RBMO_与_t_型Calder_n_Zygmund算子的交换子的有界性_英文_

RBMO_与_t_型Calder_n_Zygmund算子的交换子的有界性_英文_ MA T H EMA T ICA A P PL ICA TA () 2009 ,22 1:223,232 3 (μ) Boun dedness of Commuta tor of RBMO θ( ) Type Cal der?n2Zygmun d Opera tors witht 1 2 () () Z HA O Kai 赵凯, R EN Xiao2f a ng 任晓芳, 1 2() ) ( Z HO U Shu2j ua n 周淑娟, WA N G L ei 王磊 ( 1 . Col l e ge o f M at he m at i cs , Q i n g d ao U ni v e rs i t y , Q i n g d ao 266071 , C hi n a ; 2 . D e p t. o f )M at h . , T e ac he rs Col l e ge , Q i n g d ao U ni v e rs i t y , Q i n g d ao 266071 , C h i n a d μμAbstract :L et be a Ra ndo n mea sure o n Rw hich may be no n do ubling. The o nl y co nditio n t hat n d μ( ( ) ) must sati sf y i s t he gro wt h co nditio nB x , r?Cr fo r all x ?R, r > 0 a nd fo r so me fixed 0 < (μ) n ?d . In t hi s pap er ,under t hi s a ssump tio n ,t he bo undedne ss of t he co mmutato r s of RBMO wit h ? 1 1 θ( ) t ype Caldernó2Zygmund op erato r s f ro m (μ) to RBMO (μ) a nd f ro m (μ) to (μ) a re es2 tL Hb L tabli shed. θ( ) Key words :tt yp e Cal der nó2Zygmund op erato r ; Ha r dy sp ace ; RB MO sp ace ; Co m2mut ato r ; No n do ubli ng mea sure CLC Number :O174 . 2 AMS( 2000) Subject Cla ssif icat ion :42B20 ;42B30 () Art icle ID :100129847 20090120223210 Document code :A 1 . Introduction In rece nt yea r s ,ma ny p ap e r s p a y at t e ntio n to t he st udy of no n do ubli ng mea sure i n ha r2 mo nic a nal ysi s. So me p ap e r s ha d sho wed t hat a bi g p a r t of t he cla ssical t heo r y re mai n s vali d if μt he do ubli ng a ssump tio n o ni s sub stit ut e d by t he gro w t h co nditio n . The gro wt h co nditio n i s d t hat t he re i s a co n st a nt C > 0 such t hat fo r all x ?R, r > 0 a nd fo r so me fi xed 0 < n ? d , n μ( ( ) ) ( ) B x , r?Cr , w here B x , ri s t he ball ce nt e red at x a nd ha vi ng ra di u m r . In [ 8 ] , Tol sa (μ) X i nt ro duce d a new RB MO sp ace fo r no n do ubli ng mea sure w hic h sati sfie s all t he p rop er2 1 , ? (μ) (μ) tie s of t he u sual B MO . The p re dual of RB MO i s t he ato mic sp ace Hρw hic h i s ma deat , λλλ i a i , w herei ?R , | i |up wit h f unctio n s of t he fo r m f = ?, a nd fo r each i , ai i s< i i ?? - 1 ? μ (ρ) μ ,ad= 0 . A no t her ato mic( ?Q a f unctio n suppo r t ed o n a cube Qi , wit h ‖ai ‖L μ)i i ? 1 , ? (μ) b , w here t he f u nctio n s b= sp ace Hi s ma de up wit h f unctio n s of t he fo r m fa re atb i i i ?