Boundedness for multilinear commutator of singular integral operator with weighted Lipschitz functions

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1 Aal of the Uiverity of raiova, Matheatic ad oputer Sciece Serie Volue 40), 203, Page ISSN: Boudede for ultiliear coutator of igular itegral operator with weighted Lipchitz fuctio Guo Sheg, Huag huagxia, ad Liu Lazhe Abtract. I thi paper, the boudede for the ultiliear coutator related to the igular itegral operator with weighted Lipchitz fuctio i proved. 200 Matheatic Subject laificatio. Priary 42B20; Secodary 42B25. Key word ad phrae. Multiliear coutator; Sigular itegral operator; Weighted Lipchitz fuctio; Triebel-Lizorki pace.. Itroductio Let b be a locally itegrable fuctio o R ad T be the alderó-zygud operator. The coutator [b, T ] geerated by b ad T i defied by [b, T ]fx) = bx)t fx) T bf)x). I [6] ad [7-9], the author proved that the coutator ad ultiliear operator geerated by the igular itegral operator ad BMO fuctio are bouded o L p R ) for < p <. haillo ee [4]) proved that the coutator [b, I α ] geerated by b BMO ad the fractioal itegral operator I α i bouded fro L p R ) to L q R ), where < p < q < ad /p /q = α/. The Paluzyńki ee [6]) howed that b Lip β R )the hoogeeou Lipchitz pace) if ad oly if the coutator [b, T ] i bouded fro L p to L q, where < p < q <, 0 < β < ad /q = /p β/. Alo Paluzyńki ee [5], [0], [6]) obtai that b Lip β if ad oly if the coutator [b, I α ] i bouded fro L p to L r, where < p < r <, 0 < β < ad /r = /p β + α)/ with /p > β + α)/. O the other had, i [] ad [9], the boudede for the coutator geerated by the igular itegral operator ad the weighted BM O ad Lipchitz fuctio o L p R ) < p < ) pace are obtaied. The purpoe of thi paper i to etablih boudede for the ultiliear coutator related to the igular itegral operator with geeral kerel ee [3] ad []) ad b Lip β w)the weighted Lipchitz pace). Defiitio.. Let T : S S be a liear operator uch that T i bouded o L 2 R ) ad there exit a locally itegrable fuctio Kx, y) o R R \ {x, y) R R : x = y} uch that T f)x) = Kx, y)fy)dy R for every bouded ad copactly upported fuctio f, where K atifie: there i a equece of poitive cotat uber { k } uch that for ay k, Kx, y) Kx, z) + Ky, x) Kz, x) )dx, 2 y z < x y Received Jauary 3, 202; Revied Noveber 22,

2 BOUNDEDNESS FOR MULTILINEAR OMMUTATOR 85 ad Kx, y) Kx, z) + Ky, x) Kz, x) ) q dy 2 k z y x y <2 k+ z y k 2 k z y ) /q, ) /q where < q < 2 ad /q + /q =. Suppoe b j j =,, ) are the fixed locally itegrable fuctio o R. The ultiliear coutator of the igular itegral operator i defied by T b f)x) = b j x) b j y))kx, y)fy)dy. R j= Note that the claical alderó-zygud igular itegral operator atifie Defiitio. with j = 2 jδ ee [8] ad [8]). Alo ote that whe =, T b i jut the coutator what we etioed above. It i well kow that ultiliear operator are of great iteret i haroic aalyi ad have bee widely tudied by ay author ee [2-4], [7-9]). I [8], Pérez ad Trujillo- Gozalez prove a harp etiate for the ultiliear coutator. The purpoe of thi paper ha two-fold, firt, we etablih a weighted Lipchitz etiate for the ultiliear coutator related to the geeralized igular itegral operator, ad ecod, we obtai the weighted L p -or iequality ad the weighted etiate o the Triebel-Lizorki pace for the ultiliear coutator by uig the weighted Lipchitz etiate. 2. Notatio ad Reult Throughout thi paper, we will ue to deote a abolute poitive cotat, which i idepedet of the ai paraeter ad ot ecearily the ae at each occurret. will deote a cube of R with ide parallel to the axe. For ay locally itegrable fuctio f, the harp axial fuctio of f i defied by f # x) = up x fy) f dy, where, ad i what follow, f = fx)dx. It i well-kow that ee [8] ad [20]) f # x) up if fy) c dy. x c For p < ad 0 η <, let M η,p f)x) = up x pη/ fy) p dy) /p, which i the Hardy-Littlewood axial fuctio whe p = ad η = 0. The A p weight i defied by ee [8]) { ) } p A p = w : up wx)dx wx) dx) /p ) <, < p <, A = {w > 0 : Mw)x) wx), a.e.}, ad A = p A p. We kow that, for w A, w atifie the double coditio, that i, for ay cube, w2) w).

