Riesz Potentials, Riesz Transforms on Lipschitz Spaces in Compact Lie Groups

Transcription

1 AEN eraioal Joural o Alied Mahemaic 4:3 JAM_4_3_ Rie Poeial Rie Traorm o ichi Sace i Comac ie rou Daig Che Jiecheg Che & Daha Fa Abrac Uig he hea erel characeriaio we eablih ome boudede roerie or Rie oeial ad Rie raorm o ichi ace i a comac ie grou Key word: Rie raorm Rie oeial ichi ace Beov ace Hea erel AMS claiicaio: 43A 43A3 43B5 NTRODUCTON e be a coeced imly coeced comac emiimle ie grou o dimeio The mai uroe o hi aer i o eablih boudede o Rie raorm R ad Rie oeial o he ichi o Thee oeraor ace were udied by EM Sei i [8] by uig he hea erel o wa roved by Cowlig Maero ad Ricci i [6] ha he Rie raorm R are Calderó-Zygmud oeraor Thu a adard argume how ha R are bouded oeraor o he ebegue ace or ay Alo i i well-ow ha he Rie raorm ad R are o bouded o he ebegue ace Our reul (ee Theorem ) i hi aer however eem lile urriig ice we will how ha o a comac ie grou he Rie raorm R are bouded o he or ay i α > Our mai reul are aed i he ollowig heorem Theorem e Re() = β/ > The he Rie oeial i a liear oeraor ha ma o ichi ace Maucri received Ocober 8 revied Augu 8 Thi wor i arially uored by he NSF o Chia (ra No ) F Daig Che i wih he Dearme o Mahemaic a Jaco Sae Uiveriy Jaco Miiii 397USA ( ched@yahoocom) S Jiecheg Che i wih he Dearme o Mahemaic a Zheiag Uiveriy Haghou Chia ( cche@ueduc) T Daha Fa i wih Dearme o Mahemaic a Uiveriy o icoi-milwauee Milwauee 537 USA ( a@uwmedu) boudedly Theorem The Rie raorm are bouded o Theorem 3 Suoe ha Re() = β/ < ad α + β > The i a liear bouded oeraor rom o Our roo ue a hea erel characeriaio o he ichi ace obaied by Meda ad Pii i [7] he emi-grou roery o he hea erel ad a eimae o he hea erel i [6] Thi roo migh be a ew oe eve i he claical cae For he hioric develome o he Rie oeial o boh claical cae ad ie grou oe ca reer [8][9][] ad he reerece herei e alo wa o meio ome rece aricle [][3][4] abou harmoic aalyi o comac ie grou Thi aer i orgaied a ollowig he ecod ecio we will ree ome eceary oaio o ie grou ad deiiio o oeraor ad ace ha will be udied i he aer e will iroduce ome lemma i he hird ecio ad ree he roo o he heorem i Secio 4 hi aer we ue he oaio A B o mea ha here i a oiive coa C ideede o all eeial variable uch ha A CB e ue he oaio A B o mea ha here are wo oiive coa c ad c ideede o all eeial variable uch ha c A B c A NOTATONS AND DEFNTONS e be a coeced imly coeced comac emiimle ie grou o dimeio e g be he ie algebra o ad τ he ie algebra o a ixed maximal oru T i o dimeio m e A be a yem o oiive roo or (g τ) o ha Card(A) = (-m)/ ad le A e be he orm o g iduced by he egaive o he Killig orm B o g he comlexiicaio o g he iduce a bi-ivaria meric d o Furhermore ice hom B i odegeerae give here i a uiue H i uch ha H BH H or each H e le <> ad (Advace olie ublicaio: 7 Augu )

