Hmiomˆdec telest n sônjeshc kai pðnakec Hausdorff se q rouc analutik n sunart sewn

ARISTOTELEIO PANEPISTHMIO JESSALONIKHS SQOLH JETIKWN EPISTHMWN TMHMA MAJHMATIKWN TOMEAS MAJHMATIKHS ANALUSHS PETROS GALANOPOULOS Hmiomˆdec telest n sônjeshc kai pðnakec Hausdorff se q rouc analutik n sunart sewn Didaktorik Diatrib JessalonÐkh 22

Kefˆlaio Eisagwg. Q roi sunart sewn Parajètoume tic basikìterec ènnoiec pou qrhsimopoioôntai sth megalôterh èktash thc diatrib c aut c. SumbolÐzoume me D = {z C : z < } ton monadiaðo dðsko sto migadikì epðpedo C kai me D to sônoro tou D. To sônolo twn analutik n sunart sewn epð tou D ja to sumbolðzoume me A(D). EpÐshc me dm(z) = π dxdy = π rdrdθ paristˆnoume to kanonikopoihmèno mètro tou Lebesgue epð tou D. Gia α >, o stajmismènoc q roc Dirichlet D α apoteleðtai apì tic sunart seic f(z) A(D) gia tic opoðec isqôei f 2 D a = f() 2 + f (z) 2 ( z 2 ) α dm(z) <. D Oi D α eðnai q roi Hilbert, me eswterikì ginìmeno f, g = f()g() + f (z)g (z)( z 2 ) α dm(z). D 2

Ja doôme sto deôtero kefˆlaio ìti mða isodônamh èkfrash gia thn nìrma, ìtan α, eðnai f 2 D α ( + n) α a n 2, ìpou f(z) = a nz n D α. Eidikìtera gia α = lambˆnoume ton klasikì q ro Dirichlet D = D, pou perièqei tic analutikèc sunart seic f epð tou D twn opoðwn oi eikìnec èqoun peperasmèno embadìn (lambˆnontac up' ìyin thn pollaplìthta). Dhlad : f 2 D = f() 2 + f (z) 2 dm(z). Gia α = lambˆnoume to q ro Hardy H 2, pou perièqei tic sunart seic gia tic opoðec isqôei f 2 H = a 2 n 2 <. EÐnai fanerì ìti, gia < α < β <, D D D α D β H 2. Gia < p, o q roc Hardy H p apoteleðtai apo tic analutikèc sunart seic f A(D) pou ikanopoioôn f p H p = sup r [,) 2π f(re iθ ) p dθ <. O H eðnai o q roc twn fragmènwn analutik n sunart sewn epð tou D, dhlad f = sup z D f(z) <. Gia < p < q < isqôei H H q H p. Gia p oi H p, efodiasmènoi me th parapˆnw nìrma, eðnai q roi Banach. Perissìterec plhroforðec gia autoôc touc q rouc perièqontai sto [DU]. 3

.2 Telestèc sônjeshc JewroÔme mða analutik sunˆrthsh ϕ : D D. O telest c sônjeshc pou orðzetai apì thn ϕ eðnai C ϕ (f) = f ϕ, f A(D). Apo èna je rhma tou Littlewood (Littlewood s subordination principle, [DU, Theorem.7]) o telest c C ϕ, gia kˆje ϕ ìpwc parapˆnw, eðnai fragmènoc stouc q rouc H p kai isqôei C ϕ ( ) + ϕ() p, ϕ() p [, ). Sthn perðptwsh ìmwc twn q rwn D α den eðnai pˆnta alhjèc ìti mia analutik sunˆrthsh ϕ : D D eisˆgei fragmèno telest sônjeshc. An ìmwc h ϕ eðnai amfimonìtimh (univalent), tìte o C ϕ eðnai fragmènoc ston D α ìpwc deðqnei mða allag metablht c. Shmantikì rìlo gia tic idiìthtec twn telest n sônjeshc diadramatðzei h jèsh twn stajer n shmeðwn thc ϕ. 'Ena shmeðo β D D lègetai stajerì shmeðo thc ϕ an lim ϕ(rβ) = β. r EÐnai fanerì ìti an β D tìte ϕ(β) = β. Gia mia ϕ ìpwc parapˆnw jewroôme tic anadromèc ϕ = ϕ, ϕ 2 = ϕ ϕ,, ϕ n = ϕ ϕ n, n = 3, 4, Apì to je rhma Denjoy-Wolff [CMc, Theorem 2.5], prokôptei (ektìc thc perðptwshc ìpou h ϕ eðnai elleiptikìc automorfismìc 4

tou D, dhl. automorfismìc me stajerì shmeðo mèsa sto D), ìti upˆrqei b D D ètsi ste ϕ n b, kaj c n, omoiìmorfa sta sumpag uposônola tou D. To qarakthristikì autì shmeðo lègetai shmeðo Denjoy-Wolff thc ϕ kai eðnai ekeðno apo ta stajerˆ shmeða thc ϕ pou ikanopoieð thn lim r ϕ (rb). Sthn paroôsa melèth oi telestèc sônjeshc qrhsimopoioôntai san ergaleðo. Shmei noume ìmwc ìti h melèth twn telest n aut n, aut kajeaut parousiˆzei idiaðtero endiafèron kai èqei anaptuqjeð proc dôo kateujônseic. H mða kateôjunsh, apotelèsmata thc opoðac qrhsimopoioôme sthn paroôsa ergasða, jewreð telestèc sônjeshc se q rouc analutik n sunart sewn. AntikeÐmeno aut c thc kateôjunshc eðnai h sôndesh idiot twn thc olìmorfhc sunˆrthshc ϕ, oi opoðec idiìthtec proèrqontai apì thn Klassik Anˆlush, me tic idiìthtec tou eisagìmenou telest C ϕ, pou anafèrontai sthn JewrÐa Telest n. Perissìterec plhroforðec gia telestèc sônjeshc mporoôn na brejoôn sta biblða [CMc], [SH], [SM]. H deôterh kateôjunsh, h opoða prohgeðtai qronikˆ thc pr thc, jewreð telestèc V f(ω) = f(s(ω)) ìpou S : Ω Ω metasqhmatismìc se q ro mètrou Ω kai f metr - simh (kai sun jwc L 2 oloklhr simh sunˆrthsh). O telest c V eðnai gnwstìc wc telest c Koopman. Tètoioi telestèc eis qjhsan apì ton B. Koopman [K] se sqèsh me thn melèth thc S- tatistik c Mhqanik c. H melèth telest n Koopman epètreye argìtera thn eureða qr sh thc jewrðac Telest n sthn Statistik Mhqanik, Dunamikˆ Sust mata kai Ergodik JewrÐa. O anagn sthc mporeð na breð perissìterec plhroforðec stic ergasðec [AT], [ASS]. 5

.3 Hmiomˆdec analutik n sunart sewn 'Estw {ϕ t } t mða oikogèneia analutik n sunart sewn apì to dðsko sto dðsko pou ikanopoieð ϕ t ϕ s = ϕ t+s, t, s ϕ (z) = z h ϕ t (z) : [, ) D D eðnai suneq c sunˆrthsh. H {ϕ t } onomˆzetai hmiomˆda analutik n sunart sewn tou dðskou. EÐnai gnwstì [BP], ìti gia kˆje t h ϕ t eðnai amfimonìtimh kai ìti to ìrio ϕ t (z) G(z) = lim t + t upˆrqei omoiìmorfa sta sumpag uposônola tou D. H analutik sunˆrthsh G(z) onomˆzetai apeirostikìc genn torac thc h- miomˆdac {ϕ t }. EÔkola mporoôme na doôme ìti h sunˆrthsh G(z) ikanopoieð G(ϕ t (z)) = ϕ t(z), t, z D. t Upojètoume t ra ìti h {ϕ t } den eðnai h tetrimmènh hmiomˆda ϕ t (z) = z. Tìte, afoô oi sunart seic ϕ t eðnai klasmatikèc anadromèc thc ϕ, èpetai fusiologikˆ ìti [BP]: eðte upˆrqei èna shmeðo b D D tètoio ste ϕ t b, kaj c to t, omoiìmorfa sta sumpag uposônola tou D. eðte h hmiomˆda {ϕ t } apoteleðtai apì elleiptikoôc automorfismoôc tou D, oi opoðoi èqoun èna koinì stajerì shmeðo b D. 6