[ 8 ] so me ele me nt a r y f unctio n s ,w hich we call ato mic bloc k s. 3 Received date :May 23 ,2008 ()Foundation item : Suppo rt ed by N N SF2Chi na 10671115 Biogra phy :Z HAO Kai ,male , Ha n , Sha ndo ng ,p rof e sso r ,majo r i n ha r mo nic a nalysi s a nd wavelet . MA T H EMA T ICA A P PL ICA TA 2009 224 ρ In [ 8 ] ,we k no w t hat t hi s ato mic sp ace i s i ndep e nde nt of t he cho se n> 1 . Al so , Tol sa X 2 (μ) sho wed t hat if t he Cal der ón2Zygmund op e rato r i s bo unded o n L , t he n t he co mmut ato r of p (μ) (μ) t hi s op e rato r wit h a f unctio n of RB MO i s bo u nde d o n L , 1 < p < ?. No w , we ca n a θ( ) sk ho w a bo ut t he tt yp e Cal de r nó2Zygmu nd op erato r ? In t hi s p ap e r , we a n swe red t he (μ) θ( ) que stio n ,a nd o bt ai ned t he bo undedne ss of t he co mmut ato r s of RB MO wit h tt yp e ? 1 1 (μ) (μ) (μ) (μ) Cal der nó2Zygmund op e rato r s f ro m L to RB MO a nd f ro m Hto L .b 2 . Def in itions and Lemma s (μ) (μ) θ( ) Def in it ion 2 . 1 A li nea r op erato r T : S ? S′i s calle d a tt yp e Cal de r nó2Zyg2 mu nd op erato r fo r no n do ubli ng mea sure ,if it sati sfie s t he follo wi ng co nditio n s : 2 ) (μ) 1T ca n be e xt e nde d to a bo u nded li nea r op e rato r o n L ; d d ( ) Ω ( ) ) 2The re e xi st s a co nti nuo u s f unctio n K x , ydefi ne d o n = { x , y?R×R: x ?y} a nd so me po sitive co n st a nt C suc h t hat - n ) ( ) i| K x , y| ?C | x - y | ; d ) iiFo r x , x0 , y ?R, a nd 2 | x - x0 | < | y - x0 | , t he n |0 x - x | - n ( ) ( ) θ ( ) ( ) ?C 0 - y | , x, y| +|y , x|K K | K x , y-K y , x-0 0 | x x| - y | 0 θ( ) ) w hereti s a f unctio n defi ned o n [ 0 , + ?w hic h i s neit he r decrea se no r ne gative , sati sf yi ng 1 θ( )t θ θ θ ( ) ( ) ( ) d t < ?, a nd 0= 0 , 2 t?C t. 0?t ( ) ( ) ( ) μ( ) ) iiiT f x= K x , yf ydy, a . e . x | supp f . ?1 ρ(μ) (μ) Def in it ion 2 . 2 L etbe so me fi xed co n st a nt . We say t hat f ?L i s i n RB MO ifcl o ((μ) ) t here e xi st s so me co n st a nt C such t hat fo r a ny c ube Q ce nt e re d at so me poi nt of supp , 1 μ |f - mQQ f | d ?C , μ(ρ)Q Q? ( ) mf m f 2 . 1 - | ?C K Q R | Q , Rn d ( α β α(αfo r a ny do ubli ng cu be s Q < R Give n> 1 a nd> ,we sa y t hat so me cube Q < Ri s , d +1 β) μ(α) ( ) ( ) μβ) 2do ubli ng if Q?Q; he re ,de no t e t he 2 , 2 2do u bli ng cu be s,w here N Q , R k μ( )2 Q = 1 + KQ , R ,k n ? () l 2 Q k = 1 k d() ( )( ) a nd N i s t he fir st i nt e ger k suc h t hat l 2 Q ?Q , set N Q , RQ , R ?l Ri n ca se R = R= ?. (μ) The mi ni mal co n st a nt C i s t he RB MO no r m of f , a nd it i s de no t e d by ‖?‖.3 (μ) We no tice t hat , t he defi nitio n of RB MO sp ace i s relat e d wit h n , b ut i s no t relat ed ρρ wit h t he fi xed co n st a nt . So t hro ugho ut t he re st of t he p ap er we a ssume t hat t he co n st a nt i n (μ) t he defi nitio n of RB MO i s 2 . (μ) For RBMO space , Tol sa X int roduced so me equivalent norms and p ropo sitio ns in [ 8 ] . 