2017Ψ9 ADVANCES IN MATHEMATICS (CHINA) Sep., 2017

Λ46 Λ5Ω ff fl Π Vol. 46, No. 5 2017Ψ9 ADVANCES IN MATHEMATICS CHINA) Sep., 2017 doi: 10.11845/sxjz.2015219b Boundedness of Commutatos Geneated by Factional Integal Opeatos With Vaiable Kenel and Local Campanato Functions on Genealized Local Moey Spaces MO Huixia, XUE Hongyang School of Sciences, Beijing Univesity of Posts and Telecommunications, Beijing, 100876, P. R. China) Abstact: Suppose that T Ω,α is the factional integal opeato with vaiable kenel. We pove the boundedness fo T Ω,α on the genealized local Moey spaces LM {x 0} p,ϕ. The multilinea commutatos geneated by T Ω,α and local Campanato functions ae also consideed. Keywods: factional integal; vaiable kenel; commutato; local Campanato function; genealized local Moey space MR2010) Subject Classification: 42B20; 42B25 / CLC numbe: O174.2 Document code: A Aticle ID: 1000-09172017)05-0755-10 0 Intoduction Suppose that is the unit sphee in R n n 2) equipped with the nomalized Lebesgue measue dσ. We say that a function Ωx, z) defined on R n R n belongs to the space L R n ) L s ), if Ωx, z) satisfies the following conditions: i) Fo any x, z R n, Ωx, λz) =Ωx, z) fo all λ>0; ii) Ω L R n ) L s ) := sup x R n Ωx, z ) s dσz )) 1 s <. Then the factional integal opeato with vaiable kenel is defined by Ωx, x y) T Ω,α fx) =p.v. n α fy)dy, 0.1) R n x y whee 0 <α<n. Moeove, let b =b 1,b 2,,b m ), whee b i L loc R n )fo1 i m. Then the multilinea commutato geneated by b and T Ω,α canbedefinedasfollows: T m b Ωx, x y) Ω,α fx) =p.v. b i x) b i y)) fy)dy. 0.2) R x y n α n i=1 In 1971, Muckenhoupt and Wheeden [9] studied the p, q)-boundedness of T Ω,α. In [3], Ding studied the weak estimate fo T Ω,α with powe weights. And, in [10], Shao and Wang consideed Received date: 2015-12-22. Revised date: 2016-06-22. Foundation item: The wok is suppoted by NSFC No. 11601035, No. 11471050). E-mail: huixmo@bupt.edu.cn

756 ADVANCES IN MATHEMATICS CHINA) Vol. 46 the boundedness of T Ω,α on weighted Moey spaces. Recently, the commutatos geneated by factional integal opeatos with vaiable kenel have also attacted much attention, see [2, 4, 11 12] etc. Moeove, the classical Moey spaces M p,λ wee fist intoduced by Moey in [8] to study the local behavio of solutions to second ode elliptic patial diffeential equations. Moeove, in [1], Balakishiyev et al. intoduced the local genealized Moey spaces LM {x0} p,ϕ,anheyalso studied the boundedness of the homogeneous singula integal opeatos with ough kenel on these spaces. In [7], Mo and Xue studied the boundedness of the commutatos geneated by singula integal opeatos with vaiable kenel and local Campanato functions on genealized local Moey spaces. Motivated by the woks of [1, 7, 10], we conside the boundedness of the factional integal opeato T Ω,α with vaiable kenel on the local genealized Moey spaces LM {x0} p,ϕ. Futhemoe, we also obtain the boundedness of the commutatos geneated by T Ω,α and local Campanato functions on the local genealized Moey spaces LM {x0} p,ϕ. 1 Some Notations and Lemmas Definition 1.1 [1] Let ϕx, ) be a positive measuable function on R n 0, ) and1 p. Fo any fixed x 0 R n, a function f L q loc is said to belong to the local Moey spaces, if And, we denote f LM p,ϕ LM {x0} p,ϕ =supϕx 0,) 1 Bx 0,) 1 p f L p Bx 0,)) <. >0 LM {x0} p,ϕ R n )= { f L q loc Rn ): f {x LM 0 } < }. p,ϕ Accoding to Definition 1.1, we ecove the local Moey space LM {x0} p,λ ϕx 0,)= λ n p : M p,λ = M p,ϕ. ϕx,)= λ n p unde the choice Definition 1.2 [1] belong to the space LC {x0} f LC Let 1 q< and 0 λ< 1 n. A function f Lq loc Rn )issaio local Campanato space), if 1 =sup >0 Bx 0,) 1+λq Bx 0,) ) 1 fy) f Bx0,) q q dy <, whee Define f Bx0,) = 1 fy)dy. Bx 0,) Bx 0,) LC {x0} Rn )= { f L q loc Rn ): f {x LC 0 } < }.

