yearmonth Vol NO Journal name Journal name   English  month. year Vol. No. Continuity of higher order commutators generated by maximal Bochner-Riesz operator on Morrey space Shihong Zhu (College of Mathematics and Computer Technology,Tongling University ,Tongling 24400,China ) Abstact    In this papers ,we use the control method of the maximal fractional integral and obtain the boundedness of higher order commutator generated by maximal Bochner-Riesz operator on Morrey space. Moreover , we get it's continuty from Morrey space to Lipschtz space and from Morrey space to BMO space. Key words: BMO space, higher commutator, Morrey space,Lipschtz space CLC number:O175.14 Document code  A 1 Introduction And Main Results Let maximal Bochner-Riesz operator B∗δ [1] be B∗δf(x) = sup t>0 |Btδ(f)(x)|, whereBtδ(x) = t −nBδ(x/t) is the kernel of B t δ, which satisfies | ∂β ∂xβ Bδ(x)| ≤ C(1+|x|) −(δ+(n+1)/2) for any x ∈ Rn and multi-index β ∈ Zn+. For 0 < β < 1 ,let the homogeneous Besov-Lipschitz space Λβ [2] be the space of the functions satisfying ‖f‖Λβ = sup x,y∈Rn,x6=y |f(x) − f(y)| |x − y|β . Suppose b ∈ Λβ. For suitable function f , define the high order commutator B b,m δ,∗ generated by B∗δ and b as follows Bb,mδ,∗ f(x) = sup t>0 ∫ Rn Btδ(x − y)[b(x) − b(y)] mf(y)dy. Receive date  . Foundation item  The Natural Science Foundation of Anhui Province(kj2010b460),  Continuty of the multinear operators  , Peroject director: Zhu Shihong Biography  Zhu Shihong(1972-),male, engaged in harmonic analysis. email: zhu−shihong@126.com 2 Journal name For 1 < p < ∞, λ ≥ 0, the classic Morrey space M p,λ(Rn)[3] is the space of function f ∈ Lploc such that ‖f‖Mp,λ = sup x∈Rn, r>0 ( 1 |Q(x, r)|λ ∫ Q(x,r) |f(y)|pdy)1/p < ∞. Let 1 ≤ p < ∞. Denote the generalized Morrey space[4] as Mp,φ = {f ∈ Lploc(R n) : ‖f‖Mp,φ < ∞}, where ‖f‖Mp,φ = sup x∈Rn,r>0 ( 1 φ(x, r) ∫ Q |f(y)|pdy)1/p, φ is a integer increasing function on Rn × R+ satisfying φ(x, 2r) ≤ Dφφ(x, r), andDφis a constant ,independent of r. For 1 ≤ l < ∞ and β > 0, we call that T lβl (f)(x) = sup x∈Q ( 1 |Q|1−lβ/n ∫ Q |f(y)|l y. )1/l is the maximal fractional integral[5],where l < p < n/β and 1/q = 1/p − β/n. Let λ < 1/n and 1 < q < ∞. We define f ∈ Lqloc(R n) is belong to BMOλ,q space if f satisfies ‖f‖BMOλ,q = sup x∈Rn, r>0 ( 1 |Q(x, r)|1+λq ∫ Q(x,r) |f(y) − fQ| qdy)1/q < ∞. Also,we have ‖f‖BMOλ,q ≈ sup x∈Rn,r>0 ( 1 |Q(x, r)|1+λ ∫ Q(x,r) |f(y) − fQ|dy) ≈ ‖f‖Λβ , where fQ = 1 |Q| ∫ Q f(y)dy, λ = β/n. In this paper, x0 is the center of cube Q with the side length r. Qk is the cube with the same center as Q and side length 2k times the side length of Q. Theorem 1.1 Suppose λ ≥ 0, δ > (n− 1)/2, 0 < mβ < n, 0 < β < 1 , 1 < p < n/(mβ), 1/p − mβ/n = 1/q and b ∈ Λβ. We have ‖Bb,mδ,∗ (f)‖Mq,λ ≤ C‖f‖Mp,λ(1−βmp/n). Corollary 1.2 Let φ 1/p1 1 (r) = φ 1/p2 2 (r). Under conditions of theorem 1.1 , we get ‖Bb,mδ,∗ (f)‖Mp2,φ2 ≤ C‖f‖Mp,φ1 . Shihong Zhu : Continuity of higher order commutators generated by maximal Bochner-Riesz operator on Morrey space 3 Theorem 1.3 Let δ > (n − 1)/2, b ∈ Λβ, 0 < β < 1, 0 < mβ < n, 1/p − mβ/n = 1/q ,1 < p < n/(mβ), ζ = min{1, δ − (n − 1)/2}, and −1/q < λ < ζ .We have ‖Bb,mδ,∗ ‖BMOq,λ ≤ C‖f‖Mp,1+p(λ−mβ/n). Theorem 1.4 Suppose δ > (n−1)/2, 1/p−mβ/n = 1/q and b ∈ Λβ, 0 < β < 1 ,0 < mβ < n, 1 < p < n/(mβ). Let ζ = min{1, δ − (n − 1)/2} and 1 − (mpβ)/n < λ < 1 + (ζ − mβ)p/n . We get ‖Bb,mδ,∗ ‖Λmβ+(λ−1)n/p ≤ C‖f‖Mp,λ . 2 [Mp,λ(1−βmp/n)(Rn), M q,λ(Rn)]-Type Continuity. Lemma 2.1 Let δ > (n− 1)/2, b ∈ Λβ, 0 < mβ < n , 1 < p < n/(mβ) and 1/p−mβ/n = 1/q ,Then there is a constant C > 0 ,independent of f ,such that ‖T b,mδ,∗ f‖q ≤ C‖f‖p. Proof of lemma 2.1 For x ∈ Rn and ε > 0 with 0 < mβ − ε < mβ + ε < n, we choose a ξ > 0 such that ξ2ε = Mmβ−εf(x)/Mmβ+εf(x). Write T b,mδ,t f(x) = ∫ |x−y|<ξ Btδ(x − y)[b(x) − b(y)] mf(y)dy + ∫ |x−y|≥ξ Btδ(x − y)[b(x) − b(y)] mf(y)dy = D1 + D2. By the inequality[4] |Btδ(x − y)| ≤ |x − y| −n and b ∈ Λβ, we have |D1| ≤ C ∞ ∑ j=0 ∫ 2−j−1ξ≤|x−y|<2−jξ |f(y)| |2−jξ|n−mβ dy ≤ C ∞ ∑ j=0 (2−jξ)ε |2−jξ|n−mβ+ε ∫ |x−y|<2−jξ |f(y)|dy ≤ C ∞ ∑ j=0 2−jεξεMmβ−εf(x) ≤ Cξ εMmβ−εf(x). 