3 86 G. SHENG, H. HUANGXIA, AND L. LANZHE The Ap, r) weight i defied by ee [5]), for < p, r <, { ) /r } p )/p Ap, r) = w > 0 : up wx) r dx wx) dx) p/p ) <. Give a weight fuctio w ad < p <, the weighted Lebegue pace L p w) i the pace of fuctio f uch that ) /p f Lp w) = fx) p wx)dx <. R For a weight fuctio w, β > 0 ad p >, let F p β, w) be the weighted hoogeeou Triebel-Lizorki pace ee [2]). For 0 < β <, the weighted Lipchitz pace Lip β w) i the pace of fuctio f uch that f Lipβ w) = up fy) f w) +β/ dy <. Give oe fuctio b j Lip β w), j, we deote by j the faily of all fiite ubet σ = {σ),, σj)} of {,, } of j differet eleet ad σi) < σj) whe i < j. For σ j, et σc = {,, }\σ. For b = b,, b ) ad σ = {σ),, σj)} j, et b σ = b σ),, b σj) ), b σ = j b σi) ad b σ Lipβ ω) = b σ) Lipβ w) b σi) Lipβ w). i= Now two theore are tated out a followig. Theore 2.. Let b j Lip β w) for j, 0 < β < ad w A. Suppoe the equece {k k } l, q < p < ad r = p. The T b i bouded fro L p w) to L r +r ) w ). Theore 2.2. Let b j Lip β w) for j, 0 < β < ad w A. Suppoe the equece {k k } l, q < p <. The T b i bouded fro L p, w) to F p w ). 3. Proof of Theore I order to prove the theore, the followig lea are eeded. Lea 3.. ee [7], [9]) For 0 < β <, w A, b Lip β w) ad p, we have /p b Lipβ w) up w) β w) bx) b p wx) dx) p. B Lea 3.2. ee [7], [9]) For 0 < β <, w A, b Lip β w) ad ay cube, we have up bx) b b Lipβ w)w) +β. x Lea 3.3. ee [7], [9]) For 0 < β <, w A, b Lip β w), ay cube ad x, we have b 2 k b kw x)w2 k ) β b Lipβ w). Lea 3.4. ee [2]) For 0 < β <, w A, < p < ad > 0, we have f F, p w) up fx) f dx x L p w) up if fx) 0 dx. x 0 L p w)

4 BOUNDEDNESS FOR MULTILINEAR OMMUTATOR 87 Lea 3.5. ee [5]) Suppoe that < p < η, /r = /p η/ ad w Ap, r). The M η, f) Lr w r ) f Lp w p ). Proof of Theore 2.. I order to prove the theore, we will prove a harp fuctio etiate for the ultiliear operator. We will prove that for ay cube ad q < <, there exit oe cotat 0 uch that T b f)x) 0 dx b Lipβ w)w x) + M, f) x) + M, T f)) x)). Fix a cube = x 0, r 0 ) ad x, we write f = f +f 2 with f = fχ 2, f 2 = fχ 2) c. We firt coider the ae =. For 0 = T b ) 2 b )f 2 )x 0 ), we write The where T b f)x) = b x) b ) 2 )T f)x) T b b ) 2 )f)x). T b f)x) 0 Ax) + Bx) + x), Ax) = b x) b ) 2 )T f)x), Bx) = T b ) 2 b )f )x), x) = T b ) 2 b )f 2 )x) T b ) 2 b )f 2 )x 0 ). For Ax), by Hölder iequality ad Lea 3.2, we have Ax) dx = b x) b ) 2 dx b x) b ) 2 T f)x) dx ) ) T f)x) dx 2 up b x) b ) 2 T f)x) dx x 2 b Lipβ w)w2) + b Lipβ w) β β ) + β w2) Mβ, T f)) x) 2 b Lipβ w)w x) + β Mβ, T f)) x). ) β T f)x) r dx ) For Bx), by the type,) of T ad Lea 3.2, we obtai Bx)dx ) T b ) b )f )x) r dx R