Kai stic dôo peript seic o apeirostikìc genn torac èqei thn monadik anaparˆstash G(z) = F (z)(bz )(z b), b (.) ìpou h F A(D) èqei mh arnhtikì pragmatikì mèroc kai den eðnai h mhdenik sunˆrthsh. To b eðnai to koinì Denjoy-Wolff shmeðo twn ϕ t, sto opoðo ja anaferìmaste wc to Denjoy-Wolff shmeðo thc {ϕ t } (DW shmeðo). Shmei noume ìti an autì brðsketai mèsa ston dðsko tìte sômfwna me ta parapˆnw eðnai èna koinì stajerì shmeðo twn ϕ t. ParathroÔme ìti an oi ϕ t èqoun èna koinì stajerì shmeðo mèsa ston dðsko tìte autì eðnai aparaðthta to DW shmeðo thc hmiomˆdac. Oi hmiomˆdec {ϕ t } mporoôn na qarakthrisjoôn me bˆsh kˆpoiec amfimonìtimec analutikèc sunart seic [Si]. Sugkekrimmèna, an h : D C eðnai amfimonìtimh kai analutik tètoia ste gia kˆpoio migadikì arijmì c, na isqôei h(z) + ct h(d), (.2) gia kˆje t kai z D, tìte mporoôme na diapist soume ìti h ϕ t (z) = h (h(z) + ct) orðzei mða mh tetrimmènh hmiomˆda, thc opoðac to DW shmeðo an kei sto D. EpÐshc an h : D C eðnai amfimonìtimh kai analutik, diaforetik apì thn tautotik sunˆrthsh, tètoia ste h(d) kai gia kˆpoio migadikì arijmì c na isqôei e ct h(z) h(d) (.3) 7

gia kˆje z D kai t, tìte mporoôme na doôme ìti h sunˆrthsh ϕ t (z) = h (e ct h(z)) orðzei mða mh tetrimmènh hmiomˆda analutik n sunart sewn apì to dðsko sto dðsko me DW shmeðo to h () D. MporoÔme antðstrofa na doôme ìti kˆje mh terimmènh monoparametrik hmiomˆda {ϕ t } mporeð na parastajeð me mða apì tic dôo parapˆnw morfèc gia mða katˆllhlh amfimonìtimh kai analutik sunˆrthsh h kˆje forˆ [Si]. Sugkekrimèna, èstw {ϕ t } mia hmiomˆda me DW shmeðo b kai apeirostikì genn tora G(z). An b D tìte upˆrqei monadik amfimonìtimh h h opoða ikanopoieð ètsi ste h (z) = G(), h() =, (.4) G(z) ϕ t (z) = h (h(z) + G()t). (.5) En an to b D, upˆrqei monadik amfimonìtimh h h opoða ikanopoieð ètsi ste h (z) h(z) = G (b) G(z), h (b) = b 2, (.6) ϕ t (z) = h (e G (b)t h(z)). (.7) H sunˆrthsh h se kˆje perðptwsh ja onomˆzetai h antðstoiqh amfimonìtimh sunˆrthsh thc hmiomˆdac {ϕ t }..4 Hmiomˆdec telest n 'Estw X ènac q roc Banach. MÐa oikogèneia {T t } t fragmènwn grammik n telest n sto X lègetai hmiomˆda telest n an: 8

T = I, o tautotikìc telest c T t T s = T s+t, gia t, s. MÐa hmiomˆda telest n sto X lègetai isqurˆ suneq c (strongly continuous c -semigroup) an gia kˆje s isqôei lim T t (x) T s (x) X =, gia kˆje x X. t s An gia kˆje s isqôei h isqurìterh sunj kh lim T t T s = t s tìte lème ìti h {T t } eðnai omoiìmorfa suneq c (uniformly continuous). O apeirostikìc genn torac mðac isqurˆ suneqoôc hmiomˆdac telest n {T t } eðnai o (en gènei mh fragmènoc) telest c T t (x) x A(x) = lim, t t me pedðo orismoô { T t (x) x D(A) = x X: to ìrio lim t t } upˆrqei sto X. To sônolo D(A) eðnai èna grammikìc upìqwroc tou X, pˆntote puknìc sto X, kai o A eðnai kleistìc grammikìc telest c sto D(A). O apeirostikìc genn torac A eðnai fragmènoc an kai mìnon an h {T t } eðnai omoiìmorfa suneq c kai sthn perðptwsh aut èqome thn parˆstash T t = e ta, t. (.8) Sthn perðptwsh mðac isqurˆ suneqoôc hmiomˆdac h (.8), me katˆllhlh ermhneða, suneqðzei na isqôei. 9

'Opwc kai sthn perðptwsh twn fragmènwn telest n h sullog twn migadik n arijm n λ, gia touc opoðouc o (λi A) èqei fragmèno antðstrofo sto X, lègetai epilôon sônolo tou A kai sumbolðzetai me ρ(a). Gia λ ρ(a) o epilôwn telest c eðnai To fˆsma tou A eðnai to sônolo R(λ, A) = (λi A). σ(a) = C ρ(a) kai to shmeiakì tou fˆsma σ π (A) eðnai to sônolo twn idiotim n tou. To fˆsma eðnai pˆntote kleistì sônolo sto epðpedo. Se antðjesh ìmwc me touc fragmènouc telestèc, to σ(a) poikðllei se mègejoc kai mporeð na eðnai fragmèno mh fragmèno sônolo, to kenì akìma kai ìlo to epðpedo. To frˆgma auxhtikìthtac mðac isqurˆ suneqoôc hmiomˆdac {T t } eðnai to ìrio log T t ω = lim, t t kai isqôei ω <. An λ C me Re(λ) > ω, tìte λ ρ(a) kai o epilôwn telest c sto shmeðo autì mporeð na grafeð R(λ, A)(x) = e λt T t (x) dt, x X. (.9) Ola ta parapˆnw, kai akìma perisìterec plhroforðec gia hmiomˆdec telest n se q rouc Banach perièqontai sta [DS], [HP], [P]..5 ParousÐash twn apotelesmˆtwn 'Eqontac anafèrei ta parapˆnw mporoôme na parousiˆsoume sunoptikˆ to perieqìmeno twn epìmenwn kefalaðwn.

Sto deôtero kefˆlaio meletˆme hmiomˆdec telest n sônjeshc stouc q rouc D α, < α <. ApodeiknÔoume thn isqur sunèqeia touc kai prosdiorðzoume thn morf tou apeirostikoô genn tora. Parousiˆzoume epðshc paradeðgmata tètoiwn hmiomˆdwn. Sto trðto kefˆlaio meletˆme ton telest Cesàro C(f)(z) = ( n + ) n a k z n, (.) ìpou f(z) = a nz n. Me thn bo jeia twn hmiomˆdwn telest n sônjeshc deðqnoume ìti o C eðnai fragmènoc stouc q rouc D α, gia < α <, sumplhr nontac ètsi to kenì sth melèth autoô tou telest metaxô twn q rwn D kai H 2. Sumplhrwmatikˆ, meletˆme stouc Ðdiouc q rouc ton telest A(f)(z) = ( k=n k= a k k + o opoðoc eisˆgetai apì ton anˆstrofo pðnaka tou C. Sto tètarto kefˆlaio jewroôme mia kathgorða pinˆkwn Hausdorff twn opoðwn h genn tria akoloujða {µ n } eðnai akoloujða rop n, dhl. µ n = ) z n, t k dµ(t), n =,,... ìpou µ eðnai peperasmèno mètro Borel sto diˆsthma (, ]. Touc pðnakec autoôc touc jewroôme wc metasqhmatismoôc epð analutik n sunart sewn tou dðskou me pollaplasiasmì epð thn akoloujða twn suntelest n Taylor. Sugkekrimèna, an f(z) = a nz n A(D), jewroôme ton metasqhmatismì pou dðnetai apì th dunamoseirˆ n H µ (f)(z) = ( h n,k a k )z n, k=

