Köthe-Bochner Function Spaces

383 Pages·2012·5.756 MB·English

by  Pei-Kee Lin

Tom ya dvisWo.rJsD. a viasn dD .RLe.w is
Pei-KLeien
Kothe-BoFcuhnncetSrip oanc es
Springer-ScieMnecdei+aB,u siness
LLC
Pei-KLeien
DepartomfeM natt hematics
UniverosfMi etmyp his
MemphisT,N 3 8152
U.S.A.
LibroaCfro yn gCraetssa logingD-aitna- Publication
LinP,e i-K1e9e5,2 -
K(jthe-Bochner fsupnaccteiPsoe ni -KLeien
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p.c m.
Inclubdiebsl iographical arnedif nedreexn.c es
ISBN9 78-1-4612-648I2S-B8N 9 78-0-8176-8(1e8B8o-o3k )
DOl1 0.1007/978-0-8176-8188-3
1.N ormeldi nesapra ce2s.B. an achs pace3s.V. e ctor-vafluunecdt i4o.n s.
Operator-vfaulnucetdi Io.nT si.t le.
QA322.2.2L05023
515'.732-dc22 2003063007
CIP
AMS SubjeCclta ssificaPtiroinmsa:r4 y6:B 204,6 E4S0e;c ondary: 4466BB22842,6, 8 424,6 E302,8 B05
Printoenad c id-fpraepee r.
ISBN9 78-1-4612-6482-8
©2004Sp ringSecri ence+BusMiendeiNsaes w Y ork
Originaplulbyl isbhye dr k r in2 004
BihliuseBoston
2004
Softcover rephrairndtc o1ovsefetdtr ih tei on
Allr ighrtess ervTehdi.ws o rkm ayn otb et ransloartc eodp ieidnw holoer i np arwti thotuhtew ritten
permissoifot nh ep ublisShperri nger-ScienceM+eBduisai,n esesx cepfto rb rieefx cerpitns
LLC,
connectwiiotnrh e vieowrss cholarly anUasleyi snci osn.n ectwiiotnah n yf ormo fi nformatsitoonr age
andr etrieevlaelc,t raodnaipct atcioomnp,u tseorft waroer,b ys imiloardr i ssimmieltahro dolnoogwy
knowno rh ereaftdeerv elopiesfd o rbidden.
The uisnet hipsu blicaotfit orna dnea mest,r ademarks, msarekrsva incdse i miltaerr mesv,e ni ft heayr e
noti dentiafises du chi,sn ott ob et akeansa ne xpressoifoo pni niaosnt ow hethoerrn ott heayr es ubject
to propreirgthyt s.
9 8 76 5 4 3 2 1 SPIN1 0953037
www.birkhasuer-science.com
Contetns
Praecfe vii
Notiaotn xi
1 CalssiTchaelo rems 1
1.P1r elrii.me is.n a 1
1.B2sa iSce qu.e.n.ce.s.. . . . . . 8
1.B3na aScpchae Cso ntfao1ir n ing 29
Co
14. JmaeTsh'eso .r.em.. . . . 42
15. ContFuinncutSoipcuoaes.ns . 48
16. ThDeu onrfdt-tPPirpesoe r.t y 57
1.T7hP ee lcPzryotnp(ys*eV k.)ri 69
1.T8ne sPorro dBunacatScsph a ocfe s 74
19. CondlEi xtepictotianaonaMnda rtingales 81
10.N 1otaenRsde ma. rks 94
1.R1ee1fr e.n c.e s. . . . . . .9 6
2 ConveaxniSdt moyto hness 101
2.1S ttCr oivncexaintdy UCnovinefxoirtmy . 101
2.S2m oot..hn..e. s.s 124
2.3B naacahkP-srpS oerty 103
2.N4o taenRsde ma. rks 137
2.R5ee fre.n.ce..s. . . 193
3 Kothoehc-neBrFu nctiSopcnae s 143
3.K1o tFhuen cStpciae.os.n . .. ..... .. . 143
3.S2t raonnSdgcl laaMyre laybs lFueur nac.t ions 162
3.V3ce tMoaers u.r e. . . . . 167
3.S4o BmsaeiR ce s.u l.t s. . . . . 177
3.D5u nforOdpt-eoPr.reast. t.i s 188
3.T6hR ea don-PNriroktpoyed Ym 159
3.N7o taenRsde ma. rks 211
3.R8ee fre.nc.e.s. .. . . . . . . 216
vi CONTENTS
4 StbaitlyPi ropeIr ties 219
4.E1x tProeimanenSt dms o Pooti.hn ts 219
4.S2t rEoxntgarlneyDdm e en Ptoiin.ngt. s 226
4.S3t raonnwdg-*t lS ryo EnxgplPoyos iendt s 233
4.N4o teRse maa.nr dk s 242
4.R5ee fre.nc.e.s . . . .. . . . . .2.4.5 . .