5 88 G. SHENG, H. HUANGXIA, AND L. LANZHE b ) b x) r f x) r dx R up b x) b ) x 2 fx) dx 2 ) ) b Lipβ w)w2) + β 2 2 β 2 β ) + β w2) b Lipβ w) Mβ, f)x) 2 b Lipβ w)w x) + β Mβ, f) x). 2 fx) dx ) For x), recallig that > q, takig < p <, < t < with /p + /q + /t =, by the Hölder iequality ad Lea 3. ad 3.3, we have, for x, T b b ) 2 )f 2 )x) T b b ) 2 )f 2 )x 0 ) b y) b ) 2 fy) Kx, y) Kx 0, y) dy 2) c Kx, y) Kx 0, y) q dy k= 2 k x x 0 y x 0 <2 k+ x x 0 ) /p + b y) b ) 2 p dy y x 0 <2 k+ x x 0 k= k 2 k d) q fy) dy y x 0 <2 k+ x x 0 k= k= k= k 2 k d) q k 2 k d) q k 2 k d) q b y) b ) 2 p dy y x 0 <2 k+ x x 0 ) / b y) b ) 2 p dy 2 k+ b y) b ) 2 k+ p dy 2 k+ ) /q fy) t dy y x 0 <2 k+ x x 0 ) /p b ) 2 k+ b ) 2 p dy 2 k+ ) /p fy) t dy 2 k+ ) /p ) /t fy) t dy 2 k+ ) /p fy) t dy 2 k+ ) /t ) /t ) /t k up b k= 2 k+ y) b ) 2 k+ 2 k+ p 2 k+ β q y 2 k+ / fy) dy) + 2 k+ β k b 2 k+ k= 2 k+ ) 2 k+ b ) 2 q ) / 2 k+ p 2 k+ β fy) dy 2 k+ β 2 k+

6 thu + + BOUNDEDNESS FOR MULTILINEAR OMMUTATOR 89 k up k= y 2 k+ b y) b ) 2 k+ 2 k+ β Mβ, f) x) k b ) 2 k+ b ) 2 2 k+ β Mβ, f) x) k= k b Lipβ w)w2 k+ ) + β 2 k+ 2 k+ β Mβ, f) x) k= k kw x)w2 k+ ) β b Lipβ w) 2 k+ β Mβ, f) x) k= b Lipβ w) k= + b Lipβ w)w x) w2 k+ ) + β ) k k 2 k+ M β, f) x) k= b Lipβ w)w x) + β Mβ, f) x), w2 k+ ) ) β k k 2 k+ M β, f) x) x)dx b Lipβ w)w x) + β Mβ, f) x). Now, we coider the ae 2. We have, for b = b,, b ), T b f)x) = = = R j= j=0 σ j R j= b j x) b j y))kx, y)fy)dy [b j x) b j ) 2 ) b j y) b j ) 2 )]Kx, y)fy)dy ) j bx) b) 2 ) σ by) b) 2 ) σ ckx, y)fy)dy R = b j x) b j ) 2 ) Kx, y)fy)dy j= R + ) b j y) b j ) 2 )Kx, y)fy)dy + = + j= σ j R j= ) j b j x) b j ) 2 ) σ b j y) b j ) 2 ) σ ckx, y)fy)dy R b j x) b j ) 2 )T f)x) + ) T b j b j ) 2 )f)x) j= j= σ j j= ) j b j x) b j ) 2 ) σ )T b j b j ) 2 ) σ cf)x), thu, recall that 0 = T j= b j b j ) 2 )f 2 )x 0 ),

7 90 G. SHENG, H. HUANGXIA, AND L. LANZHE where T b f)x) T b j b j ) 2 )f 2 )x 0 ) j= b j x) b j ) 2 )T f)x) + T b j b j ) 2 )f )x) j= + j= σ j j= b j x) b j ) 2 ) σ )T b j b j ) 2 ) σ cf)x) + T b j b j ) 2 )f 2 )x) T b j b j ) 2 )f 2 )x 0 ) j= = I x) + I 2 x) + I 3 x) + I 4 x). I x) = j= b j x) b j ) 2 )T f)x), j= I 2 x) = T b j b j ) 2 )f )x), j= I 3 x) = j= σ j b j x) b j ) 2 ) σ )T b j b j ) 2 ) σ cf)x), I 4 x) = T b j b j ) 2 )f 2 )x) T b j b j ) 2 )f 2 )x 0 ). j= j= For I x), by Hölder iequality with expoet r + + we get r + I x)dx b j x) b j ) 2 ) T f)x) dx j= ) ) b j x) b j ) 2 r r j j dx T f)x) dx j= ) up b j x) b j ) 2 r j x j= j= b Lipβ w)w) b Lipβ w) T f)x) dx b j Lipβ w)w) + β ) )+ ) w) b Lipβ w)w x) + + M, T f)) x) ) + M, T f)) x) M, T f)) x). ) r = ad Lea 3.2, T f)x) dx )

