Exponential Atomic Decomposition in Generalized Weighted Lebesgue Spaces

Transcription

1 Joual of Mathatics Rsach; Vol 6 No 4; 4 ISSN E-ISSN Publish by Caaia Ct of Scic a Eucatio Exotial Atoic Dcoosio i Galiz Wight Lbsgu Sacs Nasibova NP Istut of Mathatics a Mchaics of NAS of Azbaija Cosoc: Nasibova NP Istut of Mathatics a Mchaics of NAS of Azbaija E-ail: atava8@gailco Rciv: Octob 4 Acct: Novb 3 4 Oli Publish: Novb 9 4 oi:5539/jv6437 URL: htt://xoiog/5539/jv6437 Abstact This a tats th xotial lia has syst which cosists of igfuctios of th iscotiuous ifftial oato Fa otis of this syst a stui i wight Lbsgu sacs wh th vaiabl o of suabily Kywos: syst of xots fas wight sac vaiabl xot Itouctio Ptub syst of xots t lays a iotat ol i th stuy of sctal otis of isct ifftial oatos a i th aoxiatio thoy Aatly th stuy of basis otis of ths systs ats bac to th wll-ow wo of (Paly & Wi 934) Sic th a lot of sach has b a i this fil (o tails ca b fou i Sltsii 5; Duffi & Schaff 95; Youg 98; Chists 3; Hil ) It shoul b stss that siila systs a of gat scitific st i th fa thoy as wll Sic ctly th aos a gat st i cosiig vaious obls lat to so sach fils of chaics a athatical hysics i galiz Lbsgu sacs wh a vaiabl suabily xot (fo o ifoatio s Kovaci & Raosi 99; ialig & Du ; Koilashvili & Paatashvili 6; Shaauiov 7; Koilashvili & Sao 3) Alicatio of Foui tho to th obls fo atial ifftial quatios i galiz Sobolv classs quis a goo owlg of aoxiativ otis of tub xotial systs i galiz Lbsgu sacs Aoxiatio-lat issus i ths sacs hav b fist stui by Shaauiov 7 I this wo w cosi a xotial lia has syst Th stuy of fa otis of tub xotial systs is closly lat to th o of siila otis of tub si a cosi systs Not that th lia has si a cosi systs aa wh solvig atial ifftial quatios by Foui tho Basis otis of lia has tigootic systs hav b stui i (Moisv 984; Moisv 987; Bilalov 99; Bilalov 999; Bilalov ; Bilalov 3; Bilalov 4; Moisv 998; Moisv 999) This wo is icat to th stuy of fa otis (atoic coosio fass) of th xotial icwis lia has syst i galiz wight Lbsgu sac Nful Ifoatio W will us th usual otatios N will b a st of all osiv gs; will b a st of all gs; ; will b th st of all al ubs; C will sta fo th fil of colx ubs; N R is th colx cojugat ; is th Koc sybol; A is th chaactistic fuctio of th st A Lt : b so Lbsgu-asuabl fuctio By L w ot th class of all fuctios asuabl o wh sct to Lbsgu asu Dot Lt f t I f f t t L f L I f : L () 37

2 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 Wh sct to th usual lia oatios of aio a ultilicatio by a ub L is a lia sac as su vai t Wh sct to th o f L is a Baach sac a w ot by L () Lt Thoughout this a f f if : I f WL : ( ) ( ); C t t : t t C t t l t t qt will ot th cojugat of a fuctio t : if vai t Th followig galiz Höl iqualy is tu ; q f t g t t c f g c ; Dictly fo th fiio w gt th oty which will b us i squl Poty A If f t g t a o It is asy to ov th f 38 g Statt Lt WL t t ; R Fuctio t sac L() if i i ; i Th followig facts lay a iotat ol i obtaiig i th ai sults Dot t q t Poty B [ialig & Du ] If t : th th class fi a ifily ifftiabl fuctios) is vy s i L By S w ot th sigula gal C Dfi wight class f Sf i t t is so icwis Höl cuv o C Lt : f f L : L f : f L t i i blogs to th C (class of b so wight fuctio fuish wh th o f f Th valiy of th followig statt is stablish i Koilashvili & Sao 3 Statt [Koilashvili & Sao 3 Lt bouly fo L to L if a oly if Lt b so Baach sac wh a o q Th WL Th sigula oato S is actig () will ot s cojugat wh a o By L M w ot th lia sa of th st M a M will sta fo th closu of M