ìpou h n,k ta stoiqeða tou antðstoiqou pðnaka Hausdorff. DeÐqnoume ìti kˆje tètoioc metasqhmatismìc mporeð na ekfrasjeð wc mèsoc ìroc kˆpoiwn stajmismènwn telest n sônjeshc. Qrhsimopoi ntac aut n thn morf, brðskome sunj kec epð tou mètrou ste oi pðnakec na eisˆgoun fragmènouc telestèc stouc q rouc Hardy H p, p <. 2

Kefˆlaio 2 Hmiomˆdec telest n sônjeshc stouc q rouc D α. 2. Eisagwg JewroÔme mða hmiomˆda {ϕ t } t analutik n sunart sewn tou dðskou kai orðzoume thn oikogèneia metasqhmatism n T t (f)(z) = f(ϕ t (z)), f D α. EÔkola diapist noume ìti h oikogèneia aut ikanopoieð tic sqèseic: T = I, o tautotikìc telest c T t T s = T t+s. Ja doôme ìti gia kˆje t, o T t eðnai fragmènoc telest c stouc D α, dhlad h oikogèneia {T t } t eðnai hmiomˆda fragmènwn telest n stouc q rouc autoôc. Gia thn apìdeixh, qreiazìmaste thn akìloujh ektðmhsh thc auxhtikìthtac miac sunˆrthshc pou an kei stouc D α. 3

L mma 2... 'Estw < α <. An f D α tìte gia kˆje z D isqôei ( ) 2 f(z) K f α ( z ) α Dα, (2.) 2 ìpou K jetik stajerˆ anexˆrthth tou α. Apìdeixh. Pr ta ja doôme ìti gia f(z) = a nz n D α h nìrma èqei thn akìloujh isodônamh èkfrash f 2 D α Γ(α + ) ( + n) α a n 2, ìpou Γ h gnwst sunˆrthsh Gˆmma. Prˆgmati, f 2 D α = f() 2 + f (z) 2 ( z 2 ) α dm(z) D ( 2π ) = f() 2 + f (re iθ ) 2 dθ ( r 2 ) α rdr π = a 2 + (n + ) 2 a n+ 2 r n ( r) α dr. Epeid [WW, sel.254] kai r n ( r) α dr = Γ(n + ) = n! Γ(n + )Γ(α + ) Γ(n + α + 2) katal goume f 2 D α = a 2 + (n + ) 2 a n+ 2Γ(n + )Γ(α + ) Γ(n + α + 2) = a 2 + Γ(α + ) (n + ) a n+ 2 (n + )! Γ(n + α + 2). (2.2) 4

'Omwc eðnai gnwstì [Z, sel. 77] ìti 'Ara lim n Γ(α + n + 2) =. (2.3) (n + )!(n + ) α (n + ) a n+ 2 (n + )! Γ(n + α + 2) (n + ) α a n 2 kai ètsi eôkola katal goume sto zhtoômeno. Gia thn auxhtikìthta mðac f D α ja qrhsimopoi soume thn anisìthta tou Schwarz kai thn parapˆnw isodônamh èkfrashc thc nìrmac. 'Etsi ( f(z) 2 a n z n ) 2 ( ) 2 z n = ( + n) α 2 an ( + n) α 2 ( ) ( ) ( + n) α a n 2 ( + n) α z 2n ( ) K Γ(α + ) f 2 D α ( + n) α z 2n ( = KΓ(α) ) ( + n) α z 2n f 2 D Γ(α + ) Γ(α) α ìpou K jetik stajerˆ anexˆrthth tou α. T ra, lìgw tou anaptôgmatoc ( z ) α Γ(n + α) = z n, z < Γ(α)n! 5

kai thc (2.3), katal goume ( ) 2 f(z) K α ( z 2 ) α 2 f Dα, T ra eðmaste se jèsh na broôme mða ektðmhsh thc nìrmac twn telest n sônjeshc C ϕ (f) = f ϕ, gia amfimonìtimec ϕ, stouc q rouc D α. Prìtash 2... 'Estw < α <. An ϕ : D D amfimonìtimh kai analutik sunˆrthsh, tìte ( ) ( ) α 2 2 + ϕ() f ϕ Dα K f Dα (2.4) α ϕ() ìpou K > stajerˆ anexˆrthth tou α. Apìdeixh. Apì ton orismì èqoume ìti f ϕ 2 D α = (f ϕ)() 2 + (f ϕ) (z) 2 ( z 2 ) α dm(z). Apì to l mma (2..) paðrnoume f(ϕ()) 2 K α D ( ϕ() ) α f 2 D α. (2.5) EpÐshc (f ϕ) (z) 2 ( z 2 ) α dm(z) D ( ) z = (f ϕ) (z) 2 2 α ( ϕ(z) 2 ) α dm(z) D ϕ(z) ( ) 2 α + ϕ() 2 α (f ϕ) (z) 2 ( ϕ(z) 2 ) α dm(z), ϕ() D 6

giatð apì to l mma tou Schwarz èqoume ìti gia kˆje z D z ϕ(z) + ϕ() ϕ() An t ra efarmìsoume mia allag metablht c sto teleutaðo olokl rwma (f ϕ) (z) 2 ( ϕ(z) 2 ) α dm(z) = f (z) 2 ( z 2 ) α dm(z) D ϕ(d) f (z) 2 ( z 2 ) α dm(z), dhlad (f ϕ) (z) 2 ( z 2 ) α dm(z) 2 D Sundiˆzontac tic (2.5) kai (2.6) sumperaðnoume D. ( ) α + ϕ() f 2 D ϕ() α. (2.6) f ϕ 2 D α K ( + ϕ() α ( ϕ() ) α f 2 D α + 2 ϕ() K ( ) α ( + ϕ() + ϕ() f 2 D α ϕ() α + 2 ϕ() K α ( + ϕ() ϕ() kai èqoume to zhtoômeno. ) α f 2 D α, ) α f 2 D α ) α f 2 D α Epiplèon, an ϕ() =, tìte apì thn parapˆnw diadikasða katal goume sthn f ϕ Dα f Dα. 'Amesa prokôptei apì ta parapˆnw ìti gia kˆje hmiomˆda {ϕ t }, h hmiomˆda T t (f)(z) = f ϕ t (z) apoteleðtai apì fragmènouc telestèc stouc q rouc D α, gia < α <. 7

2.2 Isqur sunèqeia Sthn parˆgrafo aut ja asqolhjoôme me thn isqur sunèqeia kai ton apeirostikì genn tora twn hmiomˆdwn telest n sônjeshc stouc q rouc D α. Pr ta ja deðxoume ìti Prìtash 2.2.. Gia kˆje < α <, ta polu numa eðnai puknˆ ston D α. Apìdeixh. Jèloume na deðxoume ìti gia kˆje f D α, upˆrqei akoloujða poluwnômwn {P n }, tètoia ste P n f Dα kaj c to n. Gia < ρ <, jewroôme thn f ρ (z) = f(ρz). 'Estw {P n } h akoloujða twn merik n ajroismˆtwn thc seirˆc Taylor thc f ρ. Epeid h f ρ eðnai analutik sto dðsko aktðnac ρ, èpetai ìti lim P n(z) = f ρ (z) n omoiìmorfa ston kleistì monadiaðo dðsko. To Ðdio isqôei kai gia tic parag gouc, dhlad lim P n(z) = f ρ(z) n omoiìmorfa ston kleistì monadiaðo dðsko. Apì aut thn omoiìmorfh sôgklish èpetai ìti P n(z) f ρ(z) 2 ( z 2 ) α dm(z) kaj c to n, dhlad D P n f ρ Dα (2.7) 8

kaj c to n. 'Estw ɛ > dojèn. Dialègoume δ (, ) ste f (z) 2 ( z 2 ) α dm(z) < ɛ. δ< z < Tìte gia kˆje < ρ < f ρ(z) 2 ( z 2 ) α dm(z) δ< z < = = 2π δ 2π δ δ< z < f (ρse iθ ) 2dθ π ( s2 ) α ρ 2 sds f (se iθ ) 2dθ π ( s2 ) α sds f (z) 2 ( z 2 ) α dm(z) < ɛ, (2.8) epeid h posìthta 2π f (re iθ ) 2 dθ eðnai aôxousa sunˆrthsh tou r. Gia ρ katˆllhla kontˆ sto f ρ(z) f (z) < ɛ gia z < δ. Tìte f ρ(z) f (z) 2 ( z 2 ) α dm(z) < Cɛ 2 z δ 9