5 StbaitlyPi roperItIi es 241
5.C1o poifie Ens( X.). . . 248
5.T2hD ei Coa z-TKhae. lo.tr oenm 257
5.T3a la'Lgs(r1 Xah)ne-doT.r em 216
5.P4r op(eV).r* t.y. .. . . . 278
5.T5hT eaa lgrSaapncd.e s . . 290
5.T6hB ena achP-orSpaekrst y 295
5.N7o teRse maa.nr dk s 307
5.R8ee fre.n.ce.s. .. . . . . 309
6 ContinFuunociutosSn pa ces 313
6.V1ce tVoarlC-uoendt Finuncutoiuosn s 313
6.T2hD ei euPdrorontipnyCneeK( ,X ) 326
6.T3h Hee reDduiontrfadr-PyPr eot.pt eirst y 331
6.P4re otcjiTvneesP orro ducts 348
6.N5o teRse ma.an rdk s 355
6.R6ee fre.n.ce.s. .. . . . . 363
Index 361
Preface
Thmiosn ogidrsea vptohaot s e pdel ac rioeBafana ascphcat eh reyoh-tKe othe­
Bohcnfeurtn icsoapnc. Te wot ypqiuceanslsit t nih oairsae rae :
Quesito1nL. e Etb eaK otfhuen csatpciaeonX nd Bana cahs pc.aeD oes
thKeot hBeo-chfnuenrcs tpciaeEo (n)Xh vaet hDeu nforpdr-oPpeietfrt tiys
boEta hn Xd h vaet hsema ep rotpyIe?tfr h aenw seirns e g,ac taiwnvefi e nd
someex tcroan doinEt a ino(dnro sX)s utchthEa ()Xh sat hDeu onrfde-tPtis
property?
Quest2i.oL ne1 t E Kao tfhuen cstpci,aeao nnXd Bana ach
:::;p :::;00 ,
sapcDeo.e s Ee ioctroh neXatr nt'a - piesnq uietfnhK ceoe t he-fBuoncchtnieorn
sapce( X)hE saa nt' ps-eqcuee?n
Tos otlhvaeeb otvowqe u ieoswntisnl ooltn gliyuv aseb etutnedsretn rading
otf hset ruurocetft hKeo the-fBuoncchstnpcieaeborsun i t wt i allsldo e velop
somuesu eltf echntihtcqa aubnee s atpopoh tleifiree d,ls dusacshh a rmonic
anaslp,yr asobibitlehioytar,yn o dep ratthoero ry.
Leutso uttlhiceno en toetfnh tes book.