3 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 Syst x N x N is sai to b uifoly iial i if Syst x N is sai to b colt i if L x x L x N Th followig cia of coltss a iialy a availabl Cio Syst x N is colt i if x N : u if Lx x u It is call iial i if f N f f Cio Syst x N is iial i has a biothogoal syst f N f x N Cio 3 Colt syst x y N is uifoly iial i su x y N is a syst biothogoal to Syst x is sai to b a basis fo if fo x N! N K : x x Syst x N is sai to b a fa if x L x N N K : x If syst x N fos a basis fo th is uifoly iial x W will also so facts about a atoic coosio a fas i Baach sacs Dfiio Lt b a Baach sac a K a Baach squc sac ix by N Lt f N g N Th g N f to K if : (i) g f N K f ; A B : A f g f B f (ii) (iii) f N K f g f f N is a atoic coosio of wh sct f ; Th coct of th fa is a galizatio of th coct of a atoic coosio Dfiio Lt b a Baach sac a K a Baach squc sac ix by N Lt g a S K N sct to K if : (i) : b a bou oato Th g g f N K f ; A B : A f g f N B f K S g f f f (ii) (iii) N A a B will b call th fa bous It is tu th followig f ; iе N S is a Baach fa fo wh Poosio Lt b a Baach sac a K a Baach squc sac ix by N wh caoical basis Lt g S L K; Th th followig statts N N a quivalt: (i) g N S (ii) g S a N is a Baach fa fo wh sct to K is a atoic coosio of wh sct to K N N 3 Wight Hay Classs wh a Vaiabl Suabily Exot z : z b a u ball o th colx la a lt b a u cicufc Dot Lt 39

4 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 La Lt WL : f L I fact lt L () h u : u i a u su u a lt th wight t f Lt us choos ( t) ( t) Lt It is ot ifficult to s that I a cosi ( t) ( t) f ( t) t f ( t) t satisfy coio () If f L () th ( t) ( t) Alyig Höl s iqualy w gt ( t) ( ) I C( ) f () f f ( ) It is absolutly cla that th latio hols if a oly if th followig iqualy is fulfill ( ) ( ) Hc w gt th coio o th aat : As li to choos q ( ) ( ) (3) th follows ictly fo latios () that is always ossibl such that th iqualis (3) a satisfi As a sult w obtai fo () that I M f ( ) Usig this la w ov th followig o La Lt : I fact lt u M ( ) is a costat La is ov ( ) WL a lt th wight (t) satisfy th coio () If u h th u h h () ( ) u( ) Assu I ( ) u ( ) u Fo La w obtai that C is a costat t : I ( ) C ( ) u (a i futh too) Cosqutly su ( ) I ( ) u C u ( ) a as a sult u h La is ov Th followig tho is tu Tho Lt WL a lt th iqualis () b satisfi If u h () th f L () : h ( ) ( ) u( ) P ( t) f ( t) t (4) 4

