Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 25(1) (2009), ISSN

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1 Aca Maheaica Acadeiae Paedagogicae Nyíregyháziensis 25) 2009), ISSN LIPSCHITZ ESTIMATES FOR MULTILINEAR COMMUTATOR OF LITTLEWOOD-PALEY OPERATOR WANG MEILING AND LIU LANZHE Absrac. In his paper, we will sudy he coninuiy of ulilinear couaor generaed by Lilewood-Paley operaor and Lipschiz funcions on Triebel-Lizorkin space, Hardy space and Herz-Hardy space.. Inroducion Le T be he Calderón-Zygund operaor, Coifan, Rochberg and Weiss see [4]) proves ha he couaor [b, T ]f) = bt f) T bf)where b BMOR n )) is bounded on L p R n ) for < p <. Chanillo see [2]) proves a siilar resul when T is replaced by he fracional operaors. In [8, 6], Janson and Paluszynski sudy hese resuls for he Triebel-Lizorkin spaces and he case b Lip β R n ), where Lip β R n ) is he hoogeneous Lipschiz space. The ain purpose of his paper is o discuss he boundedness of ulilinear couaor generaed by Lilewood-Paley operaor and b j on Triebel-Lizorkin space, Hardy space and Herz-Hardy space, where b j Lip β R n ). 2. Preliinaries and Definiions Throughou his paper, Mf) will denoe he Hardy-Lilewood axial funcion of f, and wrie M p f) = Mf p )) /p for 0 < p <, will denoe a cube of R n wih side parallel o he axes. Le f = fx)dx and f # x) = sup x fy) f dy. Denoe he Hardy spaces by H p R n ). I is well known ha H p R n )0 < p ) has he aoic decoposiion characerizaionsee [2][7][8]). For β > 0 and p >, le F p β, R n ) be he hoogeneous Tribel-Lizorkin space. The Lipschiz space Lip β R n ) is he space 2000 Maheaics Subjec Classificaion. 42B20, 42B25. Key words and phrases. Lilewood-Paley operaor; Mulilinear couaor; Triebel- Lizorkin space; Herz-Hardy space; Herz space; Lipschiz space. 85

2 86 WANG MEILING AND LIU LANZHE of funcions f such ha f Lipβ fx) fy) = sup <. x,y R n x y β x y Lea [6]). For 0 < β <, < p <, we have f F p β, sup fx) f β dx n L p sup inf fx) c dx c β n Lea 2 [6]). For 0 < β <, p, we have f Lipβ sup fx) f β dx n /p sup fx) f β dx) p. n Lea 3 [2]). For r < and β > 0, le /r M β,r f)x) = sup βr fy) dy) r, x n suppose ha r < p < n/β, and / = /p β/n, hen Lea 4 [5]). Le 2, hen M β,r f) L f L p. f f 2 f Lipβ 2 β/n. Definiion. Le 0 < p. A funcion ax) on R n is called a H p -ao, if ) supp a Bx 0, r) for soe x 0 and for soe r > 0 or for soe r ), 2) a L Bx 0, r) /p, 3) R n ax)x γ dx = 0 for all γ wih 0 γ [n/p )]. Lea 5. see [4][5]) Le 0 < p. A disribuion f on R n is in H p R n ) if and only if f can be wrien as f = λ ja j in he disribuional sense, where each a j is H p -ao and λ j are consans wih λ j p <. Moreover, ) /p f H p = inf λ j p, where he infiu ake over all decoposiions of f as above. Definiion 2. Le 0 < p, <, α R, B k = {x R n, x 2 k }, A k = B k \B k and χ k = χ Ak for k Z. L p.