2 AEN eraioal Joural o Alied Mahemaic 4:3 JAM_4_3_ deoe he ier roduc ad orm raerred rom τ o hom i by mea o hi caoical iomorhim e H exh beig he ideiy i The weigh laice P i deied by orayp : wih domia weigh deied by orayp: A Λ rovide a ull e o arameer or he euivale clae o uiary irreducible rereeaio o : or λ Λ he rereeaio U ha dimeio d A ad i aociaed characer i iw we 7 w w e iw i he eyl grou ad where w i he igaure o w e X X X be a orhoormal bai o g Form he Caimer oeraor X i i Thi i a elliic bi-ivaria oeraor o which i ideede o he choice o orhoormal bai o g The oluio o he hea euaio o d x x x x d or i give by x x where i he au-eierra erel (hea erel) i well ow ha i a ceral ucio ad oe ca wrie i a or ad > e d i eay o ee ha aiie he emi-grou roery or ay > Uig he hea erel oe ca deie variou ueul ucio ace o Oe o uch ace i he Beov ace B deied i he ollowig For ay > α ad we ay ha a ucio i i he homogeeou Beov ace hb i he hb orm o hb X d whe ; ad u X hb whe = where X X X X wih he e ay ha a ucio i i he Beov ace B i muli-idex B hb i iie Remar 4 (i) he deiiio o he Beov ace he um i ae over all he diereial moomial o order ad i a ixed big umber i eay o chec ha i he deiiio o he Beov ace oe ca ue he umber - o relace (ii) Oe ca ic ay > α i he deiiio The Beov orm obaied rom diere are euivale Aoher imora ucio ace i he ichi ace o e be a oiive ieger For ad or every eleme V g wih V = we deie he -h order dierece oeraor ceered a x wih direcio V by V x xex V > The -h order ucio -modulu o moohe o i he u V e α e ay ha a ucio i i he ichi ace V i he orm d i < ad (i = ) u (Advace olie ublicaio: 7 Augu )

3 AEN eraioal Joural o Alied Mahemaic 4:3 JAM_4_3_ i iie The ichi ace lay a imora role i udyig harmoic aalyi o ie grou i ow i [] ha oe ca ue he ichi ace o characerie he Hardy ace H o a comac ie grou Alo Meda ad Pii roved ha he Beov orm ad he ichi orm are euivale (ee [7]) hereore B Nex we recall ha oe ca ue he hea erel o deie Rie raorm ad Rie oeial o The ollowig deiiio ca be oud i Sei [8] The Rie oeial Re i deied by ' d where ' e d \ Thu i i eay o ee ha ' x dx e ca exed he deiiio o by uig he ormula o he comlex lae e are iereed i he aricular cae = / ad deie R by he Rie raorm R X d he i he cae i i i i d [6] boh oeraor i ad R are how o be Calderó-Zygmud oeraor Combiig ha wih he ² boudede roved by EM Sei [8] i ollow by a adard mehod o Calderó-Zygmud decomoiio ha boh oeraor i ad R are rog ye ( ) < < ad o wea ye ( ) hi aer we are alo i iereed i he Rie oeial Some emma e which ay H be he Sobolev ace o ucio o or X X X g X A orm o he ubace o ceral ucio i H i d H where are he Fourier coeicie o Sice he hea erel i a ceral ucio we have he ollowig eimae o emma 5 For ay muli-idex ad ay N X or ay N > uiormly or > σ Proo Ue he Hölder' ieualiy emi grou roery o ad he le ivariace o X oe ha X X X/ / X / / wih = / / H Thu he lemma ollow eaily rom he deiiio o By he Poio ummaio ormula (ee [5] or [6]) we ow ha m e e D A where iw D e w 4 Uig hi exreio o he hea erel we ca obai he ollowig eimae emma 6 For ay muli-idex wih = X uiormly or > Proo: By emma 5 we may aume ha i mall e (Advace olie ublicaio: 7 Augu )

4 AEN eraioal Joural o Alied Mahemaic 4:3 JAM_4_3_ U be a eighborhood o i τ uch ha i ralae by eleme o Λ are all dioi ad le η(x) be a C ucio uored o U radial ad ideically oe o a eighborhood o Oe deie wo modiied erel K ad V by 4 4 V e e K e e By Theorem 4 o [6] i i ow ha or ay air o ieger ad N N V K O H Alo by Theorem o [6] we ow ha give ay air o ieger ad N here i a ieger uch ha N V O H where M m D M ad D = M are diereial oeraor o order which are ivaria uder boh le ad righ ralaio Thu or σ we have X X V X V K X K V K V X K H H r or ome uiable ieger ad r Thereore or < σ X X K Recallig ha he ucio K coidered a a ucio o i uored o a mall eighborhood o o oe iroduce o hi eighborhood he regular coordiae where X hi coordiae 4 ex K e e 4 X e O i a iegrable erel o The lemma i roved By he roo o emma 6 i i eay o obai he ollowig eimae e i he roo o i Prooiio 7 e < The J X J or ay muli-idex J wih J β = Re() Proo o Theorem Proo o Theorem Sice he orm o Beov ace ad ichi ace are euivale i uice o how our heorem i he Beov orm For ay by he Miowi ieualiy ad Hölder' ieualiy ' ' d d Thu by emma 5 ad emma 6 we have Now by he deiiio o he Beov orm i uice o how hb B Fir we udy he cae = i eay o ee ha we oly eed o rove ha i > α + β he or ay muli-idex wih = oe ha u X Z u X Deoe ad he le ivariace o Uig he emi-grou roery o X we have By he roo o he emma 5 i [6] i i eay o ee ha Thu 4 X K e (Advace olie ublicaio: 7 Augu )