5 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 P ( ) is th Poisso l Vic vsa if L () cos by (4) blogs to th class h () Poof Fist w cosi th cssy Lt sults tll us (s g Kusis 984) that f th fuctio u u h () Th by La : u h f L fi Classical such that th latio (4) hols It is ow that u( ) f ( ) as a o ( ) (accoig to Fatou s la Kusis 984) a cosqutly u ( ) f ( ) as a o ( ) Th by th classical Fatou tho w obtai that i Now lt L () f ( ) f Ncssy is ov L () ( ) ( ) li u( ( ) ) ( ) f a lt th latio (4) b tu As th iqualis () hol th follows fo th sults of Koilashvili & Sao 3 that th sigula oato (4) wh Cauchy l is bou i follows ictly that th gal oato (4) wh Poisso l is uifoly bou i i P f C f f u( ) to ( ) ( ) ( ) costat it of a f Tho is ov Siilaly w fi th wight Hay classs H P ( ) By is so ub Dfi H f H f L ( ) bouay valus o of f It is absolutly cla that f blogs to th sac sac ( ) : ( ) ( ) h () Thfo ay of otis of fuctios fo () P ( t) f ( t) t L () It L () wh sct a C is a H w ot th usual Hay class f a o-tagtial H if a oly if R f a I f blog to th h stay tu fo fuctios fo Taig o accout th latioshi btw Poisso ls P ( ) a Cauchy ls usig Tho w asily gt th valiy of Tho Lt WL a lt th iqualis () b satisfi If H ( ) ( ) H K z ( t) a z F th f L () : f ( t) t F( z) K z ( t) f ( t) t (5) z Vic vsa if f L th th fuctio F fi by (5) blogs to th class () Th wight Hay class ( ) ( ) H H of fuctios which a aalytic i C \ ( ) wh thi os at ifiy is fi siilaly to th classical o Lt z fi o ( ) z at ifiy i f z) f ( z) f ( ) f is a olyoial of g ( z f z f b th aalytic fuctio i C \ of a f ( ) is a gula at of Laut sis xasio of f (z) i th ighbohoo of a ifily ot o If th fuctio ( z) f z blogs to th class ( ) H th w will say that th fuctio z Th valiy of th followig tho is ov just li i th classical cas Tho 3 Lt WL a lt th iqualis () b satisfi If z f blogs to th class f H ( ) th ( ) H 4

6 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 f ( ) f ( ) ( ) f a o-tagtial bouay valus o of f Just as w hav th followig Tho 4 Lt WL a lt th iqualis () b satisfi If f ( ) f ( ) ( ) f H ( ) f a o-tagtial bouay valus o of f fo th outsi of Lt us show th valiy a aalogu of th classical tho of Siov Assu that WL a lt H th iqualy () b fulfill Lt u a u L of u Th is ow that f L : Cosqutly f by Tho w obtai u z u a o u H f i z th u b o-tagtial bouay valu o as Hc ictly follows that f L Th Thus th followig tho is tu Tho 5 Lt WL a lt th iqualis () b satisfi If u H 4 Bass of xots i wight Hay classs Cosi th syst of xots E a ot E ( ) H u a u L th By ( ) L a L ( ) ot th stictios of classs H ( ) a H ( ) sctivly to i () L ( ) H ( ) / ; ( ) ( ) / L H Lt s show that if th iqualis () hol th th syst E fos a basis fo L ( ) Ta f L ( ) If th iqualis () hol th f( ) ( ) L L Th as is ow w (6) Mo ifoatio about this fact ca b fou i Kusis984; Pivalov 95 If th iqualis () hol th follows fo th sults i Dailyu 975 that th syst of xots E fos a basis fo L () Taig o accout (6) w obtai that f ca b xa i a sis i of th followig fo ) L () f ( f f a th biothogoal cofficits of f wh sct to th syst E It is absolutly cla that such a xasio is uiqu Cosqutly th syst that if th iqualis () hol th th syst tu () E fos a basis fo () E fos a basis fo L L ( ) ( ) Tho 6 Lt WL a lt th iqualis () b satisfi Th th syst basis fo ( ) L ( ( ) L ) 5 Ria Bouay Valu Pobl fo Cosi th followig Ria obl i ( ) H Classs ( ) H ( ) H classs Siilaly ca b ov Thus th followig tho is () E ( () E ) fos a F ( ) G( ) F ( ) f ( ) (7) f L is so fuctio By th solutio of obl (7) w a a ai of aalytic fuctios () 4

