Modulation Spaces with Exponentially Increasing Weights

Transcription

1 BULETINUL Univrsităţii Ptrol Gz din Ploişti ol LX No /8 5-9 Sri ttică - Inortică - Fizică odultion Scs with Exonntilly Incrsin Wihts ihi Pscu Univrsitt Ptrol-Gz din Ploişti Bd Bucurşti 39 Ploişti Institutul d ttică l Acdii Roân Cl Griviţi Bucurşti -il: ihipscu@irro Astrct odultion scs wr or th irst ti dind nd studid y H Fichtinr in 983 Usully th wihts usd or th dinition o odultion scs hv olynoil rowth But wihts with xonntil rowth cn lso usd Such scs r usul in th study o sudodirntil ortors o ininit ordr W shll introduc hr nw clss o odultion scs with wihts hvin xonntil rowth Ky words: odultion sc Glnd Shilov-Rouiu sc Introduction Th odultion scs wr introducd y H G Fichtinr in [] uch ltr thy wr rconizd s th riht scs or th ti-runcy nlysis Thy r lso usul in th thory o sudodirntil ortors: on cn us th to iv sil roos o clssicl thors on sudodirntil ortors such s continuity thors o Cldron illncourt ty or thors o coosition o sudodirntil ortors A corhnsiv introduction in th thory o odultion scs with wihts with olynoil rowth cn ound in [3] odultion scs with suultilictiv wihts with xonntil rowth wr studid in [4] nd [7] In [7] th odultion scs wr usd to study so clsss o ininit ordr sudodirntil ortors In our r w iv cohrnt dinition o odultion scs or so clsss o wihts which r not suultilictiv Ths scs cn usd to nlr th clss o ininit ordr sudodirntil ortors who cn studid usin odultion scs tchnius W shll cor t th nd o th nxt sction our clsss o odultion scs with thos rviously studid A Clss o odultion Scs As in [5] w shll considr suncs o ositiv rl nurs ) who stisy th ollowin ssutions: A) = ;

2 6 ihi Pscu A) + ) lorithic convxity); A3) thr xists constnt H such tht + + H ) A4) thr xists constnt H such tht H ) ; It is convnint or our uross to work with th unctions ssocitd to ths suncs : ) [ ) r) = su ln r ln ) ) r > For > N ) sunc hvin th s rortis s ) nd N its ssocitd unction w ut nd ϕ N x ) ξ ) = ϕ + ϕ ˆ S N) { ϕ S; ϕ < } = As usul ϕˆ dnots th Fourir trnsor o th unction φ nd s in [5] or silicity φ dnds on sinl vril Also sinc th suncs ) nd N ) r ixd nd consuntly thir ssocitd unction r w shll oit th runtly ro th nottions Th scs S N) r Bnch scsthir intrsction is Frécht sc i ndowd with th rojctiv liit tooloy nd will dnotd with S N) Th dul o S N) S ' N ) is sc o ultrdistriutions In ordr to introduc our odultion scs w hv to din irst so clsss o wihtd L scs W ut nd ξ ) N x ) : R R x ξ ) = ) x ξ ) R ) > L = L R ) = L N R )) = { F : R C; F x ξ ) x ξ ) < } L N L > = Th scs L < is Bnch sc with th nor / F x ) x ξ ) d dx) F L F = ; ξ ) ξ W shll nd so sil rortis o th sc L L Th sc L is invrint undr trnsltions Proo I F L nd y η) R thn

