Aal of the Uiverity of raiova, Matheatic ad oputer Sciece Serie Volue 40), 203, Page 84 94 ISSN: 223-6934 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, 202. 84
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).
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)
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
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+
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 ),
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 )
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 ) )
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)
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 )
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