1 (μ) Suppo se t hat fo r a give n f unctio n f ?L t he re e xi st so me co n st a nt C a nd a collectio nloc () of nu mbe r s { f Q } Q i . e . fo r each c ube Q ,t here e xi st s f Q?R such t hat 1 ( )μ( ) | f x - f | dx?C ,supQ Q μ(ρ)QQ ? R . The n ,we w rit e ‖f ‖= i nf C , w here3 3 a nd | f Q - f R | ?C K Q , R , fo r a ny t wo c ube s Q < t he i nfi mu m i s t a ke n o ver all t he co n st a nt s C a nd all t he numbe r s { f } sati sf yi ng t he a bo ve Q (μ)No . 1 Z HAO Kai et al . :Bo undedne ss of Co mmutato r of RBMO 225 t wo i nequalitie s. Fo r t he t wo no r m s ,we ha ve t hat ‖?‖3 a nd ‖?‖3 3 a re equivale nt . 1 ρ (μ) lo Lemma 2 . 1 Let> 1 be fixed. For a f unctio n f ?L, t he following are equivalent :c () (μ) af ?RB MO ; ( ) μ μ(ρ) bThe re e xi st s so me co n st a nt C such t hat fo r a ny c ube Q ,| f - mf | d?CQ ,Q Q ? μ(ρ) μ(ρ)Q R + , fo r a ny t wo c ube s Q < R ; a nd | mf - m f | ?C K Q R Q , R μ( ) μ( )QR () μ cThe re e xi st s so me co n st a nt C suc h t hat fo r a ny do u bli ng cu be Q ,| f -mf | Q d? Q ?μ( ) () CQa nd 2 . 1hol d s fo r a ny t wo do ubli ng c ube s Q < R . θ( ) Def in it ion 2 . 3 The co mmut ato r fo r med wit h RB MO f unctio n a nd t he tt yp e Cal der ón2Zygmund op e rato r T i s defi ne d by ) ( ( ) ) ( ( ) ( ) ( ) ) ( ( ( ) ) ( ) [ b , T ] f x= bxT f x- T bf x= K x , ybx- byf yd y . ? 1 (μ) 2 sati sfie s t he folDef in it ion 2 . 4 A f unctio n a ?L i s called a b2 ato mic block ,if itlo c lo wi ng co nditio n s : ) ( ) 1The re e xi st s so me cu be R such t hat supp a< R ; λ) 2The re a re f unctio n s aj suppo rt e d o n cu be s Qj < R a nd n umber sj ? R suc h t hat a =? - 1 ? ) λ( ) μ( ) ( ) ( ) μ( ) ( μ ( ) ydyaybydy= 0 a nd ‖aj ‖L (μ )? 2 QK.a= j a j ,j j j Q , R j ? ??j = 1 11 (μ) λ De no t e | if t here a re b 2ato mic bloc k s | | .The n we sa y t hat f ? Ha | H(μ)b = j b j ? ?1 1 asuc h t hat f = awit h | (μ) a| μ) < ?. Theno r m of f i s (i j Hi H b i = 1 i b ?? 1 1 ‖f ‖(μ) = i nf | a| (μ) ,Hi Hb i b ? w here t he i nfi mum i s t a ke n o ver all t he po ssi ble deco mpo sitio n s of f i n b 2ato mic block s. 3 . Boundedness of the Commutator 1 θ( )t εTheorem 3 . 1 If t here e xi st s> 0 , suc h t hatd t < ?, t he n t he co mmut ato r [ b , T ]1 +ε 0 ?t θ( (μ) ) fo r me d wit h tt yp e Cal de r nó2Zygmund op erato r T a nd RB MO f unctio n fo r no n do u2? (μ) (μ) bli ng mea sure i s bo unded f ro m L to RB MO . ( ) Proof He re acco r di ng to cof L e mma 2 . 1 , we o nl y need to p ro ve t he follo wi ng t wo co ncl u sio n s. ) 1The re e xi st s so me co n st a nt C such t hat ?) μ μ( ) 3 . 1 | [ b , T ] f - m( [ b , T ] f | d ?C ‖f ‖‖b ‖Q( ) Q L (μ)3 Q ? fo r a ny do ubli ng c ube Q. ) 2The re e xi st s so me co n st a nt C such t hat ? ( ) ( ) ( ) | m[ b , T ] f - m [ b , T ] f | ?C K ‖f ‖(μ )‖b ‖3 . 2 Q R Q , R L 3fo r a ny t wo do u bli ng cu be s Q < R . μ μ( () ‖b ‖?2 2 Q) To sho w 3 . 