7 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 ( F ( z); F ( z)) H ( ) H ( ) Itouc th followig fuctios sig) th u cicl sctivly: t ag G Dfi Sohotsi-Pllj foulas yil Assu z z z bouay valus of which satisfy th latio (7) alost vy i z which a aalytic isi (wh th + sig) a outsi (wh th - z x l G z t z 4 W hav x 4 z t z z t i z i z z G i z z G i t (8) Itouc th icwis aalytic fuctio z z z z z Followig th classics w call fuctio xssio fo Lt G i (7) w obtai z z z th caoical solutio of th obl (7) Substutig th (8) F F F z a fi th icwis aalytic fuctio It is ot ifficult to s that th fuctio z z z z z has h ols o zos fo z Thfo fuctios z z a z z blogs to th Hay class o so suffics to ov that L F hav th sa o at ifiy Th sults of Dailyu 975 ily ictly that th fuctio tho (Dailyu 975) H fo sufficitly sall valus of W will suos that th cofficit G ( ) satisfis th followig coios: is a icwis Höl fuctio o ) G L ( ); ) ( t) agg ( ) : Lt s s s b th cosoig jus of (t) Lt Lt us show that z H To bcaus th st will iiatly follow fo th Siov b th os of iscotiuy of th fuctio (t ) a h : h s s ; at ths os Dot h ; h 43

8 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 Assu u t h t si x 4 u t si t s As is ow (s Dailyu 975 ) th bouay valus i h t ctg s t u t u t si a xss by th foula h s u t t si It follows ictly fo th Sohotsi-Pllj foula that suva Thus th followig statio is tu fo By th fiio of solutio w hav : u t h t s si (9) ( z) H ( ) F Cosqutly F ) ( ) Thfo if h ( L ( ) ( ) L ( ) th w obtai ictly fo th Höl iqualy that ( ) L ( ) q( ) W will th followig asy-to-ov la that follows ictly fo Statt La 3 Lt C a ( t) t if fo c a ( c) Th th fuctio fo ( c) c ( t) t c blogs to L () Rst th ouct i th followig fo t l ag s l u t t a is fi by th latio i t ag i hi t si l () Taig o accout La 3 w obtai that if th iqualis ( t ) () a tu th th ouct q blogs to th sac L q() i L So if th iqualis () a tu th th fuctio (z) blogs to classs H q( ) Cosqutly accoig to th sults of Dailyu 975 (z) is a olyoial P ( z) of o Thus F ( z) P ( z) ( z) which coios th fuctio F (z) blogs to th sac H ( ) W hav Lt s fi out u 44

9 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 Cosqutly if th iqualis a tu th is cla that F ( ) L ( ) l u t t ( t ) a hc F H ( ) So if th iqualis () q( t ) t a tu th th gal solutio of th hoogous obl F G F H H ca b st as z P z z P z i classs olyoial of o F Now lt s cosi th o-hoogous obl (7) Ta f L a suos F z z Th Sohotsi-Pllj foulas ily that th bouay valus follows fo Statt that F z F z followig tho K z t f t ; H H is a abay t (3) F satisfy a th qualy (7) Moov As a sult w gt th valiy of th Tho 7 Lt b fi by () a th iqualis () () b satisfi Th th gal solutio of th Ria obl (7) i classs H H ca b st i th followig fo z P z z F z F is th caoical solutio of hoogous obl F o-hoogous obl (7) fi by (3) a 6 Atoic Dcoosio Cosi th followig xotial lia has syst P is a olyoial of o i sig t t is th aticula solutio of (4) C is a colx aat To xlo th coosio wh ga to th syst (4) w will follow th sch of Bilalov & Gusyov Cosi th cojugat obl t F G F f t t (5) t G Th aticula solutio z F of th obl (5) has th followig fo t z f t F z z t z f t t F z z Pocig as i th vious sctio w gt t R t Cosqutly t h 4 R R Th fo Tho 7 w obtai that if th iqualis q q a tu th th cojugat obl (5) has a uiqu solutio F z F z (6) R (7) ; i classs H F Also ca b ov (i th sa way as i Bilalov & Gusyov ; Bilalov & Gusiov wh 45

