Weighted BMO Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operator Wenhua GAO

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1 Alied Mechaic ad Material Olie: -- ISSN: , Vol -6, 6-67 doi:48/wwwcietificet/amm-66 Tra Tech Publicatio, Switzerlad Weighted MO Etimate for ommutator of Riez Traform Aociated with Schrödiger Oerator Wehua GAO School of Alied Mathematic, eijig Normal Uiverity Zhuhai,Zhuhai 987 PRhia Keyword: Schrödiger oerator; Weighted MO ace; Revere Hölder ieuality; ommutator Abtract I thi aer, the Schrödiger oerator o dimeio Euclid ace with the o-zero, oegative otetial fuctio atifyig the revere Hölder ieuality i coidered The weighted boudede of the commutator comoed of everal Riez traform aociated with the Schrödiger oerator ad weighted MO fuctio o weighted Lebegue itegral ace are obtaied, for ome weighted fuctio Itroductio The Schrödiger artial differetial euatio with certai otetial fuctio are mathematical model of may field, uch a fluid machiery, iformatio ytem, itelliget ytem ad o o More ad more reearcher focued o the Schrödiger oerator with otetial fuctio atifyig the revere Hölder' ieualitylet P= - + V be the Schrödiger oerator o R with We aume that V i o-zero, o-egative otetial fuctio, ad belog to revere Hölder cla V ( /) > Let = = T P V, T P V, T = P ad T P 4 = Z She [] roved that P, P ad P are alderó-zygmud oerator if V belog tov, which iclude o-egative olyomial ad allow ome o-mooth otetial Moreover, ZShe alo how L boudede for T, T, Tad T 4 whe V Recetly, ZGuo P Li ad LPeg[] tudied L boudede of commutator [ b, Tj] btj Tjb, geeralize reult i [] to weighted cae = whe b MO( R ) I thi aer we will Defiitio A o-egative locally L itegrable fuctio V o R i aid to belog to ( << ), if there exit > uch that the revere Hölder ieuality ( V ( y) dy) ( V( y) dy) () hold for every ball i R y Hölder ieuality, it i eay to ee that > ) Oe remarkable feature about ( > the cla i that, if V for ome >,the there exit ε >, which deed oly oad the cotat i (), uch that V [] It i alo well kow that, if ( >), the + ε V V ( x) dxi a doublig meaure It wa roved that if V, the T i a alderó-zygmud oerator[] However, i [], the author how thee kerel had o moothe of -Z kereldue to V for ome > > / ad they dicovered that the kerel have ome other kid of moothe All right reerved No art of cotet of thi aer may be reroduced or tramitted i ay form or by ay mea without the writte ermiio of Tra Tech Publicatio, wwwttet (ID: 67, Peylvaia State Uiverity, Uiverity Park, USA-//6,9:4:)

2 64 Seor, Meauremet ad Itelliget Material Defiitio K( x, y) i aid to atify H( m) for ome m, if there exit a cotat > uch that, l>, x, x R with x x < l, the k= k m ( ) ( K( x, y) K( x, y) dy) <, k l y x < l k l () Where / m+ / = It i eay to ee that if K( x, y) i a -Z kerel,the K( x, y ) atifie H( m ),for m I [], the author roved that thee kerel of Tj( j=,,) atify H( m), m i differet rage reectively A o-egative fuctio defied or i called weight if it i locally itegral A weighted i aid to belog to Muckehout cla A ( R ) for < <, if there exit a cotat> uch that hold for every ball xdx x dx ( ) ( ( ) ) () R ; The cla ( A ) R i defied relacig the above ieuality by ( xdx ) ( x), ae x R (4) for every ball( x ) R (ee []) It wa well kow that A, A for < ad A imlie A( < < ) We ote that A = A Defiitio [4] Let <, ad A ( R ),We will ay that a locally itegrable fuctio f( x ) belog to the weighted MO for that i u[ f( y) f ( ) ] ( ) / y dy < where the uremum i take over all ball R, Modulo cotat, the aach ace of uch fuctio i deoted by MO The mallet boud atifyig coditio above i the take to the orm off i thee ace, ad i deoted by the claical MOace A baic fact about f MO, Our mai theorem are a follow Theorem Suoe V, > /, ' < < Let ( ', ) ad > ' Aume MO Obviouly, for the cae =, the A / 'For every b MO MO (ee [4]) i that, for A, MO i f ~ MO ( ) ' A ε = for ome ', there exit a cotat uch that [ bt, ] b f L ( ) MO L ( )

