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. [2] Zhu Shihong, Continuity of of maximal Bochner-Riesz operators and its commutators[D],Anhui normal University,2006. [3] Tongseng Quek, Yang Dachun, Callderón-Zygmund-Type operators on weighted weak Hardy spaces over R n[J] , Atca Mathematica Sinica ,English Series,2000,16(1):141-160. [4] Lu Shanzhen , Yang Dachun, Weighted Herz-Hardy spaces and its Applications[J], Science in China (Series A),1995,25(3):235-245. [5] Wang Hua ,Some estimates of Bocher-Riesz operators on weighted Morrey spaces[J],2012,55(3):551-560.