Some Embedding Theorems and Properties of Riesz Potentials
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1 meica Joual o ahemaics ad Saisics 3 3(6) DOI 593/jajms336 Some Embeddig Theoems ad Poeies o iesz Poeials ahim zaev * Fuad N liyev 3 Isiue o ahemaics ad echaics o Naioal cademy o Scieces o zebaija; au; Z4 zebaija zebaija Sae Pedagogical Uivesiy; au; Z zebaija 3 au Sae Uivesiy; au; Z48 zebaija bsac I is well ow ha oeial ye iegals lay a imoa ole i eseach o he aial dieeial euaios The ese ae sudies oeies o iesz oeial i ems o local oscillaio o ucios Keywods ocal oscillaio ouded mea oscillaio oey saces iesz oeial Ioducio e be -dimesioal Euclidea sace o he ois x x x x ad a x x a be a closed ball i o adius wih he cee a oi a Deoe by loc a class o all local -owe summable ucios deied o ad by loc he class o all local bouded ucios deied o y ad we deoe we mea he usual ebesgue sace o by he coesodig om ha is x dx i ad su ess x x Deoe by P he oaliy o all olyomials o whose degees ae eual o o less ha e loc N ( N is he se o all osiive ieges) Deie he ollowig * Coesodig auho zaev@ambleu (ahim zaev) Published olie a h//joualsaubog/ajms Coyigh 3 Scieiic & cademic Publishig ll ighs eseved ucios x; i P x x ; i su x ; i x e x x x x i j ν ν ν ν be o-egaive ieges ly x x x x he ohogoalizaio ocess by he scala oduc g g o he sysem o he owe ucios x N aaged by aially lexicogahic ode [] whee E is he ebesue measue o he se E Deoe by he obaied ohogoal omed sysem e Suose ha ([][3]) I meas ha loc x ecedes he is ozeo dieece x i eihe i i is egaive o bu
2 446 ahim zaev e al Some Embeddig Theoems ad Poeies o iesz Poeials P x a a x a a a I is obvious ha P a is a olyomial degee o which is eual o less ha Deoe O a P a a O a ucio o he ball a o loc e us call local oscillaio o -h ode o he i he meic o Noe ha i he P x a a a a ad heeoe O a a a I is ow ha (c[4]) o each olyomial x he ieualiy ad each ball P C x P x x is ue whee he osiive cosa C does o deed o ad Hece i ollows ha C x x ; O x C x; I should be meioed ha he heoy o saces deied by local oscillaio has bee develoed by seveal auhos o isace FJoh ad Niebeg[5] SCamaao[6] NGeyes[7] SSae[8] J Peee[9] DSaaso[] ec (see also[][]) e loc We ioduce he ollowig deoaios ; x x x e be a class o all osiive moooically iceasig o ucios e y we deoe he se o all he ucios loc o which O We ioduce he om i he sace eualiy I by he su ad he whee is he oey sace ie loc x O x e is a osiive umbe We deoe by a se o all such ha almos deceases o I N he we deoe by he se o all ucios loc whee I we coside he class sace loc su such ha / P he as a subse i he uoie is he om o I he ioduced om he sace is a aach sace I N he we will deoe by he class o all he ucios loc ollowig elaio C a a a is valid a We deie he om o O O o which he a P C O by he eualiy a su a
3 meica Joual o ahemaics ad Saisics 3 3(6) I aicula i he O O whee O is he sace o all local summable ucios o bouded mea oscillaio The class O o he is ime was ioduced i[5] I is easy o see ha i he O ad hei oms ae euivale Coside also a class VO which was ioduced i[] VO is he class o all O o which he elaio lim su a a a a is valid Fo VO we deie e ~ y ebesgue sace o ad we deoe VO su O we mea he wea x x Poeial ye iegals lay a imoa ole i he mahemaical aalysis Fo he oeies o iesz oeials i ems o mea oscillaios we ee he eades o[9][4][6] ad he elaed aes o uhe iomaio I his ae we sudy he oeies o iesz oeial o a ucio i ems o local oscillaio o ucios o geeal oey ye saces The sucue o he ae is as ollows I secio some ieualiies ad embeddig heoems is oved The mea esuls o he ae ae give i Theoems 5 6 ad 7 which was oved i secio 3 whe belogs o ~ Some Ieualiies ad Embeddig Theoems Poosiio e loc The he ieualiy ; x x; c x is ue whee he cosa c deeds oly o ad Poo e Ieualiy we obai ; x x lyig he Hölde s x x ; x x x; Theeoe ; ; x x x The case is obvious Coollay I loc he he ieualiy ue c Poosiio e a ucio loc ; c such ha is The hee exiss ; () whee he cosa c is ideede o ad Poo e ad x x x The we have
4 448 ahim zaev e al Some Embeddig Theoems ad Poeies o iesz Poeials ; d d S S S ad similaly d ; S whee aea o S is he ui shee S is a suace S Fom hee we obai he ieualiy () Poosiio 3 e The loc ~ ( ) ad hee exiss a cosa C ideede o such ha x; C o all x ad Poo I ollows om well ow eualiy (see[7] Chae emma 4) ha (o ay osiive cosa ) x d x d x d x d d () we obai Now choosig Fom hee we obai he ieualiy () wih C ema I he case he ieualiy () i geeal is o ue Fo isace he ucio x x x belog o dx Poosiio 4 e ~ x The ~ bu x x ad hee exiss a cosa c deeds oly o such ha ; c o all Poo We have x x x x x x Theeoe su x x su x x su su ~ This meas ha ad (3)