8 BOUNDEDNESS FOR MULTILINEAR OMMUTATOR 9 For I 2 x), iilar to Bx), uig the boude of T ad Lea 3.2, we get, for < t <, I 2 x)dx t T b j b j ) 2 )f )x) t dx t R R j= j= b j x) b j ) 2 )f x) r dx b j x) b j ) 2 ) fx) dx j= b j x) b j ) 2 )) j= j= b Lipβ w) b Lipβ w)w x) 2 t t fx) dx) b j Lipβ w)w2) + β 2 ) 2 + ) + w2) M, f) x) 2 + M, f) x). 2 2 fx) dx For I 3 x), by Hölder iequality ad Lea 3.2, we get I 3 x)dx b j x) b j ) 2 ) σ T b j b j ) 2 ) σ cf)x) dx 2 j= σ j j= σ j 2 2 b j x) b j ) 2 ) σ dx 2 T b j b j ) 2 ) σ cf)x) dx 2 j= σ j ) 2 up b j x) b j ) 2 ) σ 2 r x 2 b j x) b j ) 2 ) σ c T f)x) dx 2 j= σ j up b j x) b j ) 2 ) σ c x 2 ) 2 up b j x) b j ) 2 ) σ 2 r x 2 T f)x) dx 2 ) 2 b σ Lipβ w)w2) j+ jβ 2 j b j)β j)+ σ c Lipβ w)w2) 2 + j) 2 2 ) T f)x) dx ) )

9 92 G. SHENG, H. HUANGXIA, AND L. LANZHE b Lipβ w) b Lipβ w)w x) ) + w2) M, T f)) x) 2 + M, T f)) x). For I 4 x), iilar to the proof of x) i the ae =, for < p <, < t < with /p + /q + /t =, we have T b j y) b j ) 2 )f 2 )x) T b j b j ) 2 )f 2 )x 0 ) j= 2) c j= b j y) b j ) 2 ) fy) Kx, y) Kx 0, y)) dy j= ) /q Kx, y) Kx 0, y) q dy k= 2 k x x 0 y x 0 <2 k+ x x 0 /p b y) b ) 2 p dy fy) t dy y x 0 <2 k+ x x 0 j= y x 0 <2 k+ x x 0 /p k b k= 2 k d) j y) b j ) 2 p dy q y x 0 <2 k+ x x 0 j= ) /t fy) t dy y x 0 <2 k+ x x 0 k b k= 2 k d) j y) b j ) 2 k+ p dy q 2 k+ j= + k b 2 k d) j ) 2 k+ b j ) 2 p dy q 2 k+ k= k= k 2 k q 2 k+ up j= j= x 2 k+ 2 k+ 2 k+ p 2 k+ + /p /p fy) t dy 2 k+ fy) t dy 2 k+ b j y) b j ) 2 2 k+ k+ p 2 k+ fy) dy) / + k b Lipβ w)w2 k+ ) k= k= k kw x) w2 k+ ) 2 k+ + k= 2 k+ k 2 k q fy) dy ) /t ) /t ) /t b j ) 2 k+ b j ) 2 j= ) / 2 k+ 2 k+ M, f) x) b Lipβ w) 2 k+ M, f) x)