10 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 Fbuay ) that wh th iqualis (6) a (7) fulfill th syst (4) fos a basis fo Thus th followig tho is tu L Tho 8 Lt WL a th iqualis (6)(7) b fulfill Th th xotial syst (4) fos a basis fo th galiz wight Lbsgu sac L ( ) ( ) 7 Atoic Dcoosio a Fass Dot by K E th sac of cofficits of th syst (4) It is ow that th caoical syst N fos a basis fo в K E (s g Maova ; Saigova & Maova3) N Lt K : K E L b a cofficit oato K is a isoohis btw oov f K f E K E Thus A f K E K E K a L bcaus th syst (4) fos a basis fo L a f B f f L K E A B a th costats Lt L a to E Th by fiio (s g Chists 3) th ai E ; wh ga to K E Moov th ai E ; K q E fos a Baach fa fo t b th syst cojugat is a atoic coosio of L wh ga to E K L Suos that K is so B -sac of th squcs of scalas fuish wh th o Lt th ai G ; S fo a fa fo L G g t Lq is so syst a S L K ; L is a fa oato It is ot ifficult to s that S fos isoohis btw S K Cosqutly S K S I S N fos isoohis btw K E a S K a L S K i E K a isoohic Whout loss of galy w will assu that th oato S is fi o S S L K ; L Covsly suos that th B -sac of th squcs of scalas E K a S K is isoohic to E i K a T : K is th cosoig isoohis Assu S K T It is cla that S L ; L is a isoohis W hav f SS g f T E f f L : K f ST E f Sg f f ST a E f is th valu of fuctioal L q E f f t E t t t ~ It is ot ifficult to s that g f T E f E f E f that L q g i g L W hav f ~ g f M f f L E at th lt It hc follows is th o o a ; M a th costats As a sult w obtai that th ai g ; S fos a fa fo L wh ga to So lt s cosi th gal cas wh ga to th syst (4) Assu Lt R P Th : P : (8) 46

11 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 Cosi th followig syst R q (9) t i t ; ; Pocig as i th vious cas w obtai that th syst L I this cas th cosoig cojugat obl has th followig fo t G t F t G t F f t t E fos a basis fo () If th iqualis (6) (9) a tu th this obl is uiquly solvabl i classs H H fo f L It is obvious that if th coios () a satisfi th th syst fos a basis fo isoohic to E L If th syst E fos a basis fo L wh so th is Lt K ot th sac of cofficits of th syst E wh ga to L Cosi oato T : L L fi by Tf t t f f f f f f H f H It is ot ifficult to s that i t T t Th uiqu solvabily of th cojugat obl () i classs H H ictly ilis that th oato T is a isoohis i E wh ga to coicis wh th sac L Cosqutly th sacs of cofficits of th systs L coici wh ach oth Thus th sac of cofficits of th syst K It is absolutly cla that th systs fi ub of lts Lt C b a isioal colx sac If E a E E a E iff fo ach oth by th th is cla that E E If th E is a at of E It is ot ifficult to s that th followig ict coosios hol: K C E K if ; () E C K K if () t It is absolutly cla that th syst is lialy it wh W ot th lia sa of this syst by L a itify wh C Cosi th ict su ˆ L L L C L (3) Obviously i this cas th syst E fos a basis fo L ˆ a bsis fos a fa fo Lˆ wh ga to K E E K Wh is absolutly cla that this syst is a fa squc wh ga to So w hav aiv at th coclusio that th followig tho is tu Tho 9 Lt WL th iqualis (6)(7) b fulfill a th st P b fi by (8) Th th iqualis () () hol wh ga to th sac of cofficits of th syst (4) is fi by (9) Wh th syst E fos a basis fo L ˆ th sac L ˆ is fi by th ict su (3) Wh th syst E is a fa squc wh ga to E K Th autho woul li to xss h st gatu to Pofsso BT Bilalov fo his couagt a valuabl guiac thoughout this sach Rfcs Bilalov B T (99) Basicy of So Systs of Exots Cosis a Sis Diff Uaviya 6() -6 47