1,let bbe a gro up of n umber s , sati sf yi ng| b - b| d3 3Q Q Q ? fo r a ny cu be Q , a nd | bQ - bR | ?2 KQ , R ‖b ‖3 3 fo r a ny t wo cube s Q < R . Fo r a ny cu be Q , we ( ( ( ) ) ) χ de no t e h= mT b - . A nd ,fo r a ny do ubli ng Q , beca u se bfQ Q Q d R \ 2 Q MA T H EMA T ICA A P PL ICA TA 2009 226 ( ( ) ) ( ( ( ) ( ) ) ) ( ( ) ) [ b , T ] f - T b - bf = b - bT f= b - bT ff - T b - bf ,bQ T b -Q Q - Q 1 Q 2 χ= f, f w here f = f - f . So21 1 2 Q μ ) ( m[ b , T ] f | d | [ b , T ] f -Q Q ? μ ) = |( ) T f ( ( ) f ) ( ( b) f ) + h- h- m( [ b , T ] f | d bT b - b b - - T b - Q Q 1Q 2Q Q Q - Q ? μ μ ( ) ( ( b) f ) | d d +|Q T fQ 1 bb - | T b - ?| Q Q ?? μ μ ( ( ) ) ( ) + m[ b , T ] f | h+|T b - bf - h| d +|d Q Q Q 2 Q Q Q ?? ?= I+ I+ I+ I.1 2 3 4 p (μ) Beca u se of t he H ol der i nequalit y a nd t he bo undedne ss of T o n L , p > 1 , we ca n o bt ai n 1/ p 1/ p′ p p′ μμT f | d | d bI? ||b - Q 1 Q Q ??1/ p′ 1/ p?μ( ) μ( ) ?CQ‖b ‖‖f ‖L (μ) Q 3 ?μ( ) ?CQ‖b ‖‖f ‖μ) .(3 L No w wit h t he do u bli ng p rop er t y of c ube Q a nd t he coefficie nt ?C ,KQ , 2 Q 1/ pp 1 - 1/ p μμ( ( ( ) ) ) I? | T b - bf | d2 Q 1 Q Q ?1 - 1/ pp ( ) μ( ) ?C ‖b - bf ‖(μ) QQ 1 L 1/ p p 1 - 1/ p ? ) μ( μ ?C ‖f ‖μ) Q| b - b| d( L Q 2 Q ?1/ p 1/ p p μb- b| d |Q 2 Q 2 Q 2 Q ?1 - 1/ p 1/ p 1/ p ? () ) μ( μ( μ( ) ) ?C ‖f ‖μ) Q( 4 Q ‖b ‖+ 2 Q‖b ‖KL 3 3 Q , 2 Q 1 - 1/ p1/ p ?μ( ) μ( ) ?C ‖f ‖(μ) Q4 Q‖b ‖L 3 ? μ( ) ?C ‖f ‖(μ) Q‖b ‖.L 3 ( ( ) )In o r de r to get t he e sti matio n of I, fir st ,we e sti mat e | T b - bf 3 Q 2 Fo r x , y - h| . ?Q ,Q we ca n have ) ) ( )) ) ( ) ( ( ( ( | T b - T b - bQ f 2 x- bQ f 2 y| μ( ) ) μ( ) ) ( ) ( ( ) ) ( ) ( ( ) ) f ( z d z ( y , z b z - bx , z b z - bf z d z Q ?C KQ - Kd d R\ 2 Q R\ 2 Q ?? ?( ) ( ) ( ) ?C ‖f ‖(μ)μ( )| K x , z-K y , z| | bz-b|L dz Q d R\ 2 Q ? | x - y | - n ?θ?C ‖f ‖(μ)( ) μ( )L | x - z | | bz- b| dzQ d R\ 2 Q | x - z | ?? | x - y | - n ?k +1 θ ( ) ?C ‖f ‖(μ)x - z | | bz- μ( )L | b| dz2 Q k +1 k ? 2 Q\ 2 Q| x - z | ?k = 1 ?| x -y | - n k +1 ? θ ) ( μ( | x - z | | b-+ C ‖f ‖μ)b| dzQ L 2 Q k +1 k ? 2 Q\ 2 Q | x -z | ?1k = ?( )l Q k - n k +1 ?θ ( )( ) μ( ) bz l 2 Q| ?C ‖f ‖μ)- b| dz (L 2 Q k k +1 k ?( ) 2 Q\ 2 Q l 2 Q ?k = 1 ?( )l Q k - n k +1 ? θ ( μ( ) ) + C ‖f ‖μ)l 2 Q dz ( ‖b ‖3 K2 Q , QL k k +1 k ? ( )2 Q\ 2 Ql 2 Q ?k = 1 (μ)No . 1 Z HAO Kai et al . :Bo undedne ss of Co mmutato r of RBMO 227 ??k +2 k +1 μ( μ( ))- k 2 Q - k 2 Q k +1 ? θ(θ( K2) 2 2 Q , Q) ( + ?C ‖f ‖μ)‖b ‖L 3k +2 1 k +n n ??( )( )l 2 Q l 2 Q k = 1 k = 1 ? - k k +1 ?θ( ) ‖b ‖3 K2 Q , Q2 ?C ‖f ‖(μ)L ?k = 1 ? ?C ‖f ‖(μ)‖b ‖3 .L ( ( ) ) ( ) Ta ke t he mea n val ue | T bbf xof y o n c ube Q , we o bt ai n | ?Q 2 h- - Q ? C ‖f ‖(μ )‖b ‖.Th u s L 3 ?μ( μ( ) I= |( ( ) ) ( ) ) bQ f 2 x - hQ | d x ?C ‖f ‖L (μ) ‖b ‖3 Q ;3 T b - Q ? ?