10 b Lipβ w) + b Lipβ w) BOUNDEDNESS FOR MULTILINEAR OMMUTATOR 93 k= k= b Lipβ w)w x) w2 k k+ ) + ) k 2 k+ M, f) x) w2 k k w x) k+ ) ) 2 k+ M, f) x) + M, f) x), thu, we get I 4 x)dx b Lipβ w)w x) obiig all the etiate above, we get ad T b f)x) 0 dx b Lipβ w)w x) T b f) # x) b Lipβ w)w x) + Now, chooe q < < r, by Lea 3.5, we have T b f) Lq w +q ) + + M, f) x). M, f) x) + M, T f)) x)) M, f) x) + M, T f)) x)). MT ) b f)) +q ) Lq w ) T b f)) # +q ) Lq w ) b Lipβ w) w +/ M, f) +q ) Lq w ) + w +/ M, T f)) ) L q +q ) w ) b Lipβ w) M, f) Lq w q p ) + M,T f)) Lq w q p ) ) b Lipβ w) f Lp w) + T f) Lp w)) b Lipβ w) f Lp w). Thi coplete the proof of Theore 2.. Proof of Theore 2.2. Siilar to Theore 2., for ay q < < ad cube, there exit oe cotat 0 uch that for f L p w) ad x, T b f)x) 0 dx + b Lipβ w)w x) M f) x) + M T f)) x)). Further, we have up if x c T b f)x) c dx b Lipβ w)w x) hooe q < < p ad by Lea 3.4, we obtai T b f) up if ) Fp, w x c b Lipβ w) w +/ M f) L p w ) b Lipβ w) M f) Lp w) + M T f)) Lp w)) b Lipβ w) f Lp w) + T f) Lp w)) b Lipβ w) f Lp w). + T b f)x) c dx M f) x)+m T f)) x)). L p w ) + w+ M T f)) ) L p w )

11 94 G. SHENG, H. HUANGXIA, AND L. LANZHE Thi coplete the proof of Theore 2.2. Referece [] S. Bloo, A coutator theore ad weighted BMO, Tra. Aer. Math. Soc ), [2] H.. Bui, haracterizatio of weighted Beov ad Triebel-Lizorki pace via teperature, J. Fuc. Aal ), [3] D.. hag, J. F. Li ad J. Xiao, Weighted cale etiate for alderó-zygud type operator, oteporary Matheatic ), [4] S. haillo, A ote o coutator, Idiaa Uiv. Math. J. 3982), 7 6. [5] W. G. he, Beov etiate for a cla of ultiliear igular itegral, Acta Math. Siica 62000), [6] R. R. oifa, R. Rochberg ad G. Wei, Fractorizatio theore for Hardy pace i everal variable, A. of Math ), [7] J. Garcia-uerva, Weighted H p pace, Diert. Math ), 56. [8] J. Garcia-uerva ad J. L. Rubio de Fracia, Weighted or iequalitie ad related topic, North- Hollad Math. 6, Aterda, 985. [9] B. Hu ad J. Gu, Neceary ad ufficiet coditio for boudede of oe coutator with weighted Lipchitz pace, J. of Math. Aal. ad Appl ), [0] S. Jao, Mea ocillatio ad coutator of igular itegral operator, Ark. Math. 6978), [] Y. Li, Sharp axial fuctio etiate for alderó-zygud type operator ad coutator, Acta Math. Scietia 3A)20), [2] L. Z. Liu, Iterior etiate i Morrey pace for olutio of elliptic equatio ad weighted boudede for coutator of igular itegral operator, Acta Math. Scietia 25B)2005), [3] L. Z. Liu, Sharp axial fuctio etiate ad boudede for coutator aociated with geeral itegral operator, FiloatForerly Zborik Radova Filozofkog Fakulteta, Serija Mateatika) 2520), [4] L. Z. Liu ad X. S. Zhou, Weighted harp fuctio etiate ad boudede for coutator aociated with igular itegral operator with geeral kerel, New Zealad J. of Math ), [5] B. Muckehoupt ad R. L. Wheede, Weighted or iequalitie for fractioal itegral, Tra. Aer. Math. Soc ), [6] M. Paluzyki, haracterizatio of the Beov pace via the coutator operator of oifa, Rochberg ad Wei, Idiaa Uiv. Math. J ), 7. [7]. Pérez, Edpoit etiate for coutator of igular itegral operator, J. Fuc. Aal ), [8]. Pérez ad G. Pradolii, Sharp weighted edpoit etiate for coutator of igular itegral operator, Michiga Math. J ), [9]. Pérez ad R. Trujillo-Gozalez, Sharp Weighted etiate for ultiliear coutator, J. Lodo Math. Soc ), [20] E. M. Stei, Haroic Aalyi: real variable ethod, orthogoality ad ocillatory itegral, Priceto Uiv. Pre, Priceto NJ, 993. Guo Sheg, Huag huagxia ad Liu Lazhe) Departet of Matheatic, hagha Uiverity of Sciece ad Techology, hagha, 40077, P. R. of hia E-ail addre: lazheliu@63.co