12 wwwccstog/j Joual of Mathatics Rsach Vol 6 No 4; 4 Bilalov B T (999) O basicy of systs of xots cosis a sis i Bilalov B T () O basicy of so systs of xots cosis a sis i 7-9 L Dol RAN 365() 7-8 L Dol RAN 379() Bilalov B T (3) Bass of Exots Cosis a Sis Which A Eigfuctios of Difftial Oatos Diff Uaviya 39(5) -5 Bilalov B T (4) Basis Potis of So Systs of Exots Cosis a Sis Sibisiy Mat Jual 45() Bilalov B T & Gusyov G () Basicy of a syst of xots wh a ic-wis lia has i vaiabl sacs M J Math 9(3) htt://xoiog/7/s Bilalov B T & Gusiov G () A cio fo th basis oty of tub xotial systs i Lbsgu sacs wh vaiabl xot Dolay Mathatics 83() Chists O (3) A Itouctio to Fas a Risz bass Bihaus Bosto Basl Bli Dailyu I I (975) No-gula bouay valu obls i th la M Naua 56 Duffi R J & Schaff A C (95) A class of o haoic Foui sis Tas A Math Soc 7(34) Hil C () A Basis thoy i Sig 536 Koilashvili V & Paatashvili V (6) O Hay classs of aalytic fuctios wh a vaiabl xot Poc Razaz MathIc Koilashvili V & Sao S (3) Sigula gals i wight Lbsgu sacs wh vaiabl xot Gogia Math J () Kovaci O & Raosi J (99) O sacs L a W Czchoclova Math I 4(6) Kusis P (984) Itouctio to th Thoy of Sacs H M «Mi» 364 Maova V () O basis otis of gat xotial syst Ali Mathatics htt://xoiog/436/a369 Moisv E I (984) O basicy of systs of sis a cosis DAN SSSR 75(4) Moisv E I (987) O basicy of a syst of sis Diff Uaviya 3() Moisv E I (998) O Basicy of Systs of Sis a Cosis i Wight Sac Diff uaviya 34() 4-44 Moisv E I (999) Basicy of a Syst of Eigfuctios of a Difftial Oato i a Wight Sac Diff Uaviya 35() -5 Paly R & Wi N (934) Foui Tasfos i th Colx Doai A Math Soc Colloq Publ 9 (A Math Soc Povic RI) Pivalov I I (95) Bouay Potis of Aalytic Fuctios M-L Gosthizat 336 Saigova S R & Maova V (3) Fas fo cosis wh th gat cofficits Aica Joual of Ali Mathatics a Statistics (3) 36-4 Sltsii A M (5) Classs of Aalytic Foui Tasfos a Exotial Aoxiatios Moscow Fizatl 54 x Shaauiov I I (7) So obls of aoxiatio thoy i sacs L AalMath 33() ialig F & Du () O th sacs x L a W Joual of Math Aal а Al Youg R M (98) A Itouctio to Nohaoic Foui sis Sig 46 Coyights Coyight fo this aticl is tai by th autho(s) wh fist ublicatio ights gat to th joual This is a o-accss aticl istibut u th ts a coios of th Cativ Coos Attibutio lics (htt://cativcoosog/licss/by/3/) 48