4 Journal name Similarly,we get |D2| ≤ C ∞ ∑ j=1 ∫ 2j−1ξ≤|x−y|<2jξ |2jξ|mβ |2j−1ξ|n |f(y)|dy ≤ C ∞ ∑ j=1 (2jξ)−ε |2jξ|n−mβ−ε ∫ |x−y|<2jξ |f(y)|dy ≤ C ∞ ∑ j=0 2−jεξ−εMmβ+εf(x) ≤ Cξ −εMmβ+εf(x). Thus,by the above selection of ξ we get |T b,mδ,t f(x)| ≤ C(ξ εMmβ−εf(x) + ξ −εMmβ+εf(x)) = C[Mmβ+εf(x)] 1/2[Mmβ+εf(x)] 1/2. Noting 1 < p < n/(mβ), there is an ε > 0, such that 1 < p < n/(mβ + ε). Let 1/q1 = 1/p − (mβ − ε)/n, 1/q2 = 1/p − (mβ + ε)/n, l = 2q1/q, l ′ = 2q2/q, then q1, q2 > 0,l ′ > 1 and 1/l + 1/l ′ = 1. Thus ,by inequality (1) ,we have ‖T b,mδ,t f‖ q q ≤ C ∫ Rn |Mmβ−εf(x)| q/2|Mmβ+εf(x)| q/2dx ≤ C( ∫ Rn |Mmβ−εf(x)| ql/2dx)1/l( ∫ Rn |Mmβ+εf(x)| ql ′ /2dx)1/l ′ ≤ C( ∫ Rn |Mmβ−εf(x)| q1dx)q/2q1( ∫ Rn |Mmβ+εf(x)| q2dx)q/2q2 ≤ C( ∫ Rn |f(x)|pdx)q/2p( ∫ Rn |f(x)|pdx)q/2p = C‖f‖qp. Lemma 2.2[6] Let T lβl be the maximal fractional operator , λ ≥ 0, 1 ≤ l < p < n/β and 1/p − β/n = 1/q, we have ‖T lβl (f)‖Mq,λ ≤ C‖f‖Mp,λ(1−βp/n) . Proof of Theorem 1.1 Let f = fχ2Q + fχ(2Q)c = f1 + f2. Write 1 |Q| ∫ Q |Bb,mδ,t f(y) − B b,m δ,t f2(x0)|dy ≤ C 1 |Q| ∫ Q |Bb,mδ,t f1(y)|dy + C 1 |Q| ∫ Q |Bb,mδ,t f2(y) − B b,m δ,t f2(x0)|dy = E1 + E2. We choose l satisfying 1 ≤ l < p < n/(mβ). Take a s such that 1/s = 1/l −mβ/n. By the lemma 2.1 and Hölder inequality, we have E1 ≤ C( 1 |Q| ∫ Q |T b,mδ,t f1(y)| sdy)1/s ≤ C 1 |Q|1/s ( ∫ Q |f(y)|ldy)1/l ≤ C( 1 |Q|1−mβl/n ∫ Q |f(y)|ldy)1/l ≤ CT mβll f(x). Shihong Zhu : Continuity of higher order commutators generated by maximal Bochner-Riesz operator on Morrey space 5 We turn to E2. Noticing z ∈ (2Q) c, y ∈ Q and |y − z| ∼ |x0 − z|,we have case 1 t < r |Bb,mδ,t f2(y) − B b,m δ,t f2(x0)| ≤ C ∞ ∑ k=1 | ∫ Qk+1\Qk Btδ(y − z)[b(y) − b(z)] mf(z)dz| ≤ C ∞ ∑ k=1 |2kr|mβt−n ∫ Qk+1\Qk (1 + |x − z| t )−(δ+ n+1 2 )|f(z)|dz ≤ C ∞ ∑ k=1 2−k[δ−(n−1)/2]( 1 |Qk+1|1−mβl/n ∫ Qk+1 |f(z)|ldz)1/l ≤ CT mβll f(x). case 2 t ≥ r (1)We conside δ < (n + 1)/2. By the mean value theorem, we obtain |T b,mδ,t f2(y) − T b,m δ,t f2(x0)| ≤ C ∞ ∑ k=1 |2kr|mβ| ∫ Qk+1\Qk Btδ(y − z)f(z)dz − ∫ Qk+1\Qk Btδ(x0 − z)f(z)dz| ≤ C ∞ ∑ k=1 |2kr|mβt−n−1 ∫ Qk+1\Qk (1 + |x0 − z| t )−(δ+ n+1 2 )|y − x0||f(z)|dz ≤ C ∞ ∑ k=1 ( r t ) n+1 2 −δ2−k[δ−(n−1)/2]( 1 |Qk+1|1−mβl/n ∫ Qk+1 |f(z)|ldz)1/l ≤ CT mβll f(x). (2)When δ ≥ (n + 1)/2, we also obtain |T b,mδ,t f2(y) − T b,m δ,t f2(x0)| ≤ C ∞ ∑ k=1 |2kr|mβt−n−1r ∫ Qk+1\Qk (1 + |x0 − z| t )−(δ+ n+1 2 )|f(z)|dz ≤ C ∞ ∑ k=1 2−kTmβll f(x) ≤ CT mβl l f(x). Then,by the lemma 2.2,we gain ‖Bb,mδ,t (f)‖Mq,λ ≤ C‖M(B b,m δ,t )(f)‖Mq,λ ≤ C‖(B A δ,t) ](f)(x)‖Mq,λ ≤ C‖T mβll (f)‖Mq,λ ≤ C‖f‖Mp,λ(1−mβp/n) . Taking the supreme of the left side about E1 and E2 for any t > 0 , this complets the proof of theorem 1.1. 6 Journal name Proof of Corollary 1.2 ( 1 φ2(r) ∫ Q |Bb,mδ,t f(x)| qdx)q = |Q|λ/q φ 1/q 2 (r) ( 1 |Q|λ ∫ Q |Bb,mδ,t f(x)| qdx)1/q ≤ |Q|λ/qφ1(r) 1/p φ2(r)1/q|Q|λ/p−λmβ/n ( 1 φ1(r) ∫ Q |f(x)|pdx )1/p = C ( 1 φ1(r) ∫ Q |f(x)|pdx )1/p . 3 [Mp,1+p(λ−mβ/n)(Rn),BMOλ,q(Rn)]-Type Continuity . Proof of Theorem 1.3 Write f = fχ2Q + fχ(2Q)c = f1 + f2, then ( 1 |Q|1+qλ ∫ Q |Bb,mδ,t f(x) − B b,m δ,t f2(x0)| qdx)1/q ≤ C( 1 |Q|1+qλ ∫ Q |Bb,mδ,t f1(x)| qdx)1/q + C( 1 |Q|1+qλ ∫ Q |Bb,mδ,t f2(x) − B b,m δ,t f2(x0)| qdx)1/q = F1 + F2. By the lemma 2.1,we have F1 ≤ C 1 |Q|1/q+λ ( ∫ Q |f1(x)| pdx)1/p ≤ C‖f‖Mp,1+p(λ−mβ/n). Now,we consider F2. Noticing x ∈ Q, we have y ∈ (2Q) c. case 1 t < r .We then obtain |Bb,mδ,t f2(x) − B b,m δ,t f2(x0)| ≤ C ∞ ∑ k=1 |2k+1Q|1+λtδ−(n−1)/2|2kr|−[δ+(n+1)/2]( 1 |Qk+1|1+p(λ−mβ/n) ∫ Qk+1 |f(y)|pdy)1/p ≤ C( t r )δ−(n−1)/2 ∞ ∑ k=1 2−k[δ−(n−1)/2−nλ]rnλ‖f‖Mp,1+p(λ−mβ/n) ≤ Cr nλ‖f‖Mp,1+p(λ−mβ/n) . case 2 t ≥ r. (1)When δ < (n + 1)/2, by the mean value theorem, we have |T b,mδ,t f2(x) − T b,m δ,t f2(x0)| ≤ C ∞ ∑ j=2 |2k0+j|mβt−n−1 ∫ Bk0+j+1\Bk0+j |x − x0|(1 + |x − y| t )−[δ+(n+1)/2]|f(y)|dy ≤ C ∞ ∑ k=1 2−k[δ−(n−1)/2−nλ]rnλ‖f‖Mp,1+p(λ−mβ/n) ≤ Cr nλ‖f‖Mp,1+p(λ−mβ/n). (2)When δ ≥ (n + 1)/2, we similarly gain |T b,mδ,t f2(x) − T b,m δ,t f2(x0)| ≤ C ∞ ∑ k=1 2−k(1−nλ)rnλ‖f‖Mp,1+p(λ−mβ/n) ≤ Crnλ‖f‖Mp,1+p(λ−mβ/n) . Shihong Zhu : Continuity of higher order commutators generated by maximal Bochner-Riesz operator on Morrey space 7 So,we get estimation for F2 which is F2 ≤ C‖f‖Mp,1+p(λ−mβ/n). These yields the desired result. 