'Ara f ρ f 2 D α = f ρ(z) f (z) 2 ( z 2 ) α dm(z) D f ρ(z) f (z) 2 ( z 2 ) α dm(z) z δ + 2 ( f ρ(z) 2 + f (z) 2 )( z 2 ) α dm(z) δ< z < < Cɛ 2 + 4ɛ. (2.9) Tìte apì tic (2.7), (2.9) kai thn P n f Dα P n f ρ Dα + f ρ f Dα, prokôptei to zhtoômeno. Je rhma 2.2.. 'An < α < kai {ϕ t } t hmiomˆda analutik n sunart sewn tou dðskou, tìte h hmiomˆda telest n eðnai isqurˆ suneq c ston D α. T t (f) = f ϕ t Apìdeixh. Jèloume na deðxoume ìti gia kˆje f D α kai gia kˆje s isqôei: lim t s T t (f) T s (f) Dα =. Gia P polu numo èqoume T t (f) T s (f) Dα T t (f) T t (P ) Dα + T t (P ) T s (P ) Dα + T s (P ) T s (f) Dα ( T t + T s ) f P Dα + T t (P ) T s (P ) Dα. 2

Ta polu numa eðnai puknˆ stouc q rouc D α kai ìpwc prokôptei apì thn prìtash (2..) h T t eðnai omoiìmorfa fragmènh, san sunˆrthsh tou t, sta sumpag uposônola tou [, ). 'Ara arkeð na deðxoume ìti: lim t s T t (P ) T s (P ) Dα =. An P (z) = m a nz n parathroôme ìti T t (P ) T s (P ) Dα a m (ϕ t ) m (ϕ s ) m Dα + +... + a ϕ t ϕ s Dα. ArkeÐ loipìn na deðxoume ìti gia kˆje n isqôei 'Omwc lim ϕ n t ϕ n s Dα =. t s ϕ n t ϕ n s 2 D α = = ϕ n t () ϕ n s () 2 + D (ϕ n t (z) ϕ n s (z)) 2 ( z 2 ) α dm(z) ϕ n t () ϕ n s () 2 + + 2n 2 ϕ t(z) ϕ s(z) 2 ϕ n t (z) 2 ( z 2 ) α dm(z) D + 2n 2 (z) ϕ n (z) 2 ϕ s(z) 2 ( z 2 ) α dm(z) D ϕ n t s ϕ n t () ϕ n s () 2 + + 2n 2 ϕ t(z) ϕ s(z) 2 ( z 2 ) α dm(z) D + 2n 2 (z) ϕ n (z) 2 ϕ s(z) 2 ( z 2 ) α dm(z). D ϕ n t s Dhlad arkeð na deiqjeð ìti kˆje ènac apì touc parapˆnw prosjetaðouc sugklðnei sto kaj c to t s. 2

(i) 'Opwc eðdame sthn eisagwg, kˆje hmiomˆda {ϕ t } èqei sugkekrimènh anaparˆstash anˆloga me to pou brðsketai todw shmeðo thc. Lìgw aut c thc anaparˆstashc, gia kˆje perðptwsh, isqôei lim ϕ t (z) = ϕ s (z), t s omoiìmorfa sta sumpag uposônola tou D. 'Ara ìti D lim t s ϕ n t () ϕ n s () =. (2.) (ii) Gia ton deôtero prosjetaðo ergazìmaste wc ex c. ParathroÔme ϕ t(z) ϕ s(z) 2 ( z 2 ) α dm(z) ArkeÐ na deðxoume lim ϕ t s D t(z) ϕ s(z) 2 dm(z) =. D ϕ t(z) ϕ s(z) 2 dm(z). Proc toôto parathroôme ìti oi eikìnec ϕ t (D) apoteloôn mða fjðnousa oikogèneia sunìlwn. Lìgw thc sunèqeiac thc isqôei (t, z) [, ) D ϕ t (z) D, ϕ s (D) = t<s ϕ t (D) = t>s ϕ t (D). 'Etsi apì klassikì je rhma thc JewrÐac Mètrou dhlad isodônama lim m(ϕ t (D)) = m(ϕ s (D)), t s lim ϕ t s D t(z) 2 dm(z) = 22 D ϕ s(z) 2 dm(z). (2.)

An jewr soume th sunˆrthsh f t (z) = 2( ϕ t(z) 2 + ϕ s(z) 2 ) ϕ t(z) ϕ s(z) 2 kai efarmìsoume to l mma tou Fatou katal goume ìti lim inf f t(z)dm(z) lim inf f t (z)dm(z). t s t s D IsodÔnama lim inf D t s [2( ϕ t(z) 2 + ϕ s(z) 2 )]dm(z) lim sup ϕ t(z) ϕ s(z) 2 dm(z) D t s lim inf 2( ϕ t(z) 2 + ϕ s(z) 2 )dm(z) t s D lim sup ϕ t(z) ϕ s(z) 2 dm(z). (2.2) t s Epeid ìmwc lim ϕ t s t(z) = ϕ s(z) omoiìmorfa sta sumpag uposônola tou dðskou, prokôptei ìti kai D lim inf t s ( ϕ t(z) 2 + ϕ s(z) 2 ) = 2 ϕ s(z) 2 (2.3) lim sup ϕ t(z) ϕ s(z) 2 =. (2.4) t s 'Etsi me thn bo jeia twn (2.), (2.3) kai (2.4), h (2.2) grˆfetai isodônama 4 ϕ s(z) 2 dm(z) 4 ϕ s(z) 2 dm(z) D D lim sup ϕ t(z) ϕ s(z) 2 dm(z) t s D D 23

kai ˆra lim sup ϕ t(z) ϕ s(z) 2 dm(z). t s D 'Omwc h upì olokl rwsh sunˆrthsh eðnai jetik. 'Ara lim ϕ t s D t(z) ϕ s(z) 2 dm(z) =. (2.5) (iii) Tèloc, lìgw tou ìti gia kˆje t oi sunart seic ϕ t (z) eðnai amfimonìtimec kai lim ϕ t (z) = ϕ s (z), t s omoiìmorfa sta sumpag uposônola tou D, apì to je rhma kuriarqoômenhc sôgklishc prokôptei ìti lim t s D ϕt n (z) ϕ n s (z) 2 ϕ s(z) 2 ( z 2 ) α dm(z) =. (2.6) Sundiˆzontac tic (2.), (2.5) kai (2.6) apodeiknôetai to zhtoômeno. To epìmeno je rhma prosdiorðzei ton apeirostikì genn tora twn hmiomˆdwn {T t }. Je rhma 2.2.2. 'Estw < α < kai {ϕ t } t hmiomˆda analutik n sunart sewn tou dðskou me apeirostikì genn tora G(z). An Γ α o apeirostikìc genn torac thc antðstoiqhc hmiomˆdac telest n sônjeshc {T t } t tìte me pedðo orismoô Γ α (f)(z) = G(z)f (z), (2.7) D(Γ α ) = {f D α : G(z)f (z) D α }. 24

