yearmonth Vol NO Journal name Journal name  English  month. year Vol. No. Continuity for the Maximal Bochner-Riesz operators on the weighted Weak Hardy spaces Zhu Shihong College of Mathematics and Computer ,Tongling University ,Tongling 24400,China Abstact    In this papers ,we generalize some results of other authors to weighted spaces and gain the boundedness of maximal Bochner-Riesz operator on weighted Herz-Hardy spaces,weighted Hardy spaces and weighted weak Hardy spaces ,where ω ∈ A1. Key words: Weighted Herz-Hardy spaces, Maximal Bochner-Riesz operator,Weighted weak Hardy space CLC number:O175.14 Document code  A 1 Introduction Jiang Yinsheng and other authors [1,2] discuss the boundednes of maximal Bochner-Riesz and it ′ s commutator .Recentily ,Tongseng Quek [3]proved some C-Z type operators are bounded on weighted Hardy spaces and weighted weak Hary spaces . Inspiring by the above and the paper[4] ,we will prove the boundedness of the maximal Bochner-Riesz operator on these spaces. Let maximal Bochner-Riesz operator B∗δ be T ∗δ f(x) = sup t>0 |Btδ(f)(x)|, whereBtδ(x) = t −nBδ(x/t) is the kernel of T t δ , and satisfies | ∂ β ∂xβ Bδ(x)| ≤ C(1 + |x|)(δ+(n+1)/2) for any x ∈ Rn and multi-index β ∈ Zn+. Suppose 0 < p < ∞, we denote the weak Lpω(Rn) by WLpω(Rn) and set ‖f‖WLpω(Rn) = sup λ>0 λ[ω({x ∈ Rn : |f(x)| > λ})]1/p, where, and in what follows, ω(E) = ∫ E ω(x)dx. 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,Lecturer, engaged in harmonic analysis. email: zhu−shihong@126.com 2 Journal name 2 [Kα,pq (ω1, ω2),HK α,p q (ω1, ω2)]-Type Continuity. Let us first recall the definition of Herz spaces .For k ∈ Z, let Bk = {x ∈ Rn : |x| ≤ 2k},Ck = Bk \ Bk−1andχk = χCk . Definition 1[4] Let 0 ≤ α < ∞, 1 < q < ∞,0 < p < ∞, and ω1,ω2 be non-negative weight function. The homogeneous weighted Herz spaces Kα,pq (ω1, ω2) are defined by Kα,pq (ω1, ω2) = {f ∈ Lqloc(Rn \{0}, ω2) : ‖f‖Kα,pq (ω1,ω2) < ∞}, where ‖f‖Kα,pq (ω1,ω2) = { ∞∑ k=−∞ ω1(Bk) αp n ‖fχk‖pLqω2 } 1/p. Definition 2[4] Let α ∈ Rn, 1 < q < ∞,0 < p < ∞ and ω1, ω2 be non-negative weight function. The homogeneous weighted