3 odultion Scs with Exonntilly Incrsin Wihts 7 F x y / / x ξ ) = F x ξ ) / / x + y ξ + η) [ y η )] F x ξ ) x ξ ) L Höldr inulity) I F L nd H L ) + = thn F x ) H x ξ )dxdξ F H ; ) ; ξ This l is dirct consunc o th clssicl Höldr inulity L 3 I F L N nd G L thn F G L N FH L nd Proo Lt us ssu tht F L G L Thn F x y G y η)dydη x ξ ) dxdξ / [ F x y x y G y η) y η)dydη) dxdξ ] Dinition Th odultion sc S' N) such tht F G = F G : : L N ti Fourir trnsor with window o ; / is th sc o ll th ultrdistriutions or so S N) Hr dnots th short πitξ x ξ ) = t) t ξ ) dt ) x ξ ) R whr th intrl convrs in th wk sns Our dinition xtnds th dinitions ivn in [3] [4] nd [7] In [3] wr dind odultion scs with olynoilly incrsin wihts In [4] nd [7] xonntilly incrsin wihts o ξ γ x γ th or : R R x ξ ) = ) x ξ ) R with γ wr considrd This rstriction ws iosd in ordr to nsur th suultilictivity o th wiht Our dinition llows to trt lso th cs whn γ i γ > th scs S N) r trivil) Th Corrctnss o th Dinition o odultion Scs L 4 I th sunc ) stisis A) A3) nd is its ssocitd unction thn εr) dr < ) ε > ) This l ollows ro l 3 sction chtr ro [8] L 5 For vry > thr xist so constnts > nd C = C ; ) such tht > ϕ C ϕ ) S N) ) ϕ S N)

4 8 ihi Pscu Proo I ε < in ) thn r) + ε r) H r) N r) + N εr) N H r) ) r Thror or S N) ϕ S ) w hv tht N ϕ ; ξ ) N x ) = ϕ x ξ) su x ξ ) R But ccordinly to [6] i H ξ ) N H x [ x ] ) ε ξ ) ϕ ξ ) = in ) = in ) H H thn thr xists so ositiv constnt C such tht N ε x ) su x ξ ) R ϕ x ξ ) H ξ ) N H x ) C ϕ Sinc ro L 4 it ollows tht w cn tk Th roo is colt ε ξ ) N ε x ) < = in ) = in ) H H For F L N > nd S N ) w din F y th orul < F ϕ > =< F ϕ > ) ϕ S N) Thn : L S' N) is continuous linr ortor Proosition I F L nd S N ) thn F / / Proo Lt S ) ixd window Thn s in [3] w cn s tht L N F x ξ ) C F ) x ξ ) ) x ξ ) R Sinc lso w otin s in th roo o L 3 tht thr xists ositiv constnt C such tht F C F / / ; ; ; Proosition Th dinition o th odultion sc S N ) ; is indndnt o th window

5 odultion Scs with Exonntilly Incrsin Wihts 9 Proo Lt S N) = nd lt N ; such tht so ositiv constnts nd Fro th invrsion orul [6]) w hv tht = L or Thror thr xists ositiv constnt C which dos not dnd on such tht = C / / ; / / ; ; ; Hnc L N Intrchnin th rols o nd w s tht i L L N thn lons lso to Th roo is colt Rrncs F i c h t i n r H G odultion scs on loclly coct lin rous Tchnicl rort Univrsity o inn 983 Glnd I Shilov GE - Gnrlizd unctions vol Acdic Prss Nw York-London Grő chni K Th oundtions o ti-runcy nlysis Birkhäusr rl Boston 4 Grő c h n i K Z i r n n G Scs o tst unctions vi th STFT Journl o Function Scs nd Alictions P s c u On th tooloy o Glnd-Shilov-Rouiu scs Ptrolu-Gs Univrsity o Ploişti Bulltin thtic-inortics Pysics Sris vol LX 8 6 Pscu Ti-runcy nlysis o Glnd-Shilov-Rouiu scs in rrtion 7 Piliović S Tonov N Psudo-dirntil ortors on ultrodultion scs Journl o Functionl Anlysis Rouiu Ch - Sur ulus xtnsions d l notion d distriution Annls Sciéntiius d l Écol Norl Suériur Rzut Sţii d odulţi cu ondri cu crştr xonnţilă Sţiil d odulţi u ost dinit şi studit ntru ri dtă d HFichtinr în 983 D rulă ondril olosit ntru dinir sţiilor d odulţi u o crştr olinoilă Dr ot i olosit şi ondri cu crştr xonnţilă Astl d sţii sunt util în studir ortorilor sudodirnţili d ordin ininit În cstă lucrr introduc o nouă clsă d sţii d odulţi cu ondri cu crştr xonnţilă