Itnh fiert sowtc hapwtepe rrossvo imdeeb ssairocem sefu otlrht ossst eeun dts
whdoon ohtva ea nbyca kgrionBun anadsc hpc aet heWoerp yr.e psreson otf
ofR oshelan't't-slt hme,Jom aree'stseh or(ewmh einss e rXpa lab,eK )ol'mso s
theemoN,r.R andriaan'tasehn otrtoeahtmpna rionperitasys eaprably
(V*)
deterpmrionpaeendrOd td y-e,Slh cllpurme'ctsehh otrtehtmea vesrerypba ale
refleBxnaicavhse pc aehs aa ne qui2vaRnlro emn.t
ICn hpat3ew rei ntrKoodtuhcee- fBnuocctshipncoaeeanrsn p dr osvveeer al
basic resutlhstosesp c aerse:g arding
Wep rvoet htVa it'lasel miamnaLd e be'dssog muiecnoaentrvegdte ehn­ce
L1
• oreamrs ett irliulfe r oenpel abcyae nos r der-cKoontfthuienn cutoiuosn
spcaoev aefi rn imteeae ss upcrae.
Foarn oyr detri-uncsKuo oontfhnuec tsipcoaeEna nadn Byna cahsa pce
• X,t hKeo the-fBnuocctshipnoaEencX(r e) i ( sw aek)lu ynfoirmcloyn vex
iafn odn ilbfyto ha nXda r(ew eauknliyf)co ornm.vl eyx
E
Foarn oyr detri-uncsKuo oontfhnueci tosnap cEea nadn Byna ascphc ae
• X,t hKeo thceh-nfBenuorc tsipcoaeEn X( )h ast hRea dokno-dNyim
viii PREAFCE
protpyiea frn odn lbyo Eti ahfn Xd h atvehR ea don-Npirkoopd­ym
erty.
Chpat4ei rdse vtotothe ed gepormoepotetfrrh iKtecoti e he-soB chfnnuecr­
tisopna Ocneoestf .h fenu dameqnuteasrlte igoantrshdK eiot enh-gB ochner
functsipcoaeinsts h felo lion:wg
Questi3.oL ne Et b eaK othfenu ctsipcoaean n Xd Bana acshpc .ae
Spupotshtea Ia nei xst rpeomioenft t h e bulanlit thoK efot eh-Bochner
(
)
fnuctsipcoaen X E.I s an exptotrio enftme heu nbilatol fX for
If{I gL
t
almaolsiltsn uproptIfo ?
Thaenw setroQ ues3ti idsoe npd eeonnntw hheettreh rieas m eabsluer a
seloenIc.itt ks in otwahntt e hrieasn onseBpnaaarscaphcb ael ea Xna du nit
L(/-tp,
vecIti onr X)s utchthfa oarlt, lA minso te xatnrp eomiotenfh tue n it
ILI(x/-t p,
baolXfl ,b uIti asn e xtrpeotmio enf X).O nt hoeht ehra nZdh,oi ba
Hua nBdo Lru-Lhi snh eodtw hta iifss e rXpba alteeh,ns ucmhe aabs luer a
seleecxstii.Usos tni ntge cthh,wni eisp q ruotevahetf or1 a ny and
anuyn vietc Iti oLnpr( X ),i ffo arl maoltsil tsn u proptIfo l, < (/pl t(l<l) t/00il) sx
(Lp
as troenxgplopysot eio dnXf ,te hnte hrieasl irnf neucatiFo na(lX)* )
Lp(
E
whiscthr oenxgpltoyhsu een sbil atl Xo)fa tI .
ICn hapt5we epr r esseevndeter reaepls ountl hKteots e h-Bofcnuhcnteiro n
spceas:
Buorignaeo'-sar vgaeeT heeomJr.H. o mffann-JoarngSdeK. nw aspeine n
• firpsrto tvhtefa doa rn1 y� thLee sbgeuoec-hBfnnuecrts ipcoaen
Lp(
) p <00 ,
X contaae -ioqsnuese inafcn eod n ilfy c oXn taae -iosnueseq n.c e
LatJe.Br uo,r ggaaivanen otphreoBrou ftt.eh rieas n a dvaagnteto
Buorignap'rsoi o.,if edt,.o neosrt e qIu'nitsrob e es tromnesgaulbryl.a e
Iitks n otwahnit f i asEs erpaaobrldee r-cKootnhftenuic ntsuipcooaeun,s
thdeu oatflh Keot hBeco-hnfneucrt sipcoaeEn( )X ioismsre itlcliays o­
morpthoti hscep c aeo fw eamke*a bsluXer*- av uaelfdnu ctsipcoaens .