3 ) The hoogeneous Herz space is defined where LIPSCHITZ ESTIMATES K α,p R n ) = {f L Loc Rn \{0}) : f K α,p < }, f K α,p = [ 2 kαp fχ k p L ] /p 2) The nonhoogeneous Herz space is defined by where K α,p R n ) = {f L Loc Rn ) : f K α,p < }, f K α,p = [ ] /p 2 kαp fχ k p L fχ B 0 p L. k= Definiion 3. Le α R, 0 < p, <. ) The hoogeneous Herz ype Hardy space is defined by and H α,p K R n ) = {f S R n ) : Gf) f H K α,p = Gf) K α,p ; ; K α,p R n )}, 2) The nonhoogeneous Herz ype Hardy space is defined by and HK α,p R n ) = {f S R n ) : Gf) K α,p R n )}, f HK α,p = Gf) K α,p ; where Gf) is he grand axial funcion of f, ha is Gf)x) = sup sup ϕ K x y < f ϕ y), where K = {ϕ SR n ) : sup x R n, α u ) n D α ϕu) }, ϕ x) = n ϕx/) for > 0, is a posiive ineger and SR n ) is he Schwarz class see [7, p. 88]). The Herz ype Hardy spaces have he aoic decoposiion characerizaion. Definiion 4. Le α R, < <. A funcion a on R n is called a cenral α, )-ao or a cenral a, )-ao of resric ype), if ) supp a B0, r) for soe r > 0 or for soe r ), 2) a L B0, r) α/n, 3) R n ax)x η dx = 0 for all η wih η [α n /)].

4 88 WANG MEILING AND LIU LANZHE Lea 6 [6, 5]). Le 0 < p <, < < and α n /). A α,p eperae disribuion f belongs o H K R n ) or HK α,p R n )) if and only if here exis cenral α, )-aos or cenral α, )-aos of resric ype) a j suppored on B j = B0, 2 j ) and consans λ j, j λ j p < such ha f = λ ja j or f = j=0 λ ja j ) in he S R n ) sense, and f H K α,p or f HK α,p) Definiion 5 [7]). Le α R, < p, <. ) /p λ j p. ) A easure funcion f is said o belong o hoogeneous weak Herz space W R n ), if f W K α,p K α,p = sup λ>0 λ j 2 kαp {x E k : fx) > λ} p/ ) /p < ; 2) A easure funcion f is said o belong o inhoogeneous weak Herz space W K α,p R n ), if f W K α,p = sup λ>0 λ k= 2 kαp {x E k : fx) > λ} p/ {x B 0 : fx) > λ p/ } ) /p <. Definiion 6. Le µ >, n > δ > 0 and ψ be a fixed funcion which saisfies he following properies: ) R n ψx)dx = 0, 2) ψx) x ) n δ), 3) ψx y) ψx) y x ) n2 δ) when 2 y < x. Given a posiive ineger and he locally inegrable funcion b j j ). The ulilinear couaor of Lilewood-Paley operaor is defined by [ ) ] nµ g b µ,δ f)x) = F b f)x, y) 2 dyd /2, x y n where R n [ ] F b f)x, y) = b j x) b j z)) ψ y z)fz)dz. R n j= When =, se [ g b µ,δf)x) = R n ) ] nµ F b f)x, y) 2 dyd /2, x y n

5 LIPSCHITZ ESTIMATES where F b f)x, y) = bx) bz))ψ y z)fz)dz R n and ψ x) = nδ ψx/) for > 0. Se F f)x) = f ψ x), we also define ha [ ) ] nµ g µ,δ f)x) = F f)y) 2 dyd /2, x y n R n which is he Lilewood-Paley { operaor see [8]). Le H be he space H = h : h = } hy, ) 2 dyd/ n ) /2 <, R n hen, for each fixed x R n F f)x) ay be viewed as a apping fro [0, ) o H, and i is clear ha ) nµ/2 g µ,δ f)x) = F f)y) x y and ) nµ/2 g b µ,δ f)x) = F b f)x, y) x y. Noe ha when b = = b, g b µ,δ is jus he order couaor. I is well known ha couaors are of grea ineres in haronic analysis and have been widely sudied by any auhors see [, 2, 3, 4, 8, 0, 9,, 6]). Our ain purpose is o esablish he boundedness of he ulilinear couaor on Triebel-Lizorkin space, Hardy space and Herz-Hardy space. Lea 7 [9]). Le 0 < δ < n, < p < n/δ, / = /p δ/n and w A. Then g µ,δ is bounded fro L p w) o L w). Given soe funcions b j j =,..., ) and a posiive ineger and j, we se b Lipβ = j= b j Lipβ and denoe by Cj he faily of all finie subses σ = {σ),..., σj)} of {,..., } of j differen eleens. For σ Cj, se σ c = {,..., } \ σ. For σ = {σ),..., σj)} Cj, se b σ = b σ),..., b σj) ), b σ = b σ)... b σj) and b σ Lipβ = b σ) Lipβ... b σj) Lipβ. 3. Theores and Proofs Theore. Le 0 < β < in, /2), µ > 3 /n 2δ/n, < p <, b j Lip β R n ) for j and g b µ,δ be he ulilinear couaor of Lilewood-Paley operaor as in Definiion 6. Then a) g b µ,δ is bounded fro L p R n ) o F p β, R n ) for < p < n/δ and /p / = δ/n. b) g b µ,δ is bounded fro L p R n ) o L R n ) for /p / = β δ)/n and /p > β δ)/n.