5 meica Joual o ahemaics ad Saisics 3 3(6) Fuhe we obai d d S ; S d S whee S is he ui shee S Fom hee ; S S is a suace aea o The evious oosiio shows ha i is imossible o imove he esimaio () Poosiio 5 e The he ieualiy ; C (4) is ue whee a cosa C is ideede o Poo is we coside he case The we have ; ; x dx x dx X dx x X dx x whee Fuhe alyig he iowsi s Ieualiy we have ; X x dx X x dx whee is he volume o he ui ball I he case he oo o ieualiy (4) is obvious The ollowig oosiio shows ha i is imossible o imove he esimaio (4) i he case Poosiio 6 e The hee exiss he ucio such ha o he ieualiy C (5)
6 45 ahim zaev e al Some Embeddig Theoems ad Poeies o iesz Poeials is ue whee a cosa C is ideede o ad Poo e i i I x he we obai x; x x x I x 4 o all he x; x x; x; x; Thus he ucio x ; is iegable o he esides wih esec o agume x Fuhe we obai ha i he ; ; x dx x I is obvious ha = Theeoe we obai ha C ; I he case he agumes ae simila Wih hel o Poosiio 3 Poosiio 5 ad Coollay we obai coesodigly he ollowig heoems Theoem e The ~ ad ~ C C whee Theoem e ad The C Theoem 3 e The C C ad ad whee C 3 Poeies o iesz Poeials Coside he ollowig oeial ye iegal oeao x whee K x x y K D K y X y! ydy x x i ae o-egaive ieges i N g Dg x x x X is he chaaceisic ucio o he se Oeao oeial is a ceai modiicaio o he iesz y I x dy x y I should be oed ha i ad he he iegal dies om iegal I by a olyomial owe o which is eual o less ha I
7 meica Joual o ahemaics ad Saisics 3 3(6) he he oeial I is deied o o all ucios ad oeove i iegal x he o o examle absoluely coveges almos eveywhee Noe ha modiied iesz oeial simila o he was cosideed o examle i TKuoawa[3] TShimomua ad Y izua[4] ec (see also[5]) The ollowig asseio holds ue loc Theoem 4[6] e N The he ieualiy x ad x; x; C x; is valid whee ad cosa o deed o ad x Coollay e loc N ad The ieualiy (6) C does C is valid whee (7) ad cosa C does o deed o ad ~ Theoem 5 e I ( ) he ad C C whee Poo Wih hel o Poosiio 3 ad Theoem 4 we have whee x; C C C easily us ou Fom hee he heoem saeme Coollay 3 e N The C ~ ( ) ~ ( ) x x C whee Coollay 4 e ~ ( ) The O ad C Coollay 5 I ( ) C ~ he ad C O ~ ( ) whee C Theoem 6 e N The ad whee C C I he case i addiio o
8 45 ahim zaev e al Some Embeddig Theoems ad Poeies o iesz Poeials whee Poo Taig io accou he ieualiies (4) ad (7) i is easy o obai he ieualiy C (8) whee a cosa C is ideede o Fom hee we obai ha C I he lim Theeoe i ad he om esimaio (7) i addiio we have o I ad Coollay 6 e The C I C whee I he case i I o addiio we have Coollay 7 e The VO ad VO cos Coollay 8 e N I he O ad O C C whee addiio x I o he i Theoem 7 e ad N O I C he ad C whee ad i e The Fom Theoem 7 we obai ha N he ie he oeao boudedly acs om io Coollay 9 I he ( I ) Coollay I he O Coollay I N he whee EFEENCES O [] zaev O some maximal ucios measuig smoohess ad meic chaaceisics Tas S zeb 999 v9 No5 8-4 [] DeVoe Shaley aximal ucios measuig smoohess emoi me ah Soc 984 v47 No93-5
9 meica Joual o ahemaics ad Saisics 3 3(6) [3] zaev mulidimesioal sigula iegal oeao i saces deied by codiios o he -h ode mea oscillaio Dol ad Nau (ussia) 997 v356 No (ussia) [4] zaev O some oeies o iesz oeials i ems o he highe ode mea oscillaio Poc Is ah ech NS zeb 996 v (ussia) [5] Joh F Niebeg O ucios o bouded mea oscillaio Comm Pue l ah 96 v [6] Camaao S Poiea di höldeiaia di alcue classi di uzioi Scuola Nom Su Pisa 963 v [7] eyes GN ea oscillaio ove cubes ad Hölde coiuiy Poc me ah Soc 964 v [8] Sae S Some ucio saces deied usig he mea oscillaio ove cubes Scuola Nom Su Pisa 965 v [9] Peee J O he heoy o saces J Fucioal alysis 969 v [] Saaso D Fucios o vaishig mea oscillaio Tas me ah Soc 975 v [] Tiebel H ocal aoximaio saces Zeischi ü alysis ud ihe weduge 989 v8 No [] udyj Yu Saces deied wih hel local aoximaio Tudy os a Obsh 97 v (ussia) [3] Kuoawa T iesz oeilas highe iesz asoms ad eo evi saces Hioshima ah J 988 v [4] Shimomua T izua Y Teylo exasio o iesz oeials Hioshima ah J 995 v [5] Samo S O local summabiliy o iesz oeials i he case alysis ahemaica 999 v5 5- e [6] zaev Poeies o iesz oeials i ems o # maximal ucio Embeddig heoems Hamoic aalysis Poc Is ah ad ech o NS o zeb au 7 Issue [7] Gae J ouded aalyic ucios cademic Pess Ic New Yo 98
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