3 Alied Mechaic ad Material Vol -6 6 Theorem Suoe V, > / for( )' < Let A ( )( )' A ε = for ( )' ome (( )', ) ad > ( )' Aume /( )' For every b MO, there exit uch that [ bt, ] b f L ( ) MO L ( ) Theorem Suoe V, /, / = / / ad( )' < < Let A ε, ( ) ' where ε = for ome ( ', ) ad ' > 'Aume A / ' For every b MO, there exit a cotat uch that [ bt, ] b MO f orollary Suoe V, / ad ' < < Let ome ( ', ) ad > ' Aume uch that [ bt, ] b f 4 L ( ) MO L ( ) L ( ) L ( ) A / 'For every b MO ( ) ' A ε = for ' There exit a cotat Remark The reult i[] are the ecial cae of Theorem, ad orollary, whe = Proof of Theorem Proof of Theorem, ad maily deed o the followig Prooitio We firt dicu the roblem for geeral oerator Tf( x) = K( x, y) f( y) dy Later, we will ecialize to T ( j=,,) Prooitio Let m>, < < ad Suoe K atifie H( m ) Aume ( ) A ε = for ome (, ) ad > Moreover, T i bouded o L ( ) for every (, ) tol ( ) For every b MO,the [ bt, ] i bouded from L ( ), ad there exit a cotat uch that [ bt, ] b f () L ( ) MO L ( ) We adot the Stormberg' idea Prooitio follow immediately from the followig Lemma ad a theorem of Fefferma-Stei o har fuctio Lemma Let m>, ad > Aume K atifie H( m) ad A ε,where ( ) ε = for ome (, ) Suoe Ti bouded o there exit a cotat > uch that f Lloc, b MO, # M bt f x x b MO,, L ( ) for every (, ) j The ([, ] )( ) ( ) { M ( Tf)( x) + M ( f)( x)} (6) hold Where M f M f /, ( ) = [ ( )] ad M i weighted Hardy-Littlewood maximum fuctio Proof Fix >, f Lloc, x R, ad fix a ball = ( x, l), x We oly eed tocotrol J = [ bt, ] f( y) ([ bt, ] f) dy by the right ide of (6) Let f = f+ f, where f= fχ, f= ffthe [ bt, ] f = ( bb) Tf T( bb) ft( b b) f = Af + Af + Af

4 66 Seor, Meauremet ad Itelliget Material The J Af ( y) Af dy+ ( ) A f y A f dy + ( ) A f y A f dy = J + J + J For J, by Hölder' ieuality ad (7), we ca obtai J Af ( y) dy ( b b) Tf( y) dy = ( ) ' ' / ' / ( b b ( y) dy) ( Tf( y) ( y) dy) ( ) ( ) ( x) b MO M, ( Tf)( x) oiderig J,we fix uch that < <, ad let = Sice weighted L ( ) boudede of T, we ca obtai J T( bb ) f ( y) dy ( ( y) dy) ( T(( b b ) f )( y) ( y) dy) + + ( ) ( ) ( b( y) b ( y) dy) ( f( y) ( y) dy) ( ) ( ) ( x) b M ( Tf)( x) MO, To etimate J,we et J c = K( x, z)( b( z) b ) f( z) dz, the z x > l [ K( y, z) K( x, z)]( b( z) b ) f( z) dzdy z x > l = [ K( y, z) K( x, z)]( b( z) b ) f( z) dzdy k l z x < l k= m m { (, ) (, ) } { ( ( ) ) ( ) } k k k k K y z K x z dz + + l z x < l l z x < l k= u { ( ( ) ) ( ) } u { } k k + z x l k < k b z b f z dz dy b z b + b b f z dz E + E k ( l) k For E, fix (, ), the >, ( )' = ; >,( )' = Alyig Hölder' ( ) ieuality twice, by ε, ε =, we ca get E ( b( z) b ) ( ) ( ) ( ) } { k K! k z f z z dz ( ) ( z) dz} { + + b z b k k k k z dz f z z dz k + ( ) { } { ( ) ( ) } { ( ) ( ) } ( ) ( ) ( x) b M ( Tf)( x) MO, For E ice >, ( )' =, alyig Hölder' ieuality, A ad the fact m ' f f ( ) f we ca get k MO

5 Alied Mechaic ad Material Vol ( ) E k b f z z dz z dz ( ) / ( + )( ) { ( ) ( ) } { { ( ) } } MO k + ( ) ( )( ) b M ( f)( x)( )) ( ) ombiig etimate of E ad E, we obtai MO, ( ) b M ( f)( x) MO, J = u( E + E ) u ( x) b M ( f)( x) ( x) b MO M, ( f)( x) MO, k k k Thi comlete the roof of Lemma Proof of Prooitio From Lemma, ice A imlie A, we ca get [ bt, ] f M ([ bt, ] f) # L ( ) L ( ) b f MO L ( ) Proof of Theorom Sice ZGuo, PLi ad LPeg have how thatthe kerel K j of T j ( j=,,) atifie H ( ), H(()' ) ad H ' ) i [],reectively, ad KKurata ad Sugao[] ( have how boudede of T j ( j=,,) o lebegue ace, from Prooitio, we ca how Theorem,, Proof of orollary i follow from that fact T4 = ( + V) ( )( ) = ( IT) ( ) Referece [] ZShe L etimate for Schrödiger Oerator with certai otetial, AItFourier, 4,(99),-46 [] Zihua Guo, Pegtao Li,Lizhog Peg L oudede of ommutator of Riez Traform Aociated to Schrödiger Oerator[J] J Math AalAl4()(8) 4-4 [] JGarcia-uerva, JLRubio, Weighted Norm Ieualitie ad Related Toic[M], North-Hollad Amterdam, 98 [4]JGarcia-uerva, Weighted H ace, Diert Math 6(979) [] KKurata ad Sugao, A Remark o Etimate for Uiformly Ellitic Oerator o Weighted Sace ad Morrey Sace[J], MathNachr9(),7- L

6 Seor, Meauremet ad Itelliget Material 48/wwwcietificet/AMM-6 Weighted MO Etimate for ommutator of Riez Traform Aociated with Schrödiger Oerator 48/wwwcietificet/AMM-66 DOI Referece [] K Kurata ad Sugao, A Remark o Etimate for Uiformly Ellitic Oerator o Weighted L Sace ad Morrey Sace[J], Math Nachr 9(), 7- /(SII)-66()9:<7::AID-MANA7>O;-