6 90 WANG MEILING AND LIU LANZHE Proof. a). Fixed a cube = x 0, l) and x. Se b = b ),..., b ) ), where b j ) = b jy)dy, j. Wrie f = fχ f 2 = fχ R n \ = f f 2, we have = F b f)x, y) = [b j x) b j ) ) b j z) b j ) )] ψ y z)fz)dz R n j= ) j bx) b ) σ R bz) b ) σ cψ y z)fz)dz n j=0 σ Cj = b x) b ) )... b x) b ) )F f)y) ) F b b ) )... b b ) )f)y) j= σ Cj ) j bx) b ) σ bz) b ) σ cψ y z)fz)dz R n = b x) b ) )... b x) b ) )F f)y) ) F b b ) )... b b ) )f )y) ) F b b ) )... b b ) )f 2 )y) j= σ Cj c,j bx) b ) σ F b b ) σ cf)y), hen g b µ,δ f)x) g µ,δ b ) b )... b ) b ))f 2 )x 0 ) ) nµ/2 F b f)x, y) x y ) nµ/2 F b ) b )... b ) b )f 2 )y) x 0 y ) nµ/2 b x) b ) )... b x) b ) )F f)y) x y ) nµ/2 x y bx) b ) σ F b b ) σcf)y) j= σ Cj ) nµ/2 F b b ) )... b b ) )f )y) x y

7 LIPSCHITZ ESTIMATES... 9 Thus, β/n β/n ) nµ/2 F b j b j ) )f 2 )y) x y j= ) nµ/2 F b j b j ) )f 2 )y) x 0 y j= = I x) I 2 x) I 3 x) I 4 x). β/n g b µ,δ f)x) g µ,δ b ) b )... b ) b )f 2 )x 0 ) dx I x)dx I 3 x)dx = I II III IV. I β/n 2 x)dx β/n I 4 x)dx For I, by using Lea 2, we have I sup b β/n x) b )... b x) b ) x b Lipβ β/n g β/n µ,δ f)x) dx b Lipβ Mg µ,δ f)) x). g µ,δ f)x) dx For II, aking < r < p < < n/δ, / / =, /s /s =, / = /p δ/n, ps = r by using he Hölder s ineualiy and he boundedness of g µ,δ fro L p o L and Lea 2, we ge II j= σ Cj j= σ Cj j= σ Cj β/n bx) b ) σ g µ,δ b b ) σ cf)x) dx ) / β/n bx) b ) σ dx R n g µ,δ b b ) σ cfχ )x) dx ) / b β/n σ Lipβ σ β/n / bx) b ) σ cfx)χ x) p dx R n ) /p