ByB uorignap'rsoa onafdc laslst eihocraoenBm na acshpce a,sf oarn y
order-cKootnhftenui cntusiopcouaens a nEad n Byna aschpc aeX ,t he
Kothoec-hBfnnuecrts ipcoaEen( )Xc ontaca oimnpstl eecdmo eopn.efy1
if anedi toEhn elorycr o inXft a ca oimnpstl eecdmo eopn.efy1
•
ThDei az-TKhaelot.rUo elnmt raipasrn oi dmrupatconttt o foolBr ­a
• naschpc aet heUosriytn.hug el trattp ercohd,Cnu i.Scq tuleslegh aeodw
thtta ehrieasB naascphca ewti thh Deu nforpdr-otPpwyee htrotdsiulesa
doneoshtv a te hsema ep rotpyMe.or diSfytilenlgpg'ar soB oW.f.J, ho nson
sheodtw htta ehrieass erpaabBlnaeasc phca eX sutchthfa oarno yht er
serpaabBnlaaesc phca Ye ,y* i iss omotroacp ohmipcl esmuebcnsetp ead
oXf *I.nS c eti52o.w,neu steih isd aegaat iosn h otawht.e oo (X)c on­
tianacs o mptlee.ed-m1e esqnuieanfnc oden i lXfyc ont.earsui' nnoisrf mly
comptleedm.en
PREAFCE ix
TlaangdrL'a1s( tXho)er-eBmM.. au rePyi,s ainJed.Br o uirpngr aoved
G. {i}r'=kl
• thata n1fy::; o pr anadn uyn ifbooruemndlid ysl e-quence
< 00 tn,E (}�tl)
iLnp (,1X p) theirse sutchth{ai kn contaanii 1n-s
< < 00,
susbequ.Le an,tcM ee.Tr alapgrreoadtvna hdtf oarnb yuo nedds equence
{f}=nl'� {g}=nl�
tehriaesn e ssaelnnlotyri mai-lIbilzsoeecdqk u encseu ch
tEn, {g(}n=tl�)
thfaotar l maolslt either iwse aCkualcyoh rty h eirse
{gnt)(=k} �
k( dedpieonnng) s utchha t iasn i- Isenqc.uUe esitnhgi s
w
res,hu epl rtotvhtefa doa rno yr dceorn tiKnotuhfoenuu csts ipcoaeEn ,
tehK oteh-Bofcuhnncestrpci aeEo (nXi w)se askelqytu lielacnyo mplete
ibfto hEa nXd a rwee askelqyu yec nmotplieatlel.
ThBeo uari-gnCembTrhaeno.oTr she aemp ptloiino ctfah Rea msTehye ­
• orieBnmna ascpha icaseni s m rpatontto pBinaca sciaphnct eh eHo.rPy..
Rosepnrtohtvahtelea d ev rbyuo ndseedq uieannB cnaea scaphc eei ther
contaaiw nesaC kalhuyysc uebqsueonrac nt'e lu -bssueenq.Rc eece,n tly
W.T .Go wdeirs caob vlekoRr caemdst eehoyr feoBmrna ascphce asUs. ­
intght iesc h,hn eia qlpsuroeo tvhteea dev rByna ascaphc Xe e itchoenr­
tianass u bcsepw aiYat nuh n condbiatosircio osnn atahla e idirntesa rily
inedcpoombslsaeu bcseZp ;i.a. ee,ev riyn fidniimtenena sslui bcosoepZf a
hanson ontcroimvpitlaeelsdm u ebcnseIp.nSa c eti5o.wn6eu , s Rea m­
se'Tysh eaonrgdyvi ean eceasnssdau crffiiyec nontd siuhttc iahottnh e
Lebee-sBgoucfhunnecsrtpc iaeLo (pn)X 1, p hast hBena ach­
< < 00,
Skasp rrotpye.