4 [Mp,λ(Rn),Λ[mβ+n/p(λ−1)]/n(R n)]-Type Continuity . Proof of Theorem 1.4 We write ∫ Q |Bb,mδ,t f(x) − T b,m δ,t f2(x0)|dx ≤ C ∫ Q |T b,mδ,t f1(x)|dx + ∫ Q |T b,mδ,t f2(x) − T b,m δ,t f2(x0)|dx = G1 + G2. By the Hölder inequality and the theorem 1.1 ,we get G1 ≤ C( ∫ Q |T b,mδ,t f1(x)| qdx) 1 q |Q|1−1/q ≤ C|Q|1+[mβ+(λ−1)n/p]/n‖f‖Mp,λ . Now,we estimate G2. Noting x ∈ Q, y ∈ (2Q) c, we know |x − y| ∼ |x0 − y|. case 2 t < r. We get |T b,mδ,t f2(x) − T b,m δ,t f2(x0)| ≤ C ∞ ∑ k=1 2−k[δ−(n−1)/2−mβ+(1−λ)n/p] |Q|mβ/n−1/p+λ/p‖f‖Mp,λ ≤ C|Q|mβ/n−1/p+λ/p‖f‖Mp,λ. case 2 t ≥ r. (1)When δ < (n + 1)/2,we have |T b,mδ,t f2(x) − T b,m δ,t f2(x0)| ≤ C( r t )(n+1)/2−δ ∞ ∑ k=1 2−k[δ−(n−1)/2−mβ+(1−λ)n/p]|Q|mβ/n−1/p+λ/p‖f‖Mp,λ ≤ C|Q|mβ/n−1/p+λ/p‖f‖Mp,λ . (2)When δ ≥ (n + 1)/2,we have |T b,mδ,t f2(x) − T b,m δ,t f2(x0)| ≤ C ∞ ∑ k=1 2−k[1−mβ+(1−λ)n/p]|Q|mβ/n−1/p+λ/p‖f‖Mp,λ ≤ C|Q|mβ/n−1/p+λ/p‖f‖Mp,λ . 8 Journal name We gain G2 ≤ C|Q| 1+[mβ+(λ−1)n/p]/n‖f‖Mp,λ . Together with G1 and G2, we have 1 |Q|1+[mβ+(λ−1)n/p]/n ∫ Q |Bb,mδ,t f(x) − T b,m δ,t f2(x0)|dx ≤ C‖f‖Mp,λ. Taking the supreme of the left side for any t > 0 ,we get the ideal result.This finishes the proof of theorem 1.4. References [1] Lu,S.Z.,Four lectures on real Hp spaces. World Scientific Publishing, River Edge, NI,1995. [2] M.Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochhberg and Weiss.Indiana Univ.Math.J.,1995,44:1-17. [3] Komori Y.,Shirai S.,Weighted Morrey spaces and a singular integral operator. Math Nachr., 2009,282:219-231. [4] Mizuhara T.,Boundedness of some classical operators on generalized Morrey space in Harmonic analysis[M].ICM90 Satellite proceedings Tokyo:Springer-Verlag,1991,183-189. [5] S.Chanillo,A note on commutators. Indiana Univ.Math.J.,1982,31:7-16. [6] Shihong Zhu, Estimations of the maximal multilinear Bochner-Riesz operator on Morrey space. Acta mathematicae applicatae sinica (Ser.A), 2013,36(4):631-639. Address and Email: College of Mathematics and Computer, Tongling University, 24400,Tongling City , Anhui Province ,China. email: zhu−shihong@126.com