Apìdeixh. Ex orismoô me pedðo orismoô to sônolo Γ α (f) = lim t T t (f) f t, D(Γ α ) = {f D α : to lim t T t (f) f t JewroÔme to sônolo upˆrqei ston D α }. D = {f D α : Gf D α }. Gia stajerì shmeðo z D kai gia mða sunˆrthsh f D α isqôei ìti T t (f)(z) f(z) lim = f(ϕ t(z)) t t t = G(z)f (z). t= Epeid h sôgklish stouc D α, lìgw thc prìtashc (2..), sunepˆgetai omoiìmorfh sôgklish sta sumpag uposônola tou dðskou, prokôptei ìti D(Γ α ) D. 'Ara, an jewr soume ton Γ (f)(z) = G(z)f (z), f D parathroôme ìti autìc epekteðnei ton Γ α sto sônolo D. Ja deðxoume ìti o Γ α tautðzetai me ton Γ. H perðptwsh pou h {ϕ t } eðnai tetrimmènh hmiomˆda eðnai qwrðc duskolða. 'Etsi upojètoume ìti o apeirostikìc genn torac G(z) den eðnai h mhdenik sunˆrthsh. Epeid oi stajerèc sunart seic eðnai stajerˆ shmeða gia kˆje telest T t, prokôptei ìti T t gia kˆje t. 'Epetai ìti to frˆgma auxhtikìthtac ω ikanopoieð ω <. 25

GnwrÐzoume [DS, Theorem VIII..], ìti an r > ω tìte o Γ α r èqei fragmèno antðstrofo ston D α. Eidikìtera o Γ α r eðnai epð. Upojètoume pr ta ìti h {ϕ t } èqei DW shmeðo sto D. 'Estw èna r > ω. Ja deðxoume ìti o Γ r eðnai èna proc èna. IsqÔei ω +r 2 > ω. Ara, [DS, Theorem VIII..5], upˆrqei M > ètsi ste T t Me ω +r 2 t, t. (2.8) Upojètoume ìti upˆrqei mh mhdenik f D tètoia ste Γ (f) = rf. Apì thn (.4) G(z) = G() h (z), ìpou h h antðstoiqh amfimonìtimh sunˆrthsh thc {ϕ t }. 'Ara G() h (z) f (z) = rf(z). Apì thn teleutaða sqèsh prokôptei r f(z) = k exp( G() h(z)) ìpou k C. 'Ara, lìgw thc (.5), gia kˆje t Dhlad r f(ϕ t (z)) = k exp( G() h(ϕ t(z))) = e rt f(z). e rt f Dα T t f Dα. Katal goume ètsi sto sumpèrasma e rt T t, t, to opoðo èrqetai se antðjesh me thn (2.8). 'Ara den upˆrqei mh mhdenik f D ste na isqôei Γ (f) = rf. Dhlad o (Γ r) 26

eðnai èna proc èna kai epekteðnei ton (Γ α r), o opoðoc eðnai èna proc èna kai epð stouc D α. 'Ara Γ a = Γ kai D(Γ a ) = D. Sth sunèqeia upojètoume ìti h {ϕ t } èqei DW shmeðo b D. 'Estw ènac migadikìc λ kai mða f D, ìqi h mhdenik sunˆrthsh, ètsi ste Γ (f) = λf. Apì thn (.6), isqôei G(z) = G (b) h(z) h (z), ìpou h h antðstoiqh amfimonìtimh sunˆrthsh thc {ϕ t }. 'Ara Dialègoume r tètoio ste G (b) h(z) h (z) f (z) = λf(z). (2.9) b < r < kai h f na mhn èqei rðzec ston z = r. Oloklhr nontac thn (2.9) prokôptei 2πi z =r f (z) f(z) dz = λ G (b) 2πi z =r h (z) h(z) dz = λ G (b). Apì thn arq tou orðsmatoc katal goume ìti to sônolo twn idiotim n tou Γ eðnai arijm simo. Eidikìtera upˆrqei pragmatikìc r > ω tètoioc ste o (Γ r) na eðnai èna proc èna. 'Ara kai sthn perðptwsh aut Γ α = Γ kai D(Γ α ) = D. Prìtash 2.2.2. An gia kˆpoio < α < h hmiomˆda telest n sônjeshc {T t } t eðnai omoiìmorfa suneq c ston D α tìte h {ϕ t } t eðnai h tetrimmènh hmiomˆda. 27

Apìdeixh. An h {T t } eðnai omoiìmorfa suneq c tìte o Γ α eðnai fragmènoc telest c kai gia kˆje f D α. JewroÔme tic sunart seic ìpou Γ α (f) Dα Γ α f Dα, f k (z) = zk (β α,k ) 2 IsqÔei f k Dα =, epomènwc, k =, 2,... β α,k = k2 Γ(k)Γ(α + ) Γ(k + α + ) Γ α (f k ) Dα Γ α, k =, 2,... 'Estw G(z) = a nz n, to anˆptugma Taylor tou apeirostikoô genn tora thc {ϕ t }. Tìte kai ja èqoume Γ α (f k )(z) = G(z)f k(z) Γ α (f k ) 2 D α k2 β α,k = k z k a (β α,k ) n z n 2 = k a (β α,k ) n z n+k 2 = k (β α,k ) 2 n=k 28 n=k. a n k+ z n a n k+ 2 (n + ) α. (2.2)

Epiplèon apì thn (2.3) prokôptei β α,k k α. Epomènwc Γ α (f k ) 2 D α k2 k α n=k a n k+ 2 (n + ) α. Dhlad k +α ( a 2 k α + a 2 (k + ) α + a 2 2 (k + 2) α +...) C Γ α 2 kai autì isqôei gia k =, 2,... Profan c tìte a n = gia kˆje n =,, 2,.... Dhlad G kai h {ϕ t } eðnai h tetrimmènh hmiomˆda. Sth sunèqeia perigrˆfoume to shmeiakì fˆsma tou Γ α. Prìtash 2.2.3. 'Estw {ϕ t } t hmiomˆda analutik n sunart - sewn, G(z) o apeirostikìc thc genn torac, h(z) h antðstoiqh amfimonìtimh analutik sunˆrthsh kai {T t } t h antðstoiqh hmiomˆda telest n sônjeshc. (i) An h {ϕ t } t èqei DW shmeðo b D tìte oi idiotimèc tou Γ α perièqontai sto sônolo Sugkekrimèna {kg (b) : k =,,...}. kg (b) σ π (Γ α ) h k D α. (2.2) (ii) An h {ϕ t } t èqei DW shmeðo b D tìte λg() σ π (Γ α ) e λh D α. (2.22) 29

Apìdeixh. (i) 'Estw ìti to DW shmeðo thc {ϕ t } brðsketai mèsa sto dðsko. Tìte, lìgw thc (2.7) kai thc (.6), Γ α (f)(z) = G(z)f (z) = G(b) h(z) h (z) f (z). 'Estw µ C idiotim tou Γ a. Efarmìzontac thn arq tou orðsmatoc, ìpwc sthn apìdeixh tou jewr matoc (2.2.2), prokôptei ìti µ = kg (b), ìpou k =,, 2,... Dhlad σ π (Γ α ) {kg (b) : k =,,...}. 'Estw t ra f D α, mh mhdenik sunˆrthsh, tètoia ste Γ α (f)(z) = kg (b)f(z). EÔkola mporoôme na diapist soume ìti f(z) = ch k (z), me c. Apì ta parapˆnw prokôptei h zhtoômenh isodunamða. (ii) Sthn perðptwsh pou to DW shmeðo thc hmiomˆdac eðnai sto sônoro, o apeirostikìc genn torac thc {ϕ t }, lìgw thc (.4), èqei th morf G(z) = G() h (z). Upojètoume ìti, gia kˆpoio λ C, h sunˆrthsh f(z) = e λh(z) D α. Tìte 'Ara Γ α (f)(z) = G(z)f (z) = λg()f. λg() σ π (Γ a ). 3

