAA 7 3) 205), Theorem, Lemmas 2 ad 9, T b f )x) p νx)dx M η T b f ))x) p νx)dx M # ηt b f ))x) p νx)dx R R R D α b BMOω) [Mν ωt f ) q )x)] p/q + [M ν /p ω f r r )x)] p/r + [M ν ω f q )x)] p/q) νx)dx R ) D α b BMOω) ωx) f x) R p νx)dx + ωx)t f )x) p νx)dx R ) = D α b BMOω) f x) R p µx)dx + T f )x) p µx)dx R D α b BMOω) f x) R p µx)dx. This completes the proof of Theorem 3. Proof of Theorem 4. hoose < r < p i Theorem 2 ad otice ω q A, the we have, by Lemmas 8 ad 9, T b f ) L q ω q ) M η T b f )) L q ω q ) M # ηt b f )) L q ω q ) D α b Lipβ ω) ωm β,r,ω f ) L q ω q ) = D α b Lipβ ω) M β,r,ω f ) L q ω) D α b Lipβ ω) f L p ω). This completes the proof of Theorem 4. Refereces [] S. Bloom, A commutator theorem ad weighted BMO, Tras. Amer. Math. Soc., ), [2] A. P. alderó ad A. Zygmud, O sigular itegrals with variable kerels, Appl. Aal., 7978), [3] S. haillo, A ote o commutators, Idiaa Uiv. Math. J., 3982), 7-6. [4] J. ohe ad J. Gosseli, O multiliear sigular itegral operators o R, Studia Math., 72982), [5] J. ohe ad J. Gosseli, A BMO estimate for multiliear sigular itegral operators, Illiois J. Math., 30986), [6] R. R. oifma, R. Rochberg ad G. Weiss, Factorizatio theorems for Hardy spaces i several variables, A. of Math., 03976), [7] Y. Dig ad S. Z. Lu, Weighted boudedess for a class rough multiliear operators, Acta Math. Siica, 7200), [8] J. Garcia-uerva, Weighted H p spaces, Dissert. Math., 62979). [9] J. Garcia-uerva ad J. L. Rubio de Fracia, Weighted orm iequalities ad related topics, North- Hollad Math., 6, Amsterdam, 985.

14 iog he / FAA 7 3) 205), [0] Y. X. He ad Y. S. Wag, ommutators of Marcikiewicz itegrals ad weighted BMO, Acta Math. Siica hiese Series), 5420), [] B. Hu ad J. Gu, Necessary ad sufficiet coditios for boudedess of some commutators with weighted Lipschitz spaces, J. of Math. Aal. ad Appl., ), [2] S. Jaso, Mea oscillatio ad commutators of sigular itegral operators, Ark. Math., 6978), [3] L. Z. Liu, The cotiuity for multiliear sigular itegral operators with variable alderó-zygmud kerel o Hardy ad Herz spaces, Siberia Electroic Math. Reports, 22005), [4] L. Z. Liu, Good λ estimate for multiliear sigular itegral operators with variable alderó-zygmud kerel, Kragujevac J. of Math., ), [5] L. Z. Liu, Weighted estimates of multiliear sigular itegral operators with variable alderó- Zygmud kerel for the extreme cases, Vietam J. of Math., ), 5-6. [6] S. Z. Lu, D.. Yag ad Z. S. Zhou, Oscillatory sigular itegral operators with alderó-zygmud kerels, Southeast Asia Bull. of Math., 23999), [7] M. Paluszyski, haracterizatio of the Besov spaces via the commutator operator of oifma, Rochberg ad Weiss, Idiaa Uiv. Math. J., 44995), -7. [8]. Pérez, Edpoit estimate for commutators of sigular itegral operators, J. Fuc. Aal., 28995), [9]. Pérez ad R. Trujillo-Gozalez, Sharp weighted estimates for multiliear commutators, J. Lodo Math. Soc., ), [20] E. M. Stei, Harmoic aalysis: real variable methods, orthogoality ad oscillatory itegrals, Priceto Uiv. Press, Priceto NJ, 993. [2] A. Torchisky, Real variable methods i harmoic aalysis, Pure ad Applied Math. 23, Academic Press, New York, 986. [22] H. Xu ad L. Z. Liu., Weighted boudedess for multiliear sigular itegral operator with variable alderó-zygmud kerel, Africa Diaspora J. of Math., 62008), -2.