( ) μ( ) μ μ( ) I?| h+ m[ b , T ] f | Q?C| [ b , T ] f + h| d?C ‖f ‖(μ) ‖b ‖Q.4 Q Q Q L 3 Q ? () Therefo re ,we ha ve t he i nequatio n 3 . 1. () No w ,we t ur n to p ro ve 3 . 2. Fo r a ny x ?Q , y ? R , w rit e ( ) ( ) | [ b , T ] f x- [ b , T ] f y| ( ( )( ( ( ( )( ( ) ( ) ) ) ( ) ) ( ) ) ) ( ) = | bxT f xf xbT f yf y|bbb- T b -- by+ T b -- - Q Q Q Q ( ( )?| bx) ( ) ( ( ) ) ( ) - bT f x| +| by- bT f y|Q Q ) ) ( )( ( ) ) ( ) ( ( +| T b - bQ f x- T b - bQ f y| ?=J + J + J . 1 2 3 Fo r J a bo ve e sti matio n , we ca n get t hat ( ( ) ), acco r di ng to t he ?1 m|b - bT f| Q Q ? C ‖f ‖L (μ) ‖b ‖3 . Si mila rl y μ( ) ( ( ) ) ( ) T f y | d y | b yQb- R ?1/ p 1/ p′ p p′ μμb| d ? | T f | d| b -Q R R ??1/ p′ 1/ p′ p′ p′ 1/ p ?μμ) μ( d ‖f ‖(μ) R?C | b - b| d+ |b- b| L R R Q R R ??1/ p′1/ p ?μ( μ( ) ) ?CRR‖b ‖ L 3‖f ‖ (μ) KQ , R ? μ( ) ?CR‖b ‖‖f ‖μ)3 L ( K.Q , R ? ( ( ) ) Therefo re m R | b - bQ T f | ?C ‖b ‖3 ‖f ‖L (μ) K. Q , R Fo r J , w rit e 3 ( ( ) ) ( ) ( ( ) ) ( ) T b - bf x- T b - bf yQ Q ( ( ) χ) ( )( ( ) χ) ( )= T b - bfx+ T b - bfd xQ Q Q R\ Q R R ( ( ) χ) ( )( ( ) χ) ( )- T b - bfy- T b - bfd y Q Q Q R\ Q R R d ( ( ) ( )= [ T b - ) χx ( ( ) χ) ( ) bf- T b - bfd y]Q Q R\ Q R\ Q R R ( ( ) χ ) ( )( ( ) χ) ( )- T b - bfy+ T b - bfxQ Q Q Q R R ?= J + J + J .31 32 33 No w , d ( ( d ) χ) χ ) ( )( ( ) ( ) J | ?| [ T b -bf- bf| x T b - y] | 31 Q Q R\ Q R\ Q R R ( ) μ( ) ( ) ( ( ) ( ) ( ( ) ) ) ( ) μ( ) K y , zbz-bf zdzQ d R \ Q ?R R ?μ( )( )( ) ( ) ?C ‖f ‖(μ)| K x , z- K y , z| | bz- b| dzL Q d R\ Q ? R MA T H EMA T ICA A P PL ICA TA 2009 228 |x - y | - n ? θ ( ) ) ( μ( ?C ‖f ‖μ)| x - z | | b| b z - dz L Q d R\ Q | x - z | ? R ?| x - y | - n ?k +1 θ ?C ‖f ‖(μ)z | | ( ) b μ( )| x - L bz- 2 Q| dz k +1 k R ? 2 Q \ 2 Q |?x - z | k = 1R R ?| x -y | - n k +1 ? θ ) ( x - z | | b- μ( + C ‖f ‖μ)| b Q | dz L 2 Qk +1 k R ? 2 Q \ 2 Q| x - z | ?1k = R R ?)( l R - n k ?k +1 θ ( ) ( )μ( ) l 2 Q | bz ?C ‖f ‖(μ)- b | dz R L 2 Qk R k +1 k ? ( )2 Q \ 2 Ql 2 Q R ?R R k = 1? ( )l R k - n k +1 ? θ ( ) μ( ) l 2 Q dz, Q + C ‖f ‖(μ )‖b ‖K R L 32 Qk R k +1 k ?( ) 2Q \ 2 Q l 2 Q R ?R Rk = 1? ?k +2 k +1 μ( )μ( )2 Q R 2 Q R - k - k ? θ( )K +2 θ( ( ( ) ) K2?C ‖f ‖μ)‖b ‖+ Q , R L 3k +2 n k +1 n ?? ( )( l 2 Q ) l 2 Q R k = 1 R k = 1 ? 1 θ( )t - k ??( ?C ‖f ‖μ))θ( ) ‖b ‖Kd t ( ‖b ‖3 K +L K2 ?C ‖f ‖μ)3 Q , R(Q , R L ? 0?t k = 1 ? ?C ‖f ‖(μ)‖b ‖3 KQ , R .L ( ) ( ) A bo ve i nequalitie s u si ng t he f act t hat l QR ? l Ra ndk +1 k +1 ( ( ) ) K?C K+ K+ K?C K+K. 2 Q, Q Q , R R , Q QQ , R , 2 QR R R R A nd ) χ) ( ( ( ) J 32 ?| T b - bQ fx|Q R ( ) ( ( ) ) ( ) μ( ) ?