AntÐstrofa upojètoume ìti to λg() σ π (Γ α ). 'Estw f D α, mh mhdenik sunˆrthsh, h opoða ikanopoieð thn Γ α (f)(z) = λg()f(z). Apì thn teleutaða sqèsh prokôptei 'Ara f (z) = λh (z)f(z). f(z) = ce λh(z), me c. 'Etsi apodeðqjhke to zhtoômeno. 2.3 Merikˆ paradeðgmata Sthn parˆgrafo aut ja anaferjoôme se sugkekrimèna paradeðgmata hmiomˆdwn {ϕ t } kai ja melet soume ta basikìtera qarakthristikˆ twn hmiomˆdwn telest n sônjeshc pou eisˆgontai apì autèc. Parˆdeigma. 'Estw h hmiomˆda ϕ t (z) = ( z) e t. Aut èqei antðstoiqh amfimonìtimh analutik sunˆrhsh apeirostikì genn tora h(z) = log z, G(z) = ( z) log 3 z

kai DW shmeðo b =. To shmeiakì fˆsma tou apeirostikoô genn tora Γ α thc antðstoiqhc hmiomˆdac telest n sônjeshc eðnai σ π (Γ α ) = {,, 2,...} gia kˆje < α <. Proc toôto arkeð na deðxoume ìti gia thn h(z) = log z isqôei h k (z) D α, k =,, 2,... gia kˆje < α <. Prˆgmati, èstw < α <. Apì to [Z, Theorem 2.3] èqoume ìti to anˆptugma Taylor èqei suntelestèc (log 2 z )k = A κ nz n A κ n k logk (n + ) n + kaj c to n. 'Epetai ìti (log 2 z )k 2 D α k 2 = k 2 log 2(k ) (n + ) (n + ) 2 ( + n) α log 2(k ) (n + ) (n + ) +α <, dhlad T ra epeid (log 2 z )k D α, k =,, 2,... log z = log 2 z + log 2 32

ja èqoume (log z )k = k j= ( ) k (log 2 j 2 )k j (log z )j, kai h grammikìthta tou q rou D α ja exasfalðzei ìti (log z )k D α gia kˆje k =,, 2,... MporoÔme epð plèon na diapist soume ìti to fˆsma tou Γ α tautðzetai me to shmeiakì fˆsma. Proc toôto ja qrhsimopoi soume èna apotèlesma twn MacCluer kai Shapiro, to opoðo diatup noume prosarmosmèno stic dikèc mac anˆgkec: Je rhma ([MS]). Upojètoume ìti ϕ : D D eðnai analutik kai den èqei peperasmènh gwniak parˆgwgo se kanèna shmeðo tou D. Upojètoume epiplèon ìti o telest c sônjeshc C ϕ eðnai fragmènoc sto D γ, gia kˆpoio γ >. Tìte o C ϕ eðnai sumpag c telest c ston D α gia kˆje α > γ. H gwniak parˆgwgoc mðac ϕ : D D sto shmeðo ζ D lème ìti upˆrqei, an upˆrqei η D, ètsi ste to ìrio ϕ(z) η lim z ζ z ζ na upˆrqei san peperasmènoc migadikìc arijmìc, kaj c to z ζ mèsa se kˆje gwniakì tomèa me koruf to ζ kai ˆnoigma mikrìtero tou π, [SH, sel. 56]. Sthn perðptwsh mac, gia kˆje < β <, h sunˆrthsh ϕ(z) = ( z) β, 33

eðnai eôkolo na elegqjeð ìti den èqei gwniak parˆgwgo se kanèna shmeðo tou D kai epiplèon eisˆgei fragmèno telest sônjeshc C ϕ sto klasikì q ro Dirichlet (γ = ), kaj c eðnai amfimonìtimh. 'Ara o C ϕ eðnai sumpag c stouc D α, < α <. Epomènwc gia kˆje t o telest c sônjeshc T t, pou eisˆgetai apì th ϕ t (z) = ( z) β, eðnai sumpag c stouc D α. Apì th genik jewrða twn hmiomˆdwn èpetai ìti to fˆsma σ(γ α ) apoteleðtai apokleistikˆ apì idiotimèc [P, Corollary 3.7]. Parˆdeigma 2. JewroÔme thn hmiomˆda ϕ t (z) = e t z ( e t )z, h opoða èqei antðstoiqh amfimonìtimh sunˆrthsh thn z h(z) = ( z), apeirostikì genn tora thn sunˆrthsh G(z) = z( z) kai DW shmeðo b =. 'Estw < α <. An Γ α o apeirostikìc genn torac thc antðstoiqhc hmiomˆdac telest n sônjeshc ston D α, tìte Γ α (f)(z) = z( z)f (z). EÐnai fanerì apì to anˆptugma Taylor thc h(z) k = h(z) k D α k =. Epomènwc to shmeiakì fˆsma tou Γ α eðnai σ π (Γ α ) = {}. zk ( z) k Gia ton prosdiorismì tou fˆsmatoc tou Γ α qreiazìmaste to parakˆtw ìti 34

L mma 2.3.. 'Estw < α < kai λ C. Tìte ( z) λ D α an kai mìno an Re(λ) > α 2. Apìdeixh. Upojètoume pr ta ìti o λ eðnai paragmatikìc arijmìc. An λ, èpetai ˆmesa ìti h ( z) λ D α. An λ <, lìgw tou anaptôgmatoc ( z) = c λ n z n ìpou c n n λ, [?, sel. 53], prokôptei ìti mìno an c n 2 ( + n) α ( z) λ D α an kai n (2λ++α) <, (2.23) dhlad an kai mìno an λ > α 2. Sthn perðptwsh pou o λ = λ + iλ 2 C, parathroôme ìti (( z) λ ) 2 ( z 2 ) α dm(z) = D = λ 2 ( z) λ 2 ( z 2 ) α dm(z) D = λ 2 e 2Re{(λ ) log( z)} ( z 2 ) α dm(z) D = λ 2 e 2(λ ) log z e 2λ2Arg( z) ( z 2 ) α dm(z) D ( z) λ 2 ( z 2 ) α dm(z) D giatð gia kˆje pragmatikì arijmì λ 2 upˆrqoun jetikèc stajerèc C, C 2, anexˆrthtec tou z, ètsi ste C < e 2λ 2Arg( z) < C 2 35

gia kˆje z D. 'Ara h ( z) λ D α an kai mìno an h ( z) λ D α, dhlad an kai mìno an λ > α 2. Qrhsimopoi ntac mèjodo ìmoia me aut tou [Si2], eðmaste se jèsh na broôme perissìterec plhroforðec gia to σ(γ α ). 'Estw λ C me Re(λ) α 2. JewroÔme ta polu numa n ( ) n λ( ) m P n (z) = + m m zm kai tic sunart seic ParagwgÐzontac prokôptei ParathroÔme ìti m= f n,λ (z) = ( z) λ e P n(z). (2.24) f n,λ(z) = ( z) λ e P n(z) (P n(z)( z) λ). (λ Γ α )(f n,λ )(z) = λf n,λ (z) + z( z)f n,λ(z) [ ] = ( z) λ λe Pn(z) + ze Pn(z) P n(z)( z) λze P n(z) = ( z) λ+ e P n(z) (λ + zp n(z)). EpÐshc elègqoume ìti (λ + zp n(z)) = λ + z n m= ( n )λ ( )m m m mzm = λ( z) n. 'Ara (λ Γ α )(f n,λ ) = λf n,n+λ+, 36

dhlad h diaforik exðswsh (λ Γ α )(y) = λf n,n+λ+ èqei lôsh sto dðsko thn f n,λ, h opoða eðnai monadik. An dialèxoume fusikì n tètoio ste tìte Re(n + λ + ) > α 2 f n,n+λ+ D α. An upojèsoume ìti to λ ρ(γ α ), tìte o antðstrofoc (λ Γ α ) upˆrqei. 'Ara f n,λ = (λ Γ α ) (f n,n+λ+ ) D α giatð èqoume upojèsei ìti Re(λ) α 2. Apì ta parapˆnw èpetai ìti, ìtan < α <, èqoume {λ C : Re(λ) α 2 } σ(γ α). (2.25) Sto epìmeno kefˆlaio ja doôme poiì akrib c eðnai to fˆsma tou Γ α. Parˆdeigma 3. JewroÔme thn hmiomˆda automorfism n ϕ t (z) = (et + )z + e t (e t )z + e t +, t, me antðstoiqh amfimonìtimh sunˆrthsh apeirostikì genn tora h(z) = 2 log + z z, G(z) = 2 ( z2 ), 37