No. 5 Mo H. X. and Xue H. Y.: Commutatos Geneated by Factional Integal Opeatos 757 Remak 1.1 [1] Obviously, CBMO q R n )=LC {0} q,0 Rn ), CBMO {x0} q R n )=LC {x0} q,0 Rn ). Moeove, one can imagine that the behavio of CBMO {x0} q R n ) may be quite diffeent fom that of BMOR n ), since thee is no analogy of the John-Nienbeg inequality of BMO fo the spaces CBMO {x0} q R n ). Lemma 1.1 [1] 1 Bx 0, 1 ) 1+λq Let 1 <q<, 0 < 2 < 1 and b LC {x0}.then Bx 0, 1) ) 1 bx) b Bx0, 2) q q dx C 1+ln ) 1 b {x LC 0 }. 2 And, fom this inequality, we have b Bx0, 1) b Bx0, 2) C 1+ln ) 1 Bx 0, 1 ) λ b {x LC 0 }. 2 In this section, we use the following statement on the boundedness of the weighted Hady opeato: H w gt) := gs)ws)ds, 0 <t<, t whee w is a fixed non-negative and measuable function on 0, ). Lemma 1.2 [5 6] Let v 1,v 2 and w be positive almost eveywhee and measuable functions on 0, ). The inequality ess sup v 2 t)h w gt) C ess sup v 1 t)gt) 1.1) holds fo some C>0and all non-negative and non-deceasing g on 0, ) if and only if ws) B := ess sup v 2 t) t ess sup v 1 τ) ds<. s<τ < Moeove, if C is the minimum value of C in 1.1), then C = B. Lemma 1.3 [9] Let Ω L R n ) L s ), s>1, and fo any x R n, Ωx, z )dσz ) =0,wheez = z z and z Rn \{0}. Assume that 0 <α<n,1 s <p< n α and 1 <q<, such that 1 q = 1 p α n, then T Ω,α is bounded fom L p R n )tol q R n ), whee s = s s 1 is the conjugate exponent of s. We fomulate ou main esults in Sections 2 and 3. 2 Factional Integal Opeatos With Vaiable Kenel In this section, we conside the boundedness of the factional integal opeatos with vaiable kenel on genealized local Moey spaces. Theoem 2.1 Let Ω L R n ) L s ),s>1, and fo any x R n, Ωx, z )dσz ) =0, whee z = z z and z Rn \{0}. Assume that 0 <α<n,1 s <p< n α and 1 <q<, such that 1 q = 1 p α n. Then the inequality T Ω,α f L q Bx 0,)) n q f L p Bx 0,t))t n q 1

758 ADVANCES IN MATHEMATICS CHINA) Vol. 46 holds fo any ball Bx 0,)andallf L p loc Rn ). Poof Let B = Bx 0,). We wite f = f 1 + f 2, whee f 1 = fχ 2B and f 2 = fχ 2B) c. Thus, we have T Ω,α f Lq B) T Ω,α f 1 Lq B) + T Ω,α f 2 Lq B). Since T Ω,α is bounded fom L p R n )tol q R n ) see Lemma 1.3), then we have T Ω,α f 1 L q B) f L p 2B) n q f L p Bx 0,t)) t n q. 2.1) +1 Moeove, it is obvious that Ωx, x ) Ls Bx 0,t)) = t+ x x0 0 Ωx, u) s du B0,t+ x x 0 ) n 1 d Ωx, u ) s dσu ) Ω L L s ) B0,t+ x x 0 ) 1 s. ) 1 s ) 1 s 2.2) Note that 1 2 x 0 y x y 3 2 x 0 y fo x B, y 2B) c. Thus, by 0.1), 2.2), the Fubini theoem and Hölde s inequality, we have fy) Ωx, x y) T Ω,α f 2 x) 2B) c x 0 y n α dy fy) Ωx, x y) dy 2B) c x 0 y tn+1 α fy) Ωx, x y) dy x 0 y t Bx 0,t) fy) Ωx, x y) dy. f Lp Bx 0,t)) Ωx, x ) Ls Bx 0,t)) Bx 0,t) 1 1 p 1 s f L p Bx 0,t)) B0,t+ x x 0 ) 1 s Bx 0,t) 1 1 p 1 s f L p Bx 0,t)) t n q +1. 2.3) Theefoe, T Ω,α f 2 Lp B) n q So, combining 2.1) and 2.4), we have f Lp Bx 0,t)) t n q. 2.4) +1 T Ω,α f L p B) n q f L p Bx 0,t)) t n q +1. Thus, we complete the poof of Theoem 2.1.