| K x , y b y - bf y | d yQ Q ?R ? ‖f ‖(μ)L ( )| by | μ( )?C - b Q dy n Q( ) ?l QR R ? ‖f ‖(μ )L ( ) μ( ) μ( )?C | b y - b - b Q | dy+ | Q Q | dy b n RR Q Q ) ??l RR ? ( ‖f ‖L μ) ?C ‖b ‖μ( )Q , Q R K 2 Q 3 n R) l ? ?C ‖f ‖(μ) ‖b ‖K.L 3 Q , R ? () The e sti matio n of J i s si mila r . He nce J ?C ‖f ‖(μ) ‖b ‖KQ , R . Thi s p ro ve d 3 . 2,a nd33 3 L 3 e nds t he p roof . θ( ) Lemma 3 . 1 If T i s t hett yp e Cal de r nó2Zygmund op e rato r fo r no n do ubli ng mea sure ,1 θ( )t (μ) ε b ?RB MO , a nd t he re e xi st s> 0 suc h t hat d t < ?, t he n t he co mmut ato r [ b , T ]ε 1 + 0 ?tp (μ) i s bo unde d o n L , 1 < p < + ?. 1 θ( )t ε Theorem 3 . 2 If t here e xi st s> 0 suc h t hatd t < ?, t he n t he co mmut ato r [ b , T ]ε 1 + 0 ?tθ( ) fo r me d wit h t hett yp e Cal de r nó2Zygmu nd op e rato r T a nd t he RB MO f unctio n fo r no n do u21 1 (μ) (μ) bli ng mea sure i s bo unded f ro m Hto L . b ( ) Proof A s we k no w ,o nl y to sho w t hat fo r a ny b 2ato mic bloc k a x,t he re i s 1 μ ( ) ( ) ( ) | [ b , T ] a x| d x?C | a | H 3 . 3 (μ) . d b R? ( ) No w , suppo se t hat a xi s a b 2ato mic block . Writ e (μ)No . 1 Z HAO Kai et al . :Bo undedne ss of Co mmutato r of RBMO 229 μ μ ( ) μ( )x|[ b , T ] a dx | d 2 Q R\ 2 Q ? + I I . ?= I I 12 Fi r st ,we sho w t hat 1 μ( ) = |3 . 4 ( ) | d x ?C |( ) I I [ b , T ] a x a | (μ) . 1Hb 2 Q ? In f act ( ) μ( ) I I = | [ b , T ] a x | d x1 2 Q ? λ( ) μ( )? | | | [ b , T ] ax|dx j j ? 2 Q?j λλ? | || |( ) μ( )+ ( ) μ( )j j | [ b , T ] ax|dx | [ b , T ] ax|dx j j ?? 2 Q\ 2 Q 2 Q ??j j jj ?= I I + I I .11 12 2 (μ) U si ng t he Hol de r i nequalit y a nd t he bo u nde dne ss of co mmut ato r o n L ,we o bt ai n λ( ) μ( )I I 12 = | j | | [ b , T ] aj x| dx ? 2 Q ?j j 1/ 22 1/ 2λ? | |( ) μμ( ) [ b , T ] ax| d2 Q j j j | ? 2 Q ?j j 1/ 21/ 22 λ?C | |μ( ) μ2 Q j | a| dj j ? Q ?j j - 1 - 1 ?μ λμ( )λμ( ) ?C | | ‖a‖(μ) 2 Q?C | | 2 QK( ) j j L j j j , Q2 QQ j j ??j j λ?C | j | . ?j 1 λ( ) μ( )(μ) . Fo r I I , o ne ha s a | So I I 12 = | j | | [ b , T ] aj x| dx?C |H11 b ? 2 Q ?j j λ( ) I I = | | μ( )[ b , T ] ax|11 j dx | j ? 2 Q\ 2 Q ?j jN Q , Q j λ? | |( ) μ( ) | [ b , T ] ax| dx.j j k +1 k ?? 2 Q \ 2 Q?j j j k = 1 ) ( )( )) ) ( ) ( ( ) ( ( Si nce [ b , T ] ax mbax, t he n = bx- mbTa xT b - j QQ j QQ j - j j N Q , Q j λ( ) | I11 ? | j |μ( )[ b , T ] aj x|dx k +1 k ?? 2 Q \ 2 Q?k = 1 j j j N Q , Q j λ( )( ) μ( ) bx- mQQ b | | Ta j x| dx? | j || j k +1 k ?? 2 Q \ 2 Q?j k = 1 j j N Q , Q j λ( ( ) ) ( ) μ( )+ | j || T b - mQ Q baj x| dx j k +1 k ?? 2 Q \ 2 Q?j j j k = 1 ?= I111 + I I 112 . k +1 x be t he ce nt e r of cu be Q. Acco r di ng to t he ca ncellatio n of a, a nd | mb - L et j j j QQ mb |?2 QQ j j Ck ‖b ‖3 , we o bt ai nN Q , Q j λ( ) μ( ( ) )I I 111 = | j || bx- mQQ b | | Ta j x| dx jk +1 k ?? 