kai DW shmeðo b =. H antðstoiqh hmiomˆda telest n sônjeshc lìgw thc (2.4) ikanopoieð T t (f) = f ϕ t, T t Dα D α Kα /2 e α 2 t, epomènwc èqei fragma auxhtikìthtac ω α 2. Ex ˆllou, eðnai eôkolo na doôme, ìpwc sto Parˆdeigma 2, ìti ( ) λ + z D α α z 2 < Re(λ) < α 2 sunep c to shmeiakì fˆsma tou apeirostikoô genn tora Γ α eurðsketai 'Epetai ìti ω = α 2 kai σ π (Γ α ) = {λg() : e λh(z) D α } { ( ) } λ λ + z = 2 : 2 Dα z = {λ : α 2 < Re(λ) < α 2 }. σ(γ α ) {z : Re(z) α 2 }. ParathroÔme t ra ìti o Γ α eðnai o apeirostikìc genn torac thc S t (f) = f ψ t ìpou ψ t (z) = ϕ t (z) = ϕ t (z), 38

kai èqome thn perðptwsh omˆdac telest n. H {ψ t } èqei DW shmeðo b =, antðstoiqh amfimonìtimh sunˆrthsh kai apeirostikì genn tora h ψ (z) = 2 log + z z, G ψ (z) = 2 ( z2 ). To frˆgma auxhtikìthtac thc {S t } upologðzetai ìpwc prohgoumènwc kai eðnai ω ψ = α 2 epomènwc σ( Γ α ) {z : Re(z) α 2 } SugkrÐnontac me thn antðstoiqh sqèsh gia ton Γ α brðskoume σ(γ α ) = σ π (Γ α ) = {z : α 2 Re(z) α 2 }. 39

Kefˆlaio 3 O telest c Cesàro stouc q rouc Dirichlet 3. Eisagwg 'Opwc anafèrjhke sthn eisagwg, o metasqhmatismìccesàro orðzetai gia analutikèc sunart seic f(z) = a nz n wc ex c: C(f)(z) = z f(ζ) z ζ dζ n = ( a k )z n. (3.) n + EÐnai gnwstì ìti o periorismìc tou C stouc q rouc H p, < p <, eðnai fragmènoc telest c [Si2], [M], kaj c epðshc kai stouc q rouc Bergman A p [Si4]. Sthn enìthta aut ja deðxoume ìti o C eðnai fragmènoc telest c stouc stajmismènouc q rouc Dirichlet D α, < α <. 'Opwc anafèrjhke parapˆnw, o C eðnai fragmènoc ston D = H 2 kai eðnai eôkolo na diapist soume ìti den eðnai fragmènoc ston D = D, diìti gia thn stajer sunˆrthsh f(z) D, k= C(f)(z) = z log z 4 / D.

EÐnai gnwstì ìti o C sundèetai me thn stajmismènh hmiomˆda telest n sônjeshc ìpou S t (f)(z) = ϕ t(z) f(ϕ t (z)) (3.2) z ϕ t (z) = e t z ( e t )z, eðnai h hmiomˆda analutik n sunart sewn tou dðskou, tou paradeðgmatoc 2, tou prohgoômenou kefalaðou. H hmiomˆda {S t } qrhsimopoi jhke gia thn melèth tou C stouc q rouc Hardy [Si2] kai stouc q rouc Bergman [Si4]. H qr sh thc {S t } sthn melèth tou C basðzetai sto gegonìc ìti o C eðnai o antðstrofoc tou apeirostikoô genn tora thc {S t }. QrhsimopoioÔme ed thn Ðdia mèjodo thc [Si2] gia thn apìdeixh ìti o C eðnai fragmènoc stouc D α. 3.2 O C eðnai fragmènoc stouc D α. Pr ta ja deðxoume ìti L mma 3.2.. 'Estw < α <. Gia kˆje t S t Dα D α K ìpou K jetik stajerˆ anexˆrthth tou α. Apìdeixh. 'Estw f D α. Grˆfoume ( ) 2 e α 2 t (t + ) 2, (3.3) α w t (z) = ϕ t(z). z 4

'Etsi S t (f) 2 D α = S t (f)() 2 + D (w t (z)f(ϕ t (z))) 2 ( z 2 ) α dm(z). Gia thn ektðmhsh tou oloklhr matoc èqoume (w t (z)f(ϕ t (z))) 2 ( z 2 ) α dm(z) D 2 w t(z)f(ϕ t (z)) 2 ( z 2 ) α dm(z) D + 2 w t (z)(f(ϕ t (z)) 2 ( z 2 ) α dm(z) D = 2I + 2I 2. (3.4) T ra lìgw thc (2.) I = e t ( e t 2 ) D ( ( e t )z) 2 f(ϕ t (z)) 2 ( z 2 ) α dm(z) K e 2t ( e t ) 2 ( ) z 2 α dm(z) f 2 α D ( e t )z 4 D ϕ t (z) α K2α e 2t ( e t ) 2 ( ) α z dm(z) f 2 α ( e t )z 4 D ϕ t (z) α D Allˆ gia kˆje z D kai t isqôei ( ) α z ( e t )z α (3.5) ϕ t (z) opìte to olokl rwma thc teleutaðac anisìthtac gðnetai e 2t ( e t ) 2 D ( e t )z 4 ( e t )z α dm(z) e 2t ( e t ) 2 = ( e t )z 2 α ( e t )z 2dm(z) D 42

'Omwc sunep c ( e t )z e t e 2t ( e t ) 2 ( e t )z 2 α e 2t ( e t ) 2 e (2 α)t = e αt ( e t ) 2 (3.6) kai ja èqoume e 2t ( e t ) 2 D ( e t )z 2 α ( e t )z dm(z) 2 e αt ( e t ) 2 ( e t )z dm(z). 2 AnaptÔsontac se seirˆ ( e t )z = ( e t ) n z n brðskoume D ( e t )z dm(z) = ( e t ) 2n 2 n + ( e t ) 2(n+) = ( e t ) 2 n + = = = D ( e t ) log 2 ( e t ) 2 ( e t ) log 2 e t ( e t ) log ( e t et ) 2 t ( e t ) 2 43

'Ara gia to I èqoume sunolikˆ I K2α α e αt ( e t ) 2 t ( e t ) 2 f 2 D α = K α te αt f 2 D α (3.7) Gia thn ektðmhsh tou oloklhr matoc I 2 qreiazìmaste thn akìloujh anisìthta, pou prokôptei apì thn (3.5), ( ) z ( z 2 ) α 2 α = ( ϕ ϕ t (z) 2 t (z) 2 ) α ( ) α ( ) α + z z = ( ϕ t (z) 2 ) α + ϕ t (z) ϕ t (z) ( ) α z 2 α ( ϕ t (z) 2 ) α ϕ t (z) 2 α ( e t )z α ( ϕ t (z) 2 ) α. To olokl rwma I 2 gðnetai I 2 2 α w t (z) 2 (f(ϕ t (z))) 2 ( e t )z α ( ϕ t (z) 2 ) α dm(z) D = 2 α e 2t D ( e t )z 2 α f (ϕ t (z)) 2 ϕ t(z) 2 ( ϕ t (z) 2 ) α dm(z) 2 α e αt f (ϕ t (z)) 2 ϕ t(z) 2 ( ϕ t (z) 2 ) α dm(z) D (me allag metablht c) 2 α e αt f (z) 2 ( z 2 ) α dm(z) D 2e αt f 2 D α. (3.8) 44