No. 5 Mo H. X. and Xue H. Y.: Commutatos Geneated by Factional Integal Opeatos 759 Theoem 2.2 Let Ω L R n ) L s ),s>1, and fo any x R n, Ωx, z )dσz ) =0, whee z = z z and z Rn \{0}. Assume that 0 <α<n,1 s <p< n α and 1 <q<, such that 1 q = 1 p α n. If functions ϕ, ψ : Rn 0, ) 0, + ) satisfy the inequality ess inf t<τ < ϕx 0,τ)τ n p t n q +1 Cψx 0,), 2.5) whee C does not depend on, then the opeato T Ω,α is bounded fom LM {x0} p,ϕ to LM {x0} q,ψ. Poof Take v 1 t) =ϕx 0,t) 1 t n p,v2 t) =ψx 0,t) 1,gt) = f Lp Bx 0,t)) and wt) = t n q 1, then fom 2.5) we have ws)ds ess sup v 2 t) t ess sup v 1 τ) <. s<τ < Thus, fom Lemma 1.2, it follows that Theefoe, ess sup T Ω,α f LM q,ψ = f {x LM 0 }. p,ϕ Thus we complete the poof of Theoem 2.2. v 2 t)h w gt) C ess sup v 1 t)gt). =supψx 0,) 1 Bx 0,) 1 q TΩ,α f L q Bx 0,)) >0 sup ψx 0,) 1 f L p Bx 0,t)) >0 sup ϕx 0,) 1 n p f L p Bx 0,)) >0 3 Commutatos of Factional Integal Opeatos With Vaiable Kenel In this section, we conside the boundedness of the multinea commutatos geneated by factional integal opeatos with vaiable kenel and Campanato functions on genealized local Moey spaces. Theoem 3.1 Let Ω L R n ) L s ),s>1, and fo any x R n, Ωx, z )dσz ) =0, whee z = z z and z Rn \{0}. Let 0 <α<n,1 s <p< n α and 1 <q,p 1,p 2,,p m <, such that 1 q = m i=1 1 p i + 1 p α n and m i=1 1 p i + 1 p 1 s. Let 1 p = 1 p α n,x 0 R n and b i LC {x0} p i,λ i fo 0 <λ i < 1 n,i=1, 2,,m. Then the inequality T b Ω,α f L q Bx 0,)) m i=1 b i LC p i,λ i n q holds fo any ball Bx 0,), whee λ = λ 1 + λ 2 + + λ m. t n q +1 1+ln t ) m n nλ f L pbx0,t))t p 1

760 ADVANCES IN MATHEMATICS CHINA) Vol. 46 Poof Without loss of geneality, it is sufficient fo us to show that the conclusion holds fo m =2. Let B = Bx 0,). And, we wite f = f 1 + f 2, whee f 1 = fχ 2B,f 2 = fχ 2B) c. Thus, we have b T 1,b 2) Ω,α f Lq B) b T 1,b 2) Ω,α f Lq 1 B) + b T 1,b 2) Ω,α f Lq 2 =: I + II. B) Let us estimate I and II, espectively. It is easy to see that b T 1,b 2) Ω,α f Lq 1 B) = b 1 b 1 ) B )b 2 b 2 ) B )T Ω,α f 1 L q B) + b 1 b 1 ) B )T Ω,α b 2 b 2 ) B )f 1 ) L q B) + b 2 b 2 ) B )T Ω,α b 1 b 1 ) B )f 1 ) L q B) 3.1) + T Ω,α b 1 b 1 ) B )b 2 b 2 ) B )f 1 ) Lq B) =: I 1 +I 2 +I 3 +I 4. Since 1 p = 1 p α n, it is obvious that 1 q = 1 p 1 + 1 p 2 + 1 p. And, fom Definition 1.2, it is easy to see that b i b i ) B L p ib) C n p +nλ i i b i {x LC 0 } p i,λ i fo i =1, 2. 