8 92 WANG MEILING AND LIU LANZHE j= σ Cj j= σ Cj β/n b σ Lipβ σ β/n /)/ps ) δps/n)/ps b Lipβ M δ,r f) x). ) /ps bx) b ) σ c ps dx δps/n ) /ps fx) ps dx β/n b σ Lipβ σ β/n b σ c Lipβ σc β/n M δ,r f) x) For III, we choose < r < p < < n/δ, / = /p δ/n, r = ps, by he boundness of g µ,δ fro L p R n ) o L R n ) and Hölder s ineualiy wih /s /s =, we have ) / III g β/n µ,δ b j b j ) )fχ )x) dx R n j= ) /p b β/n / j x) b j ) ) p fx)χ x) p dx R n j= δps/n)/ps) /)/ps b β/n j x) b j ) ) ps dx j= ) /ps fx) ps dx δps/n b Lipβ M δ,r f) x). ) /ps For IV, by he Minkowski s ineualiy and by he ineualiy a /2 b /2 a b) /2 for a b 0, we have ) nµ/2 F b j b j ) )f 2 )y) x y j= ) nµ/2 F b j b j ) )f 2 )y) x 0 y j= [ ) nµ/2 ) nµ/2 R n ) x y x c 0 y ) 2 /2 dyd b j z) b j ) ψ y z) fz) dz j= n

9 c R n dyd ] /2 dz n c j= LIPSCHITZ ESTIMATES nµ/2 x x 0 /2 j= b ) 2 jz) b j ) ψ y z) fz) x y ) nµ)/2 b j z) b j ) fz) x x 0 /2 R n ) n /2 dyd dz. y z ) 2n2 2δ ) nµ x y Se B = Bx, ), hen ) nµ n dy R x y y z ) n 2n2 2δ ) ) nµ n dy Bx,) x y y z ) 2n2 2δ ) ) nµ n dy k= 2 k B\2 k B x y y z ) 2n2 2δ n 2 2n2 2δ dy 2 y z ) 2n2 2δ k= n Bx,) 2 k B\2 k B Bx,) 2 k)nµ) k= n n 2 k dy x z ) 2n2 2δ k= ) nµ 2 k)2n2 2δ) dy 2 k2n2 2δ) dy 2 k B x z ) 2n2 2δ ) 2 knµ) 2 k2n2 2δ) 2 k ) n ) 2 k3n nµ 2δ) k= C x z ) 2n2 2δ 2 k y z ) 2n2 2δ ) x z ) 2n2 2δ ) x z ) 2n2 2δ

10 94 WANG MEILING AND LIU LANZHE and noice ha x z x 0 z for x and z R n \. We obain b j z) b j ) fz) x x 0 /2 so c j= c j= k= k= R n ) ) nµ /2 n dyd dz x y y z ) 2n2 2δ b j z) b j ) fz) x x 0 /2 2 k \2 k 2 k/2 x 0 x /2 x 0 z n/2 δ) fz) 2 k δ/n 2 k 0 d x z ) 2n2 2δ ) /2 dz b j z) b j ) dz j= r b j y) b j ) ) dy j= ) /r fy) r dy 2 k δ/n 2 k 2 kβ /2) b Lipβ β/n M δ,r f) x) k= b Lipβ β/n M δ,r f) x), IV b Lipβ M δ,r f) x). /r We pu hese esiaes ogeher, by using Lea and aking he supreu over all such ha x, we obain g b µ,δ f) F b β, Lipβ Mg µ,δ f)) M δ,r f) L ) b Lipβ Mg µ,δ f)) L M δ,r f) L ) b Lipβ g µ,δ f) L M δ,r f) L ) b Lipβ f L p. This coplees he proof of a). b). By soe arguen as in proof of a), we have g b µ,δ f)x) g µ,δ b ) b )... b ) b )f 2 )x 0 ) dx I x)dx I 2 x)dx I 3 x)dx I 4 x)dx = V V 2 V 3 V 4.