2 Q \ 2 Q?k = 1 j j j MA T H EMA T ICA A P PL ICA TA 2009 230 N Q , Q j λ( ) ( ) ( ) ( ) μ( ) μ( )mb | | ay| | K x , y-?C | j |K x , x | dxdy QQ j | bx-j 1 k k + j? ? Q 2Q \ 2 Q ??j j jk = 1 jN Q , Q j - k k - n λθ( )) μ( )( ) ( ) μ( )?C | j |( | ay| dy| bx- mQQ b | 2 l 2 Q j dxj k +1 k j ?? Q Q \ 2 Q 2??j j jk = 1 j N Q , Q j - k k - n ? λθ( ) ( ) ) μ( μ( ) ?C | |2 l 2 Q ‖a‖μ)Q| b x - mb | d x j j j L ( ) ( ) jQQ j k +1 k ?? 2 Q \ 2 Q?= 1 k j j j N Q , Q j - k k - n ? λθ( ) ( ) μ( ?C | |2 l 2 Q ‖a‖μ) Qj j j L ( j) ??j k = 1 k +1 k +1 μ( ) ( ) μ( )| mb - mb | dxQQ 2 QQ k +1 k j j QQ2 \ 2?j j j j N Q , Q j - k k - n k +2 ?λθ( ) ( ) μ( )μ( ) (μ) Q?C | j |k2 l 2 Q j ‖aj ‖L j 2 Qj ‖b ‖3 ??j k = 1 N Q , Q j - k ?λθ( ) μ( )(μ) ?C | j |k2 ‖aj ‖L ‖b ‖3 2 Qj ??j k = 1 1 θ( )t ? λμ( ) ‖a‖(μ )‖b ‖2 Qd t?C | j |j L 3 j 1 +ε ? 0?t j 1 λ(μ) ?C | j |?C |a | ,H b ?j a nd N Q , Q j λI= | |( ( ) ( μ( )) ) | T b - mbax| dx112 j QQ j k +1 k j ?? 2 Q \ 2 Q?k = 1 j j j N Q , Q j 1 1 λ?C | |( ( ) ‖bx-) ( ) μ( ) j mbax‖(μ) dxQQ j L n k +1 k j ?? 2 Q \ 2 Q?| x | j k = 1 j j N Q , Q j k - n k +1 ?λ( ) μ( )μ( )?C | |l 2 Q ‖a‖(μ) ‖b ‖2 Q2 Qj j j L 3 j j ??k = 1 j N Q , Q j k +1 μ( )2 Q j - 1 λ?C | j |KQ , Q ‖b ‖3 k +1 n j??( )l 2 Qj k = 1 j 1 λ ?C | j |?C | a | (μ) .H b ?j Therefo re 1 λ( ) | |b , T ] ax|[ μ( ) I I = j | j a | (μ) . dx?C |11H b ? 2 Q\ 2 Q ?j j He nce 1 μ( ) ( ) I I a | = |[ b , T ] a x | d x ?C |(μ) , 1Hb 2 Q ? () i . e . 3 . 4hol ds. ) ( ( ) ) ( ) ( ) ( No w ,fo r I I , we w rit e [ b , T ] a x= b -bQQ Ta -T b - bQQ ax. Th u s2 μ( ) ( ) d x I I = |[ b , T ] a x | 2 d R\ 2 Q ? (μ)No . 1 Z HAO Kai et al . :Bo undedne ss of Co mmutato r of RBMO 231 ( ( ) ) ( ) ( ) μ( ) μ( ) ?|b - b| | Ta x | d x +| d x QQ | T b - ba xQQ d d R\ 2 Q R\ 2 Q?? ?= I I + I I .21 22 L et x be t he ce nt er of c ube Q , r be t he ra di u m of c ube Q.The n Q μ( ) b - b| | Ta ( x ) | d x I I = |QQ 21 d R\ 2 Q ? μ( ) μ( ) ( ) ( ) ( ) ( ) ?|a y | d y d x b x - b| |K x , y - K x , x | | QQ Qd Q R\ 2 Q ?? μ( ) μ( ) ( ) ( ) ( ) ( ) ?C|| d x d y x , x b|a y | K x , y - K | | b x - Q QQ d Q R\ 2 Q?? ? ) ( ( ) ( ) ( ) μ( ) μ( )| K x , y- bx- b| K x , x Q | |dxdy QQ j +1 j 2 r ?| x - x |?2 r ?j = 1Q ?- j j - n b| ( ) θ( ) ( ) ( ) QQ μ( ) μ( )2 2 r| bx- dxdy j +1 j ?2 2 r ?| x - x | r?j = 1Q ?j - j - n θ(μ( ( )) () ( ) ) | bx - b| μ( )QQ dx j j +1 ? 2 r ?| x - x | ?2 r?j = 1 Q ? - j j - n θ(μ( ) () μ( )( ) ) ( ) b| dx| bx- QQ j +1 ? 2 Q?j = 1 ? - j j - n θ(( ) ) μ( ) () d y 2 2 r ?C|a y | ? Q ?j = 1 j +1 j +1 μ( ) μ( ) ) ?|( b| d x +|b- b| d x b x - 2 QQ QQ 2 QQ j +1 j +1 2 Q 2 Q ?? ?- nj +2 - j j θ() μ( ) μ( ) ) ( ( ) d y j 2 2 r 2 Q ‖b ‖3?C|a y | ? Q ?j = 1 ?- j 1 θ( ) ?C ‖a ‖(μ) j 2 ‖b ‖L 3 ?j = 1 1 θ( )t 1 1 ?C |a | μ)| (μ) . (a d t ?C |HHε1 + b b 0 t? ( ) μ( ) ( ) ( ) μ( ) Beca u se of a ydy= a ybydy= 0 , o ne ha s ?? ( ( μ( ) ) ) ( ) I I = | T b - ba x | d x22 QQ d R\ 2 Q ? ) ( ) μ( )μ( )( ( )( ) ) ( ( )- ba ydy- dx ?CK x , y K x , x byQQ Q d R\ 2 Q Q?? μ( )( )( ) ( ) ( ) μ( ) - K x , x | dxK x , y| Q d R\ 2 Q ?? μ( ( ) ( ) ) μ( )( ) ( ) dx | K x , y- K x , x |Q k +1 k ? 