3.2) Thus, using Hölde s inequality, Lemma 1.3 and 3.2), we have I 1 b 1 b 1 ) B L p 1 B) b 2 b 2 ) B L p 2 B) T Ω,α f 1 L p B) b 1 b 1 ) B L p 1 B) b 2 b 2 ) B L p 2 B) f Lp 2B) b 1 b 1 ) B L p 1 B) b 2 b 2 ) B L p 2 B) ñ p b 1 LC n q 1+ln t f L p Bx 0,t)) t ñ p +1 ) 2 t λ1+λ2)n n p 1 f L p Bx 0,t)). 3.3) Moeove, fom Lemma 1.1, it is easy to see that b i b i ) B L p i2b) C n p i +nλ i b i LC p i,λ i fo i =1, 2. 3.4) And, let 1 < p, q < such that 1 q = 1 p 1 + 1 q and 1 p = 1 p 2 + 1 p. It is easy to see that 1 q = 1 p α n. Then similaly to the estimate of 3.3), we have I 2 b 1 b 1 ) B L p 1B) T Ω,α b 2 b 2 ) B )f 1 ) L q B) b 1 b 1 ) B L p 1B) b 2 b 2 ) B )f 1 L p R n ) b 1 b 1 ) B L p 1B) b 2 b 2 ) B L p 2 2B) f Lp 2B) b 1 {x LC 0 } b 2 {x p 1,λ LC 0 } n q t λ1+λ2)n n p 1 f Lp Bx 1 0,t)).

No. 5 Mo H. X. and Xue H. Y.: Commutatos Geneated by Factional Integal Opeatos 761 Similaly, I 3 b 1 LC n q t λ1+λ2)n n p 1 f Lp Bx 0,t)). Moeove, let 1 < q < such that 1 q = 1 q n α. It is easy to see that 1 q = 1 p 1 + 1 p 2 + 1 p. Thus, by Lemma 1.3, Hölde s inequality and 3.4), we obtain I 4 = T Ω,α b 1 b 1 ) B )b 2 b 2 ) B )f 1 ) L q B) b 1 b 1 ) B )b 2 b 2 ) B )f 1 L q R n ) b 1 b 1 ) B L p 12B) b 2 b 2 ) B L p 22B) f Lp 2B) b 1 {x LC 0 } b 2 {x p 1,λ LC 0 } n q t λ1+λ2)n n p 1 f 1 L p Bx 0,t)). Theefoe, combining the estimates of I 1, I 2, I 3 and I 4, we have I b 1 LC n q t λ1+λ2)n n p 1 f L p Bx 0,t)). Let us estimate II. It is analogue to 3.1). We have b T 1,b 2) Ω f L 2 q B) = b 1 b 1 ) B )b 2 b 2 ) B )T Ω,α f 2 Lq B) + b 1 b 1 ) B )T Ω,α b 2 b 2 ) B )f 2 ) L q B) + b 2 b 2 ) B )T Ω,α b 1 b 1 ) B )f 2 ) L q B) + T Ω,α b 1 b 1 ) B )b 2 b 2 ) B )f 2 ) Lq B) =: II 1 +II 2 +II 3 +II 4. Since 1 p = 1 p n α, it is easy to see that 1 q = 1 p 1 + 1 p 2 + 1 p. Thus, using Hölde s inequality and 2.4), we have II 1 b 1 b 1 ) B L p 1B) b 2 b 2 ) B L p 2B) T Ω,α f 2 L p B) b 1 b 1 ) B L p 1B) b 2 b 2 ) B L p 2B) ñ p b 1 LC n q 1+ln t f Lp Bx 0,t)) t ñ p +1 ) 2 t λ1+λ2)n n p 1 f Lp Bx 0,t)).