11 LIPSCHITZ ESTIMATES For V, aking /s = /r δ/n, by he boundedness of g µ,δ fro L r R n ) o L s R n ), so we have V sup b x) b )... b x) b ) x b Lipβ β/n b Lipβ β/n /s = C b Lipβ βδ)r/n b Lipβ M βδ,r f) x). g µ,δ f)x) s dx ) /r fx) r dx ) /s ) /r fx) r dx g µ,δ f)x) dx For V 2, aking /s /s =,/s = /r δ/n, by using he Hölder s ineualiy and he boundedness of g µ,δ fro L r R n ) o L s R n ), we ge V 2 j= σ Cj j= σ Cj j= σ Cj bx) b ) σ g µ,δ b b ) σ cf)x) dx ) /s bx) b ) σ s dx ) /s g µ,δ b b ) σ cfχ )x) s dx R n /s bx) b ) σ s dx ) /r ) /s bx) b ) σ cfx)χ x) r dx R n /s/r b σ Lipβ σ β/n b σ c Lipβ σc β/n j= σ Cj b Lipβ ) /r fx) r dx βδ)r/n b Lipβ M βδ,r f) x). ) /r fx) r dx

12 96 WANG MEILING AND LIU LANZHE For V 3, by he boundness of g µ,δ fro L r R n ) o L s R n ) and Hölder s ineualiy we ge ) /s V 3 g µ,δ b j b j ) )fχ )x) s dx R n /s R n j= b j x) b j ) ) r fx)χ x) r dx j= /s β/n /r b Lipβ b Lipβ M βδ,r f) x). For V 4, siilar o he proof of IV in a), we ge b j z) b j ) fz) x x 0 /2 ) c j= R n ) /r fx) r dx ) /r ) ) nµ /2 n dyd dz x y y z ) 2n2 2δ b j z) b j ) dz x 0 x /2 x 0 z n/2 δ) fz) ) c k= 2 k/2 2 k δ/n 2 k j= r b j y) b j ) ) dy j= ) /r fy) r dy 2 k δ/n 2 k /r fy) dy) r 2 k b βδ)r/n Lipβ 2 k b Lipβ M βδ,r f) x). So we have g b µ,δ f)x) g µ,δ b ) b )... b ) b )f 2 )x 0 ) dx Thus, g b µ,δ f)) # b Lipβ M βδ,r f). By using Lea 3 and he boundedness of g µ,δ, we have /r b Lipβ M βδ,r f) x).

13 LIPSCHITZ ESTIMATES g b µ,δ f) L g b µ,δ f)) # L This coplees he proof of b) and he heore. b Lipβ M βδ,r f) L b Lipβ f L p. Theore 2. Le 0 < β, µ > 3 /n 2δ/n, n/n β) < p, / = /p β δ)/n, b j Lip β R n ) for j. Then g b µ,δ is bounded fro H p R n ) o L R n ). Proof. I suffices o show ha here exiss a consan C > 0 such ha for every H p -ao a, g b µ,δ a) L. Le a be a H p -ao, ha is ha a suppored on a cube = x 0, r), a L /p and R n ax)x γ dx = 0 for γ [n/p )]. When = see [9]. Now, consider he case 2. Wrie g b µ,δ a)x) L x x 0 2r ) / g b µ,δ a)x) dx x x 0 >2r g b µ,δ a)x)dx ) / = I II. For I, choose < p < n/β δ) and such ha / = /p β δ)/n. By he boundedness of g b µ,δ fro L p R n ) o L R n ) see Theore ), he size condiion of a and Hölder s ineualiy, we ge I g b µ,δ a) L r n/ / ) b Lipβ a L p r n/ / ) b Lipβ r n /p/p ) r n/ / ) b Lipβ. For II, le τ, τ N such ha τ τ =, and τ 0. We ge F b a)x, y) b x) b x 0 ))... b x) b x 0 )) j= σ Cj bx) bx 0 )) σ c b Lipβ x x 0 β C b Lipβ ττ = B B x x 0 τβ x x 0 β b Lipβ y x 0 ) n2 δ B ψ y z) ψ y x 0 ))az)dz bz) bx 0 )) σ ψ y z)az)dz ψ y z) ψ y x 0 ) az) dz B B z x 0 τ β ψ y z) az) dz x 0 z az) dz