Chpat6ei rds e evodtt oC (,XK )a nXd® YS.mo ei nterreesssat urilent g
proivtneh dci ahsp .t er
ThCee mbraens-otEsel-ltDaoiuns-dtKe-nlOle mTI.hn Se eocnr6t .io3,
• wea ppRlayme syT'hse oargeaamin wndep rotvahteaB naascph aXc e
hatsehh ereDduifntoarerdty-pt Priospi efra tfnwyyoe ra nkulslyel q uence
{n}X=1� {x}=n�l {x}=nl�
inX ,e itheri asn uslelq uoern cec ontaa ins
eou-bsseq.Uu seitnnhcrgeee s,wu epl rtotvhteCa (,XK )h as hteh­ree d
irtyDa uonrfd-pPreotptieiafrsn ot dny il bfyo CtK(h) a nXd h atvhee
heirteDaduronyrf d-pPreot.pt eirs ty
ThBeu -DiTehorseet.Ome nelq uesatobiuBotnan a scphca eiss :
•
QuestioLne Xt a nYd b et wBoa nascpchaew si tthhRe a don­
4.
Nikopdryomp. D eoretsyp rtohjete ecntpsirovoredX ®u Ych ta tvhe e
Radon-Npirkoyop?dIe ytfrmh ta en siwnsee rtg eiac,va wne afin nodn ­
triliac vondoinXt ainoYdn s utchhtXa ®Yh ast hRea don-Nikodym
proyp?e rt
JB.o uragnadiP niG s.si heortw hettad h eirasB e na acsahpc Xe w ith
tehR adoond-yNpmir kopseurtcthhtyXa 0Xc ontaca oiopnefys o ( so
X®Xd eonso htva et hRea don-Npirkoop.dIe Ynrm ts tyeh)cpeoa nordft
Sceti6o.wn4ep , r otvhteia E fi asr eflveeKx toihfeu ncstpciaeao nnXd
PREAFCE
x
iasB naascaphc wei tthhe Radopnr-oNpitekhrtoethdnpyey r, mo i jeve ct
tnesporruo cdEt@ Xh sat ehR adon-Npirkotopyde.yr m
Toh etlhpree ae drk eterpa ocwfk ht ai ksn wonw,eh aavodep ttehndeo ­
ttaoin( Pro1bo)lr(e Pmr o3b.l2fe.omsr5 voe )ldp robalne(dumQ ess 1toi)ro n
(uQes3t.i2of.onor 5p )qe une s.t ions
Thaeu tthhnoakrhs ifr si eanndcdso llwehapogio unetoseua dtnc do rrected
sommeia sktaenstd y potshp eri enl riyvmr eisni.aTo hnesisnlecue di,np r a­
tircHu,. Hl uazdiJkJ.,ma is,No .nKJal.t ,Ao .nK ami,nE s.Ok dea,lN l. Ran­
driatnoaannaidnHn uai,yS iunPngra. t irclhuyeol, wa veses rpyel tc hniakatso
JD.i easntDdeav ldiK arm,ew rhroe tarhdo uogumhra nupstac nrodeiff red
suggesctoirorne(scf otoaumirnr iod sn tsaa ntkdpye osas n)Md .T aalgrwahnod
L1
expltahiieedn oaetf dh e etx esnioohfni s( X-h)teosrtoe htma cao rprreocotf
coublepd r eseWnea tlelsdiot.k o tea h ntkot hree fearntedhe Eesxe cvuet i
EditBoirru ksohBefiior s ,Mt aotnhetmiaacnsPdy h sc,is AKnoansn ,tt tfhoeri r
suggeasnctdoi motmn.ses n
Ofc ou,r sateuhteih rsoe orsn psfiobrlm eia asnktyie tnsh bioso k.
PeKie-Le in
Octo2b42e,0r 0 3