2 Q\ 2 Q?k = 1 ?- k k - n μ( θ( ) ( ) μ( )( ) ( ) ) 2 l 2 Q dx k +1 k ? 2 Q\ 2 Q?k = 1 ?k k +1 - n- k θ() μ( μ( ) ( ) )( ) ( ) d y 2 l 2 Q 2 Q ?C| bQQ | |b y - a y | ? Q ?k = 1 ( ) ( ) μ( ) ?C|a y | d y b y - b| | QQ Q ? MA T H EMA T ICA A P PL ICA TA 2009 232 λ( )?C | | | ( ) μ( )j byay| dyb| |- j QQ ? Q ?j j ? λ?C | |μ( ) μ( )( ) dy+ | dy (μ) j | by- b|b- b|‖a‖QQ QQ QQ j L jj ? Q Q ??j jj - 1 - 1 λμ( ) μ( ?C | | 2 Q‖b ‖) λj j 3K2 QK?C | | .Q , Q j Q , Q j j j ??j j 1 1 () Therefo r I I ?C | a | (μ) . No w ,we al so p ro ve d I I ?C | a | μ) . Thi s mea n s t hat 3 . 3i s(22 H2 Hb b t r ue . The t heo re m i s p ro ve d. Ref erences : [ 1 ] Préez C. Endpoint e stimat es fo r co mmutato r s of si ngular integral op erato r s [ J ] . J . Funct . A naly. , 1995 , 128 :163,185 . Mateu J ,Mat tila P ,Nocolau A ,O ro bit g J . BMO fo r no n do ubling mea sures [J ] . Duke Mat h. J . ,2000 ,102 : [ 2 ] 533,565 . [ 3 ] Nazaro v F , Treil S ,Vol ber g A . Tb2t heo rem o n no nho mo geneo us sp aces [J ] . Acta Mat h . ,2003 ,190 :151,239 . Stein E M . Ha r mo nic A nalysi s : Real2variable Met ho ds , O rt ho go nalit y , a nd O scillato r y Integral s [ M ] . [ 4 ] Princeto n : Princeto n U niv. Pre ss ,1993 . Tol sa X. Co tlar’ s inequalit y wit ho ut t he do ubling co nditio n a nd exi st ence of p rincip al val ues fo r t he [ 5 ] Ca uchy int egral of mea sures [J ] . J . Reine A ngew . Mat h . ,1998 ,502 :199,235 .2 Tol sa X. L 2bo undedne ss of t he Ca uchy integral op erato r fo r co ntinuo us mea sure s [ J ] . Duke Mat h . J . , [ 6 ] 1999 ,98 :269,304 . () Tol sa X. A T 1t heo rem fo r no n2do ubling mea sures wit h ato ms [ J ] . Proc . Lo ndo n Mat h . Soc . ,2001 ,82 [ 7 ] () 3:195,228 . 1 [ 8 ] Tol sa X. BMO , Ha nd Calderón2Zygmund operato r s fo r no n do ubling mea sures [ J ] . Mat h . A nn. ,2001 , 319 :89,149 . ) ( Tol sa X. L it tlewoo d2Paley t heo r y a nd t he T 1t heo rem wit h no n2do ubling mea sure s [ J ] . A dv. Mat h. , [ 9 ] 2001 ,164 :57,116 . () [ 10 ] Yabuta K. Generalizatio ns of Calderón2Zygmund operato r s [J ] . St udia Mat h . ,1985 ,82 1:17,31 . μ) ( ) θ RBMO(与t型 Cal derón2Zygmun d 算子的交换子的有界性 1 2 1 2 赵凯,任晓芳,周淑娟,王磊 () 1 . 青岛大学数学科学学院 ,山东 青岛 266071 ;2 . 青岛大学师范学院数学系 ,山东 青岛 266071 d μμ( ( ) ) 摘要 :设是 R上的 Ra ndo n 测度 ,其唯一需要满足的条件是增长条件 : B x , r?n d (μ) Cr 对任意 x ?R, r > 0 成立 , 0 < n ?d. 本文中 ,在这种非双倍测度下了 RB MO 与? θ( ) (μ) (μ) t型 Cal de r nó2Zygmu nd 算子的交换子是 L 到 RB MO 有界的 , 同时还建立了该交1 1 (μ) (μ) 换子 H到 L 的有界性.b θ( ) 关键词 :t型 Cal der nó2Zygmund 算子 ; Ha r dy 空间 ; RB MO 空间 ;交换子 ;非双倍测度