762 ADVANCES IN MATHEMATICS CHINA) Vol. 46 Fo x B, using Lemma 1.1, it is analogue to 2.3). We have T Ω,α b 2 b 2 ) B )f 2 )x) fy) b 2 y) b 2 ) B Ωx, x y) n α dy 2B) c x 0 y b 2 y) b 2 ) B Ωx, x y) fy) dy < x 0 y <t Bx 0,t) b 2 y) b 2 ) B Ωx, x y) fy) dy b 2 b 2 ) B L p 2 Bx0,t)) Ωx, x ) L s Bx 0,t)) f Lp Bx 0,t)) Bx 0,t) 1 1 p 2 1 s 1 p 1+ln t ) t nλ2 ñ p 1 f L p Bx. 0,t)) Let 1 < q <, such that 1 q = 1 p 1 + 1 q. Then, using Hölde s inequality and 3.5), we have 3.5) So, II 2 b 1 b 1 ) B L p 1 B) T Ω,α b 2 b 2 ) B )f 2 ) L q B) b 1 {x LC 0 } b 2 {x p 1,λ LC 0 } n q t λ1+λ2)n n p 1 f Lp Bx 1 0,t)). Similaly, we have II 3 b 1 LC n q t λ1+λ2)n n p 1 f L p Bx 0,t)). Let us estimate II 4. Fo x B, using Hölde s inequality and Lemma 1.1, it is analogue to 3.5). We have T Ω,α b 1 b 1 ) B )b 2 b 2 ) B )f 2 )x) fy) b 1 y) b 1 ) B b 2 y) b 2 ) B Ωx, x y) n α dy 2B) c x 0 y b 1 y) b 1 ) B b 2 y) b 2 ) B Ωx, x y) fy) dy < x 0 y <t b 1 b 1 ) B L p 1 b 2 b 2 ) B L p 2 Bx0,t)) Ωx, x ) L s Bx 0,t)) f Lp Bx 0,t)) Bx 0,t) 1 1 p 1 1 p 1 2 s 1 p b 1 LC 1+ln t ) t nλ2 ñ p 1 f L p Bx 0,t)). 3.6) II 4 = T Ω,α b 1 b 1 ) B )b 2 b 2 ) B )f 2 ) Lq B) b 1 LC n q t λ1+λ2)n n p 1 f Lp Bx 0,t)).

No. 5 Mo H. X. and Xue H. Y.: Commutatos Geneated by Factional Integal Opeatos 763 Theefoe, combining the estimates of II 1, II 2, II 3 and II 4, we have II b 1 LC q 1,λ 1 q 2,λ 2 n q t λ1+λ2)n n p 1 f L p Bx 0,t)). So, fom the estimates of I and II, we obtain b T 1,b 2) Ω,α f L q Bx 0,)) b 1 {x LC 0 } b 2 {x p 1,λ LC 0 } n q t λ1+λ2)n n p 1 f Lp Bx 1 0,t)). Thus, we complete the poof of Theoem 3.1. Theoem 3.2 Let Ω L R n ) L s ),s>1, and fo any x R n, Ωx, z )dσz ) = 0, whee z = z z fo any z R n \{0}. Let 0 < α < n, 1 s < p < n α and 1 < p 1,p 2,,p m,q <, such that 1 q = m i=1 1 p i + 1 p α n and m i=1 1 p i + 1 p 1 s. Let 1 p = 1 p α n, x 0 R n and b i LC {x0} p i,λ i fo 0 <λ i < 1 n,i=1, 2,,m. If functions ϕ, ψ : R n 0, ) 0, + ) satisfy the inequality 1+ln t ) m ess inf ϕx 0,s)s n p t<s< t ñ p nλ+1 Cψx 0,), whee λ = m i=1 λ i and the constant C>0doesnotdepend on. Then the commutato T b Ω,α is bounded fom LM {x0} p,ϕ to LM{x0} q,ψ. Poof Take v 1 t) =ϕx 0,t) 1 t n p,v2 t) =ψx 0,t) 1,gt) = f L q Bx 0,t)) and wt) = 1+ln t )m t nλ ñ p 1. It is easy to see that ws)ds ess sup v 2 t) t ess sup v 1 τ) <. s<τ < Thus, by Lemma 1.2, we have ess sup v 2 t)h w gt) C ess sup v 1 t)gt). So, T b Ω,α f {x LM 0 } q,ψ =supψx 0,) 1 Bx 0,) 1 q TΩ,α f L q Bx 0,)) >0 m b i {x LC 0 } sup ψx 0,) 1 1+ln t ) m t nλ n p 1 f L p i,λ i p Bx 0,t)) = i=1 m b i {x LC 0 } i=1 >0 sup p i,λ i >0 m b i {x LC 0 } f {x p i,λ LM 0 }. p,ϕ i i=1 ϕx 0,) 1 n p f L p Bx 0,))

764 ADVANCES IN MATHEMATICS CHINA) Vol. 46 Thus, we complete the poof of Theoem 3.2. Acknowledgements The authos expess thei gatitude to the efeees fo thei vey valuable comments. Refeences [1] Balakishiyev, A.S., Guliyev, V.S., Gubuz, F. and Sebetci, A., Sublinea opeatos with ough kenel geneated by Caldeón-Zygmund opeatos and thei commutatos on genealized local Moey spaces, J. Inequal. Appl., 2015, 2015: 61, 18 pages. [2] Chen, Y.P., Ding, Y. and Li, R., The boundedness fo commutato of factional integal opeato with ough vaiable kenel, Potential Anal., 2013, 381): 119-142. [3] Ding, Y., Weak type bounds fo a class of ough opeatos with powe weights, Poc. Ame. Math. Soc., 1997, 12510): 2939-2942. [4] Ding, Y., Chen, J.C. and Fan, D.S., A class of integal opeatos with vaiable kenels on Hady spaces, Chinese Ann. Math. Se. A, 2002, 233): 289-296 in Chinese). [5] Guliyev, V.S., Local genealized Moey spaces and singula integals with ough kenel, Azeb. J. Math., 2013, 32): 79-94. [6] Guliyev, V.S., Genealized local Moey spaces and factional integal opeatos with ough kenel, J. Math. Sci. N. Y.), 2013, 1932): 211-227. [7] Mo, H.X. and Xue, H.Y., Commutatos geneated by singula integal opeatos with vaiable kenels and local Campanato functions on genealized local Moey spaces, Commun. Math. Anal., 2016, 192): 32-42. [8] Moey, C.B.J., On the solutions of quasi-linea elliptic patial diffeential equations, Tans. Ame. Math. Soc., 1938, 431): 126-166. [9] Muckenhoupt, B. and Wheeden, R.L., Weighted nom inequalities fo singula and factional integals, Tans. Ame. Math. Soc., 1971, 161: 249-258. [10] Shao, X.K. and Wang, S.P., Boundedness of factional opeatos with vaiable kenels on weighted Moey spaces, J. Anhui Univ. Nat. Sci., 2015, 391): 21-24 in Chinese). [11] Zhang, P. and Chen, J.C., A class of integal opeatos with vaiable kenels on the Hez-type Hady spaces, Chinese Ann. Math. Se. A, 2004, 255): 561-570 in Chinese). [12] Zhang, P. and Zhao, K., Commutatos of integal opeatos with vaiable kenels on Hady spaces, Poc. Indian Acad. Sci. Math. Sci., 2005, 1154): 399-410. ν &flοßμχß)1.$ Campanato fi '»ο"ffi1/ff,$ Moey %ψο-#* 435, 62 ±ffiξ fiφfiffl, ±, 100876) 0+ R T Ω,α T=8NC>@U<F@Va. 7W`QO T Ω,α _A[K: Moey MG LM {x0} p,ϕ >]IY, JZ9DLPO T Ω,α ^K: Campanato BUS;>?XYHEa_A[ K: Moey MG>]IY. ρ! @U<F@; 8NC; HEa; K: Campanato BU; A[K: Moey MG