14 98 WANG MEILING AND LIU LANZHE C b Lipβ ττ = x x 0 τβ y x 0 ) n δ B z x 0 τ β az) dz b Lipβ y x 0 ) n2 δ rn /p) x x 0 β C b Lipβ y x 0 ) r τ βn /p) x x n δ 0 τβ. Thus, so g b µ,δ a)x) b Lipβ C b Lipβ ττ = 0 0 x x 0 ) n2 δ ττ = x x 0 ) n δ r τ βn /p) x x 0 τβ ) ) 2 /2 d r n /p) x x 0 β ) /2 ) 2 d b Lipβ x x 0 n δ) r n /p) x x 0 β C b Lipβ x x 0 n δ) r τ βn /p) x x 0 τβ, II b Lipβ r n /p) we ge C b Lipβ = J J 2, ττ = ττ = x x 0 >2r r τ βn /p) J b Lipβ r n /p) k= ) / x x 0 n δ β) dx 2 k \2 k x x 0 >2r ) / x x 0 n δ τβ) dx 2 k r) n δ β) dx ) / b Lipβ r n /p) 2 kn δ β) r n δ β) 2 k)n r n dx b Lipβ b Lipβ. k= k= 2 kn δ β n/) r n /p) n δ β)n/ ) /

15 LIPSCHITZ ESTIMATES For J 2, siilar o J, we have J 2 b Lipβ. Cobining he esiaes for I and II, hen leads o he desired resul. Theore 3. Le 0 < β, µ > 3 /n 2δ/n, 0 < p <, <, 2 <, / / 2 = β δ)/n, n / ) α < n / ) ε, ε < in, β), b j Lip β R n ) for j. Then g b α,p µ,δ is bounded fro H K R n ) o K α,p 2 R n ). α,p Proof. By Lea 7, le f H K R n ) and f = λ ja j, supp a j B j = B0, 2 j ), a j be a cenral α, ) ao, and λ j p <. g b µ,δ f) α,p p K 2 C k 2 2 kαp 2 kαp j=k λ j g b µ,δ a j )χ k L 2 ) p λ j g b µ,δ a j )χ k L 2 ) p = I II. For II, by he boundedness of g b µ,δ on L, L 2 ) see Theore ), we have II b p Lip β b p Lip β b p Lip β b p Lip β For I, we have 2 kαp j=k j=k λ j a j L ) p λ j 2 k j)α ) p λ j p j j λ j p. 2 k j)αp, 0 < p λ j p 2 k j)αp/2 ) j 2 k j)αp /2 ) p/p, < p < F b a j ))x, y) b x) b 0)) b x) b 0)) ψ y z) ψ y))a j z)dz B j bx) b0)) σ c bz) b0)) σ ψ y z)a j z)dz B j j= σ Cj b Lipβ x β B j ψ y z) ψ y) a j z) dz

16 00 WANG MEILING AND LIU LANZHE C b Lipβ ττ = x τβ B j z τ β ψ y z) a j z) dz x β b Lipβ y ) n2 δ 2jn / ) α) C x τβ b Lipβ y ) n δ 2jτ βn / ) α). ττ = Thus, and g b µ,δ a j )x) b Lipβ x β 2 jn / ) α) C b Lipβ 0 ττ = x ) n δ x τβ 2 jτ βn / ) α) 0 ) ) 2 /2 d x ) n2 δ b Lipβ x β x n δ) 2 jn / ) α) C b Lipβ x τβ x n δ) 2 jτ βn / ) α), ττ = ) ) 2 /2 d g b µ,δ a j )χ k L 2 b Lipβ 2 jn / ) α) x n βδ)) 2 dx B k \B k ) /2 C b Lipβ 2 jτ βn / ) α) x n τβ δ) 2 dx B k \B k ττ = b Lipβ 2 jn / ) α) kn / )) 2 jτ βn / ) α) kτ βn / )) ) ττ = b Lipβ 2 kα 2 j k)n / ) α) 2 j k)βn / ) α) ), ) /2 so