Herz-Hardy spaces are defined by HKα,pq (ω1, ω2) = {f ∈ S ′ : G(f) ∈ Kα,pq (ω1, ω2)} and ‖f‖HKα,pq (ω1,ω2) = ‖G(f)‖Kα,pq (ω1,ω2), where G(f) is usually called the grand maximal function of f. Definition 3[4] Let ω1, ω2 ∈ A1, n(1− 1q ) ≤ α < ∞ and s ≥ [α+n( 1q −1)] be non-negative integral. The function a(x) is called a center atom of (α, q, ω1, ω2) − type,if a(x) satisfies (1) supp ak ⊂ Bk = B(0, 2kr), (2) ‖ak‖Lqω2 ≤ ω1(Bk) −α n , (3) ∫ a(x)xβdx = 0, |β| ≤ s. Theorem 1 Let δ > n−12 , 0 < p < ∞, 1 < q < ∞, and n(1 − 1q ) ≤ α < n(1 − 1q ) + ε, ε = min{1, (δ − n−12 )}, ω1, ω2 ∈ A1 ,Then T ∗δ is a bounded operator from HK α,p q (ω1, ω2) to Kα,pq (ω1, ω2). Proof of Theorem 1 Since ω1, ω2 ∈ A1(Rn) , a temperate distribution of f can be written as f = ∑∞ −∞ λjaj and ( ∑∞ −∞ |λj |p) 1 p ≤ C‖f‖HKα,pq (ω1,ω2), where aj is the central atom of (α, q, ω1, ω2)− type. Let n(1− 1q ) ≤ α < n(1− 1q )+ ε and f ∈ HK α,p q (ω1, ω2). Then we have ‖T tδ (f)‖HKα,pq (ω1,ω2) = { ∞∑ k=−∞ ω1(Bk) αp n ‖T tδ (f)χk‖pLqω2} 1 p ≤ C{ ∞∑ k=−∞ ω1(Bk) αp n ( k−3∑ j=−∞ |λj |‖T tδ (aj)χk‖Lqω2 ) p} 1 p + C{ ∞∑ k=−∞ ω1(Bk) αp n ( ∞∑ j=k−2 |λj |‖T tδ (aj)χk‖Lqω2 ) p} 1 p = D1 + D2. Zhu Shihong : Continuity for the Maximal Bochner-Riesz operators on the weighted Weak Hardy spaces 3 For D2, when 0 < p ≤ 1. We notice ω(E)ω(B) ≤ C( |E| |B|) ξ as E ⊂ B and 0 < ξ < 1. By the boundedness of operator T ∗δ on L q ω2(R n)[5] , we have D2 ≤ { ∞∑ k=−∞ ω1(Bk) αp n ( ∞∑ j=k−2 |λj |p‖T tδ (aj)χk‖pLqω2 )} 1 p ≤ C{ ∞∑ j=−∞ |λj |p[ j−1∑ k=∞ ( ω1(Bk) ω1(Bj) ) αp n + 1 + ( ω1(Bj+1) ω1(Bj) ) αp n + ( ω1(Bj+2) ω1(Bj) ) αp n ]} 1 p ≤ C{ ∞∑ j=−∞ |λj |p[ j−1∑ k=∞ ( |Bk| |Bj | ) αpξ n + 1 + 2αp + 22αp]} 1 p ≤ C{ ∞∑ j=−∞ |λj|p} 1 p .. When p > 1, by Hölder ′ s inequality , we have D2 ≤ { ∞∑ k=−∞ ω1(Bk) αp n [ ∞∑ j=k−2 |λj |ω1(Bj) −α n ]p}1/p ≤ { ∞∑ k=−∞ ( ∞∑ j=k−2 |λj |pω1(Bj) −αp 2n ω1(Bk) αp 2n )( ∞∑ j=k−2 ω1(Bj) −αp ′ 2n ω1(Bk) αp ′ 2n ) p p ′ } 1 p ≤ { ∞∑ k=−∞ [ ∞∑ j=k−2 |λj|p( ω1(Bk) ω1(Bj) ) αp 2n [ ∞∑ j=k+1 2 (k−j)αp ′ ξ 2 + 1 + 2 αp ′ 2 + 2αp ′ ] p p ′ } 1 p ≤ { ∞∑ j=−∞ |λj |p( j+2∑ k=−∞ ( ω1(Bk) ω1(Bj) ) αp 2n )} 1 p ≤ { ∞∑ j=−∞ |λj |p j+2∑ k=−∞ 2 (k−j)αpξ 2 } 1 p ≤ C{ ∞∑ j=−∞ |λj |p} 1 p Now,let us turn to the estimate for D1. For any j respect to a fixed k satisfied j ≤ k − 3 ,let Bj = B(0, 2 jr) = B(0, r1) , Bk = B(0, 2 k−jr1). As x ∈ Bk, y ∈ Bj , we know |x−y| ∼ |x−0| .We will consider two cases for D1. case 1 t < r1. By Hölder ′ s inequality and the condition of core B tδ, we have { ∫ Bk | ∫ Bj Btδ(x − y)aj(y)dy|qω2(x)dx} 1 q ≤ C{ ∫ Bk |t−n ∫ Bj (1 + |x − y| t )−(δ+ n+1 2 )aj(y)dy|qω2(x)dx} 1 q ≤ Ctδ−n−12 (2(k−j)r1)−(δ+ n+1 2 )|Bj |( ω2(Bk) ω2(Bj) ) 1 q ( ∫ Bj |aj(y)|qω2(y)dy) 1 q ≤ C2(j−k)(δ+ n+1 2 −n q )‖aj‖Lqω2 ≤ C2 (j−k)(δ+ n+1 2 −n q ) ω1(Bj) −α n . 4 Journal name When 0 < p ≤ 1 ,we gain D1 ≤ C{ ∞∑ k=−∞ ( k−3∑ j=−∞ |λj |p2(j−k)p(δ+ n+1 2 −n q ) ( ω1(Bk) ω1(Bj) ) αp n )} 1 p ≤ C{ ∞∑ j=−∞ |λj |p( ∞∑ k=j+3 2 (j−k)p(δ+ n+1 2 −n q −α)} 1 p ≤ C{ ∞∑ j=−∞ |λj|p} 1 p . When p > 1 .By Hölder ′ s inequality , we have D1 ≤ C{ ∞∑ j=−∞ ( ∞∑ k=j+3 |λj |p2(j−k) p 2 (δ+ n+1 2 −n q −α))( ∞∑ k=j+3 2(j−k) p ′ 2 (δ+ n+1 2 −n q −α)) p p ′ } 1 p ≤ C{ ∞∑ j=−∞ |λj |p( ∞∑ k=j+3 2 (j−k) p 2 (δ+ n+1 2 −n q −α) )} 1 p ≤ C{ ∞∑ j=−∞ |λj |p} 1 p . case 2 t ≥ r1. By the mean value theorem and the vanishing moment condition of atom aj , we have ( ∫ Bk | ∫ Bj Btδ(x − y)aj(y)dy|qω2(x)dx) 1 q = ( ∫ Bk | ∫ Bj [Btδ(x − y) − Btδ(x − 0)]aj(y)dy|qω2(x)dx) 1 q ≤ C{ ∫ Bk |t−(n+1) ∫ Bj (1 + |x − 0| t )−(δ+ n+1 2 )yaj(y)dy|qω2(x)dx} 1 q (1) When δ < n+12 . ( ∫ Bk | ∫ Bj Btδ(x − y)aj(y)dy|qω2(x)dx) 1 q ≤ C(r1 t ) n+1 2 −δ2(j−k)(δ+ n+1 2 )[ ω2(Bk) ω2(Bj) ] 1 q ‖aj‖Lqω2 ≤ C2(j−k)(δ+ n+1 2 −n q ) ω1(Bj) −α n . When 0 < p ≤ 1,we obtain D1 ≤ C{ ∞∑ j=−∞ |λj |p( ∞∑ k=j+3 2(j−k)p(δ+ n+1 2 −n q −α)} 1 p ≤ C{ ∞∑ j=−∞ |λj |p} 1 p . When p > 1. By Hölder ′ s inequality , we also have D1 ≤ C{ ∞∑ k=−∞ ( k−3∑ j=−∞ |λj |p2(j−k) p 2 (δ+ n+1 2 −n q ) ( ω1(Bk) ω1(Bj) ) αp 2n )} 1 p ≤ C{ ∞∑ j=−∞ |λj |p} 1 p . Zhu Shihong : Continuity for the Maximal Bochner-Riesz operators on the weighted Weak Hardy spaces 5 (2) When δ ≥ n+12 . By the mean value theorem and the vanishing moment condition of atom aj , we have ( ∫ Bk | ∫ Bj Btδ(x − y)aj(y)dy|qω2(x)dx) 1 q ≤ C2(j−k)(n+1− n q )ω1(Bj) −α n As the above, we consider D1 under the condition of 0 < p ≤ 1 at first.We obtian D1 ≤ C{ ∞∑ j=−∞ |λj |p( ∞∑ k=j+3 2(j−k)p(n+1− n q −α)} 1 p ≤ C{ ∞∑ j=−∞ |λj|p} 1 p . Also, we secondly consider D1 under the condition of p > 1.By Hölder ′ s inequality , we have D1 ≤ C{ ∞∑ k=−∞ ( k−3∑ j=−∞ |λj |p2(j−k) p 2 (n+1−n q −α)} 1 p ≤ C{ ∞∑ j=−∞ |λj|p} 1 p . Thus,we have ‖T tδ (f)‖Kα,pq (ω1,ω2) ≤ C‖T t δ (f)‖HKα,pq (ω1,ω2). We take the supreme of the left side respect to the inequality above for any t > 0, and obtain desirable result. 3 [Lpω(R n) ,Hpω(R n)]-Type Continuity Definition 4 Let ω∞(R n) and p ∈ (0, 1]. The weighted Hardy spaces Hpω(Rn) is defined by Hpω(R n) = {f ∈ S ′(Rn) : φt ∗ (f)(x) = sup t>0 |φt ∗ f(x)| ∈ Lpω(Rn)}, where φ ∈ S(Rn) is a fixed function with ∫ Rn φ(x)dx 6= 0 and φt = t−nφ(y/t) for any t > 0. Moreover ,we define‖f‖Hpω(Rn) = ‖φ ∗ (f)‖Lpω(Rn). Definition 5 Let ω ∈ A∞, p ∈ (0, 1]. A p-atom with respect to ω is a function a supported in a ball B such that ‖a‖L∞(Rn) ≤ ω(B)− 1 p ∫ Rn a(x)xαdx = 0 for every multi-index α with |α| ≤ [n(qω/p− 1)], where ,and in what follows, [s] denotes the greatest integer less than or equal to s. Theorem 2 Let ω ∈ A1, δ > n−12 , min{ nδ+(n+1)/2 , nn+1} < p ≤ 1, Then T ∗δ is a bounded map from Hpω(Rn) into L p ω(Rn). Proof of Theorem 2 We only need to show that for any p-atom a with respect to ω, ‖T ∗δ (a)‖Lpω(Rn) ≤ C with C independent of a .Suppose supp a ⊂ B(x0, r).Let ω ∈ Aq0(Rn) with . We choose p0 > 1, and write ‖T tδ (a)‖pLpω(Rn) ≤ ∫ B(x0 ,4r) |T tδ (a)(x)|pω(x)dx + ∫ Rn\B(x0 ,4r) |T tδ (a)(x)|pω(x)dx = L1 + L2. 6 Journal name By the boundedness of operator T ∗δ on L q ω [5] , we then have L1 ≤ C( ∫ B(x0,4r) |T tδ (a)(x)|p0ω(x)dx) p p0 ( ∫ B(x0,4r) ω(x)dx) 1− p p0 ≤ C( ∫ B(x0 ,r) |a(x)|p0ω(x)dx) p p0 ω(B(x0, r)) 1− p p0 ≤ C, where C is independent of a. Noticing y ∈ B(x0, r) and x ∈ Rn \ B(x0, 4r), we gain |x − x0| ∼ |x − y|. case 1 t < r By the vanishing moments of a-atom,we gain T tδ (a)(x) ≤ Ct−n ∫ B(x0,r) (1 + |x − y| t )−(δ+ n+1 2 )|a(y)|dy ≤ Ctδ−n−12 |x − x0|−(δ+ n+1 2 )‖a‖∞|B| ≤ Ctδ− n−1 2 |x − x0|−(δ+ n+1 2 )ω(B) − 1 p |B| L2 ≤ Ctp(δ− n−1 2 )ω(B)−1|B|p ∫ Rn\B(x0,4r) |x − x0|−p(δ+ n+1 2 )ω(x)dx ≤ Ctp(δ−n−12 )ω(B)−1|B|p ∞∑ k=2 ∫ Bk+1\Bk |x − x0|−p(δ+ n+1 2 )ω(x)dx ≤ C( t r )p(δ− n−1 2 ) ∞∑ k=2 ( ω(Bk+1) ω(B) )2−kp(δ+ n+1 2 ) ≤ C ∞∑ k=2 2−k[p(δ+ n+1 2 )−n] ≤ C case 2 t ≥ r.By the vanishing moments of a-atom and the mean value theorem ,we have T tδ (a)(x) ≤ Ct−(n+1) ∫ B(x0 ,r) (1 + |x − x0| t )−(δ+ n+1 2 )|y − x0||a(y)|dy When δ < n+12 , we obtain L2 ≤ C( r t )p( n+1 2 −δ) ∞∑ k=2 2−kp(δ+ n+1 2 ) ω(Bk+1) ω(B) ≤ C ∞∑ k=2 2−k[p(δ+ n+1 2 )−n] ≤ C. When δ ≥ n+12 , we obtain T tδ (a)(x) ≤ |x − x0|−(n+1)rn+1‖a‖∞ L2 ≤ rp(n+1)ω(B)−1 ∞∑ k=2 ∫ Bk+1\Bk |x − x0|−p(n+1)ω(x)dx ≤ C ∞∑ k=2 2−k[p(n+1)−n] ≤ C. Thus,we have ‖T tδ (a)‖pLpω ≤ C For any t > 0,we take the supreme of the left side and can gain the desire result. Zhu Shihong : Continuity for the Maximal Bochner-Riesz operators on the weighted Weak Hardy spaces 7 4 [WLpω(R n), WHpω(R n)]-Type Continuity . Lemma 1 Let p ∈ (0, 1] and ω ∈ A∞(Rn). For f ∈ WHp∞(Rn), there exists a sequence {fk}∞k=−∞ of bounded measurable functions such that f = ∞∑ k=−∞ fk inS ′ (Rn). (1) Each fk can be further decomposed into fk = ∑ i bki, where the sequence {bki}i satisfies suppbki ⊂ QkiandQkiisacube, (2) ∑ i ω(Qki) ≤ c12−kp; ∑ i χQki(x) ≤ c1, χE being the characteristic function of the set E, c1 a constant and c1 ≤ C‖f‖pWHpω(Rn); ‖bki‖L∞(Rn) ≤ C2k and ∫ Rn bki(x)x αdx = 0, for|α| ≤ [n(qω/p − 1)]. Conversely, if f ∈ S ′(Rn) has a decomposition satisfying (1) and (2) ,then f ∈ WH pω(Rn) and ‖f‖p WHpω(Rn) ≤ Cc1, where C is a constant. Theorem 3 Let δ > n−12 , max{ nδ+(n+1)/2 , nn+1} < p ≤ 1, ω ∈ Aq ,q ≥ 1, Then T ∗δ is a bounded operator from WHpω(Rn) to WL p ω(Rn). Proof of Theorem 3 For any λ > 0, let k0 ∈ Z such that 2k0 ≤ λ < 2k0−1. Let ω ∈ Aq(Rn) and f ∈ WHpω(Rn). Then by Lemma 1,we write f = ∞∑ k=−∞ fk = ∞∑ k=−∞ ∑ i bki = k0∑ k=−∞ ∑ i bki + ∞∑ k=k0+1 ∑ i bki = F1 + F2, where b,k,is are as in Lemma 1. Suppose Ak = suppfk,then Ak = ∪iQki and ω(Ak) ≤∑ i ω(Qki) ≤ C2−kp‖f‖ p WHpω .Since Bδ∗ is bounded on L q ω spaces, we have ω({x ∈ Rn : |T tδ (F1)(x)| > λ}) ≤ C‖T tδ (F1)‖2L2ω/λ 2 ≤ C‖F1‖2L2ω/λ 2 ≤ C[ k0∑ k=−∞ ‖fk‖L2ω ] 2/λ2 ≤ C[ k0∑ k=−∞ ( ∑ i ∫ Bki |bki|2ω(x)dx) 1 2 ]2/λ2 ≤ C{ k0∑ k=−∞ 2k[ ∑ i ω(Bki)] 1 2 }2/λ2 ≤ C[ k0∑ k=−∞ 2k(1−p/2)]2‖f‖p WHpω /λ2 ≤ C‖f‖p WHpω /λp. Now set Q∗ki = Q(xki, (5 √ n)rki) and Ak0 = ⋃∞ k=k0+1 ⋃ i Q ∗ ki, where Q ∗ ki is the cube with the same xki-center as Qki and side length 5 √ n times the side length of Qki. We get ω(Qk0) ≤ ∞∑ k=k0+1 ∑ i ω(Q∗ki) ≤ ∞∑ k=k0+1 ∑ i ω(Qki) ≤ ∞∑ k=k0+1 2−kp‖f‖p WHpω ≤ ‖f‖p WHpω /λp. 8 Journal name To finish the proof, we still need to estimate ω({x ∈ Ack0 : |T t δ (F2)(x)| > λ)}) ≤ C‖T tδ (F1)‖pLpω/λ p. For x ∈ (A∗k0) c,y ∈ Qki, we know |x − y| ∼ |x − xki|. case 1 t < rki ,We then obtain ∫ Ac k0 |T tδ (bki)(x)|pω(x)dx = ∫ Ac k0 { ∫ Qki t−n(1 + |x − y| t )−(δ+ n+1 2 )|bki(y)|dy)}pω(x)dx ≤ C ∞∑ j=2 ∫ Qj+1 ki \Qj ki tp(δ− n−1 2 )|x − xki|−p(δ+ n+1 2 2kp|Qki|pω(x)dx ≤ C ∞∑ j=2 2−jp(δ+ n+1 2 )( t rki )p(δ− n−1 2 )2kpω(Qj+1ki ) ≤ C ∞∑ j=2 2−j[p(δ+ n+1 2 )−n]2kpω(Qki) ≤ C2kpω(Qki), where Qjki is the cube with the same xki-center as Qki and side length 2 j times the side length-rki of Qki. case 2 t ≥ rki. we consider δ < n+12 at first. By the vanishing moments of bki and the mean value theorem, we have ∫ Ac k0 |T tδ (bki)(x)|pω(x)dx = ∫ Ac k0 | ∫ Qki [Btδ(x − y) − Btδ(x − xki)]bki(y)dy|pω(x)dx ≤ C ∞∑ j=2 ∫ Qj+1 ki \Qj ki tp(δ− n+1 2 )(2jrki) −p(δ+ n+1 2 )rpki2 kp|Qki|pω(x)dx ≤ C ∞∑ j=2 ( rki t )p( n+1 2 −δ)2−j[p(δ+ n+1 2 )−n]2kpω(Qki) ≤ C2kpω(Qki) Secondly,we consider δ ≥ n+12 ,∫ Ac k0 |T tδ (bki)(x)|pω(x)dx = ∫ Qc k0 | ∫ Qki [Btδ(x − y) − Btδ(x − xki)]bki(y)dy|pω(x)dx ≤ C ∞∑ j=2 ∫ Qj+1 ki \Qj ki ( ∫ Bki |x − xki|−(n+1)rki|bki(y)|dy)pω(x)dx ≤ C ∞∑ j=2 2−j[p(n+1)−n]2kpω(Qki) ≤ C2kpω(Qki) These yields the desired results . Acknowledgements    References Zhu Shihong : Continuity for the Maximal Bochner-Riesz operators on the weighted Weak Hardy spaces 9 [1] Jiang Yinsheng ,Tang Lin , Continuity for maximal commutators of Bochner-Riesz operators with BMO functions[J],Acta Mathematica. Scientia.2002,22B(3):339-349. 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