5 AEN eraioal Joural o Alied Mahemaic 4:3 JAM_4_3_ X Z X xd dx X xd dx X x d dx J J J3 For ay < σ whe > α + β by he Miowi ieualiy we ee ha J i domiaed by X ddx Thu by emma 5 we have J e be he cougae idex o By Hölder' ieualiy we have ' J3 X x ddx d u X Similarly by Hölder' ieualiy he ecod iegral J₂ i domiaed by u o a oiive coa ' X d X u Combiig he eimae o J₁ J₂ ad J₃ we obai he eimae () which rove he cae = Nex we how he cae < Checig he roo or he cae = we oly eed o how ha boh iegral X x d dx d 3 X xd dx d are bouded by B u o a coa mulile Uig he Hölder eualiy we eaily obai ha he iide iegral i () C X x d dx X which imlie ha he iegral () i bouded by B Now or > α + β + ad ay wih = we wrie X X X uch ha ₁ = ₁ > α + β ₂ = ₂ > α + (β/) + The he iegral (3) i bouded by d X x d dx d ' ' X X d d ' X d B The roo o he heorem i comleed Proo o Theorem By checig he roo o Theorem oe eaily ee ha R hb B To rove he heorem ow i uice o how ha R B By he boudede o R or < < roved i [CM] clearly we oly eed o dicu he cae = ad = = ad = by emma 5 i i eay o ee ha (Advace olie ublicaio: 7 Augu )

6 AEN eraioal Joural o Alied Mahemaic 4:3 JAM_4_3_ R X x d dx Uig egrae by ar; he ir erm above i bouded by (u o a coa mulile) lim X x dx X x d dx By emma 6 lim X x dx lim i X x dx lim i X x Alo X xd dx X x d dx egraig by ar ime or a eve > α we ge X x d dx B B X x d dx d he ame way we ca rove he cae = = Nex we coider he cae = ad < hi cae we have R X d dx B The roo or he cae = ad < < i imilar ad i ied The heorem i roved Proo o Theorem 3 ihou lo o geeraliy we may aume = - β/ Fir we rove ha C 4 B or ay B By he deiiio o he Rie oeial i i eay o ee ha x x d or ome aiyig - α/ > By emma 5 i i eay o ee ha d he = ad le ε be a mall oiive umber uch ha ε i uiciely mall i a way o be deermied The x d x d Thu u o a oiive coa i domiaed by u d e ow have B by chooig a mall ε > i bouded by he = ad = C d d B B he < ad Uig he Miowi ieualiy we have d For = (Advace olie ublicaio: 7 Augu )

7 AEN eraioal Joural o Alied Mahemaic 4:3 JAM_4_3_ d d hb For < < we ue Hölder' ieualiy o obai C C d hb where ' ' C d Thu (4) i roved To comlee he roo o Theorem 3 i remai o how ha C hb B e i he roo ice i i imilar o hoe o Theorem Reerece [] B Bla ad D Fa Hardy ace o comac ie grou A Fac Sci Touloue Mah 6 (997) No [] J Che ad D Fa Ceral Mulilier o Comac ie rou Mah Z DO:7/ [3] J Che ad D Fa A Biliear Fracioal egral o Comac ie rou o aear i Caad Mah Bull [4] J Che D Fa ad Su Hardy Sace Eimae or ave Euaio o Comac ie rou Jour Fuc Aalyi 59 () [5] J Clerc Bocher-Rie mea o H^{} ucio (<<) ecure Noe i Mah 34 (987) 86-7 [6] M Cowlig AM Maero ad F Ricci Poiwie eimae or ome erel o comac ie grou Red Circ Ma Palerma XXX (98) [7] S Meda ad R Pii ichi ace o comac ie grou Moah Mah 5 (988) 77-9 [8] EM Sei Toic i Harmoic Aalyi A o Mah Sudie #63 (97) Priceo Uiveriy Pre Priceo NJ [9] EM Sei Harmoic Aalyi: Real Variable Mehod Orhogoaliy ad Ocillaory egral Priceo Uiv Pre New Jerey 993 [] EM Sei Sigular egral ad Diereiabiliy Proerie o Fucio Priceo Uiv Pre New Jerey 97 (Advace olie ublicaio: 7 Augu )