17 I b p Lip β When 0 < p, When p >, I b p Lip β k 2 LIPSCHITZ ESTIMATES... 0 ) p λ j 2 j k)n / ) α) C b p Lip β C b p Lip β b p Lip β b p Lip β I b p Lip β k 2 k 2 λ j p C b p Lip β k 2 b p Lip β k 2 λ j 2 j k)βn / ) α)) p. λ j p 2 pj k)n / ) α) k 2 λ j p k=j2 λ j p. k 2 k=j2 2 p j k)n / ) α)/2 ) λ j p 2 pj k)βn / ) α) 2 pj k)n / ) α) 2 pj k)βn / ) α) ) λ j p 2 pj k)n / ) α)/2 k 2 2 p j k)βn / ) α)/2 Fro I and II, we have λ j p. ) p/p ) λ j p 2 pj k)βn / ) α)/2 ) p/p ) ) g b µ,δ f) K α,p 2 b Lipβ λ j p ) /p f H K α,p

18 02 WANG MEILING AND LIU LANZHE This coplees he proof of Theore 3. When α = n / ) ε, ε < in, β), his kind of boundedness fails. Now, we give an esiae of weak ype. Theore 4. Le 0 < β, 0 < p, <, 2 < / 2 = / β δ)/n, b j Lip β R n ) for j. Then g b µ,δ aps H K n / )ε,p R n ) coninuously ino W K n / )ε,p 2 R n ), where 0 < ε < in, β). Proof. We wrie f = λ ka k, where each a k is a cenral n / )ε, ) ao suppored on B k and λ k p <. Wrie g b µ,δ W K n / )ε,p 2 sup λ{ 2 ln /)ε)p {x E l : g b µ,δ λ>0 sup λ>0 l= λ{ = G G 2. l= 2 ln / )ε)p {x E l : g b µ,δ k=l 3 l 4 λ k a k )x) > λ/2} p/ 2 } /p λ k a k )x) > λ/2} p/ 2 } /p By he L, L 2 ) boundedness of g b µ,δ and an esiae siilar o ha for I in Theore 3, we ge G p 2 lpn /)ε) g b µ,δ λ k a k )x)χ l p 2 b p Lip β λ k p. l= k=l 3 To esiae G 2, le us now use he esiae g b µ,δ a k )x) b Lipβ x β x n δ) 2 kn / ) α) C b Lipβ x τβ x n δ) 2 kτ βn / ) α), ττ = which we ge in he proof of Theore 3. Noe ha when x E l, λ < 2 l 4 b Lipβ λ k g b µ,δ a k )x) l 4 C b Lipβ α = n / ) ε, l 4 λ k 2 l ) βδ n l 4 λ k ττ = 2 k ) n / ) α l 4 2 l ) τβδ n 2 k ) τ βn / ) α

19 b Lipβ l 4 b Lipβ 2 lβδ n ε) LIPSCHITZ ESTIMATES λ k 2 l ) βδ n ε λ k p ) /p, for λ > 0, le l λ be he axial posiive ineger saisfying 2 lλnε β δ) b Lipβ λ λ k p ) /p, hen if l > l λ, we have so we obain {x E l : g b µ,δ G 2 sup λ>0 sup λ>0 λ{ λ{ l λ l= l λ l= l 4 λ k a k ) > λ/2} = 0, 2 ln / )ε)p 2 l ) np/ 2 } /p 2 l ) nε β δ) } sup λ2 l λnε β δ) λ>0 b Lipβ λ k p ) /p. Now, cobining he above esiaes for G and G 2, we obain g b µ,δ f) W K n / b )ε,p Lipβ λ k p ) /p. 2 Theore 4 follows by aking he infiu over all cenral aoic decoposiions. Acknowledgeen The auhors would like o express heir graiude o he referee for his coens. References [] J. Álvarez, R. J. Bagby, D. S. Kurz, and C. Pérez. Weighed esiaes for couaors of linear operaors. Sudia Mah., 042):95 209, 993. [2] S. Chanillo. A noe on couaors. Indiana Univ. Mah. J., 3):7 6, 982. [3] W. Chen. A Besov esiae for ulilinear singular inegrals. Aca Mah. Sin. Engl. Ser.), 64):63 626, 2000.

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