Epiplèon, afoô < α <, èqoume S t (f)() 2 e 2t f() 2 e 2αt f 2 D α. (3.9) Sundiˆzontac tic (3.4),(3.7),(3.8),(3.9), katal goume ìti S t (f) 2 D α e 2αt f 2 D α + 2 K α e αt t f 2 D α + 4e αt f 2 D α K α e αt (t + ) f 2 D α, ìpou K jetik stajerˆ anexˆrthth tou α. Prìtash 3.2.. 'Estw < α <. H stajmismènh hmiomˆda telest n {S t } t eðnai isqurˆ suneq c ston D α. Apìdeixh. Qrhsimopoi ntac thn isodônamh èkfrash gia thn nìrma tou D α, kai lìgw tou jwr matoc tou kleistoô graf matoc, eðnai eôkolo na diapistwjeð ìti o telest c pollaplasiasmoô M z : f(z) zf(z) eðnai fragmènoc ston D α. Me anˆlogo epiqeðrhma diapist netai ìti an f D α kai f() = tìte f(z) z D α kai upˆrqei stajerˆ C α ste f(z) z D α C α f Dα Gia t, s kai f D α èqoume S t (f) S s (f) Dα = ϕ tf(ϕ t ) ϕ s f(ϕ s ) Dα z C α ϕ t f(ϕ t ) ϕ s f(ϕ s ) Dα = C α T t (M z (f)) T s (M z (f)) Dα 45

ìpou T t, T s oi antðstoiqoi mh stajmismènoi telestèc sônjeshc. Apì thn isqur sunèqeia thc hmiomˆdac {T t } èpetai to zhtoômeno. O apeirostikìc genn torac thc stajmismènhc hmiomˆdac {S t } prokôptei, me mèjodo anˆlogh gia tic mh stajmismènec hmiomˆdec (je rhma 2.2.2), ìti eðnai o telest c me pedðo orismoô α (f)(z) = ( z)(zf(z)), D( α ) = {f D α : ( z)(zf(z)) D α }. Apì th morf tou apeirostikoô genn tora sumperaðnoume ìti h {S t } den eðnai omoiìmorfa suneq c ston D α. Prˆgmati o α mporeð na grafeð α (f)(z) = z( z)f (z) ( z)f(z) = Γ α (f)(z) M z (f)(z) ìpou Γ α o apeirostikìc genn torac thc antðstoiqhc mh stajmismènhc hmiomˆdac T t : f f ϕ t kai M z o telest c pollaplasiasmoô f(z) ( z)f(z). Epeid o Γ α eðnai mh fragmènoc telest c (Prìtash 2.2.2), en o M z eðnai fragmènoc ston D α, èpetai ìti o α eðnai mh fragmènoc. Sunep c h {S t } den eðnai omoiìmorfa suneq c. UpenjumÐzoume ìti an Γ α ìpwc parapˆnw, tìte apì to Parˆdeigma 2 tou prohgoômenou kefalaðou èqoume {λ C : Re(λ) α 2 } {} σ(γ α). (3.) H epìmenh prìtash prosdiorðzei epakrib c ta fˆsmata twn Γ α kai α. 46

Prìtash 3.2.2. 'Estw < α <. 'Estw epðshc α o apeirostikìc genn torac thc stajmismènhc hmiomˆdac telest n sônjeshc{s t } t kai Γ α o apeirostikìc genn torac thc antðstoiqhc mh stajmismènhc hmiomˆdac. Tìte kai σ( α ) = {λ C : Re(λ) α }. (3.) 2 σ(γ α ) = {λ C : Re(λ) α 2 } {}. Apìdeixh. Apì thn (3.3), gia to frˆgma auxhtikìthtac thc {S t }, prokôptei ω α 2. 'Ara {λ C : Re(λ) > α 2 } ρ( α). San sunèpeia autoô kai thc (3.) èpetai σ( α ) {λ C : Re(λ) α 2 } σ(γ α). (3.2) EpÐshc, gia kˆje g D α isqôei g(z) g() z D α. An dialèxoume èna λ ρ( α ), ìqi to mhdèn, upˆrqei monadik sunˆrthsh l (z) D( α ), tètoia ste JewroÔme thn sunˆrthsh Lìgw thc isìthtac (λ α )(l (z)) = g(z) g() z l(z) = zl (z) + g() λ D α. z α (f)(z) = Γ α (zf). 47

isqôei (λ Γ α )(l(z)) = (λ Γ α )(zl (z)) + g() = z(λ α )(l (z)) + g() = g(z). Dhlad to λ ρ(γ α ). 'Ara σ(γ α ) {} σ( α ). Apì th teleutaða sqèsh kai tic (3.), (3.2), katal goume ìti σ( α ) = {λ C : Re(λ) α 2 }, kai σ(γ α ) = {λ C : Re(λ) α 2 } {}. Apì thn prohgoômenh prìtash prokôptei ìti ρ( α ) kai epomènwc o epilôwn telest c R(, α ) : D α D α eðnai fragmènoc. Gia f D α, h sunˆrthsh g = R(, α )(f) = ( α ) (f) lambˆnetai wc h analutik sto D lôsh thc α (g) = f. IsodÔnama ( z)(zg(z)) = f(z), 48

kai ètsi brðskoume g(z) = z z f(ζ) ζ dζ. Dhlad o epilôwn telest c tou apeirostikoô genn tora thc {S t } sto λ = eðnai o telest c Cesàro. 'Epetai ˆmesa ìti o C eðnai fragmènoc ston D α, < α <. H epìmenh prìtash dðnei epiplèon mða ektðmhsh thc nìrmac kai epðshc to fˆsma tou C. QrhsimopoioÔme ton sumbolismì C α gia ton C ston q ro D α. Prìtash 3.2.3. 'Estw < α <. Tìte 2 α C α B α 2 (3.3) ìpou B > stajerˆ anexˆrthth tou α. EpÐshc σ(c α ) = {λ C : λ α }. (3.4) α Apìdeixh. Sumfwna me to fasmatikì je rhma [DS] kai thn (3.) σ(c α ) = { z : z σ( α)} = { z : Re(z) α 2 } = {λ : λ α α } kai apodeðqjhke o isqurismìc gia to fˆsma. Gia thn ektðmhsh thc nìrmac, èpetai ˆmesa ìti C α sup λ σ(cα ) λ = 2 α, (3.5) 49

dhlad h arister anisìthta. Gia thn ˆllh anisìthta qrhsmopoioôme thn anaparˆstash tou C α = R(, α ) wc metasqhmatismoô Laplace thc antðstoiqhc hmiomˆdac C α (f)(z) = Epeid C α (f)() = f(), C α (f) 2 D α = f() 2 + D S t (f)(z)dt. C α (f) (z) 2 ( z 2 ) α dm(z). Gia to olokl rwma, apì thn genikeumènh anisìthta Minkowski [WZ, sel.43], èqoume 2 S t (f) (z) dt ( z 2 ) α dm(z) D [ ( ) ] 2 S t (f) (z) 2 ( z 2 ) α 2 dm(z) dt D ( S t (f) Dα dt ) ( ) 2 ( K 2 α ) ( e α 2 t (t + ) 2 dt ) 2 f 2 D α ( K α ( ) [ ( 2 = K α α ) 4 ( Γ K ( 2 α e α 2 u u 2 du ) 2 f 2 D α ) 3 2 ( 2 e s s 2 ds ] 2 f 2 D α )) 2 f 2 D α. 5

Apì ta parapˆnw prokôptei C α (f) 2 D α f() 2 + K ( 2 α )4 f 2 D α B α 4 f 2 D α. (3.6) Sundiˆzontac tic (3.5) kai (3.6) prokôptei h (3.3). 3.3 O suggen c telest c A. Ja proqwr soume t ra sth melèth tou metasqhmatismoô A, pou orðzetai wc ( ) a k A(f)(z) = z n (3.7) k + ìpou f(z) = a nz n. O metasqhmatismìc autìc sqetðzetai ˆmesa me ton telest Cesàro C, diìti o pðnakac pou antistoiqeð ston A ston q ro Hardy H 2 eðnai o anˆstrofoc tou antðstoiqou pðnaka gia ton C. Sthn perðptwsh mˆlista tou H 2 o A eðnai o suzug c telest c (me thn ènnoia thc suzugðac se q rouc Hilbert) tou C. Sthn genik perðptwsh miac analutik c sunˆrthshc f, h seirˆ sthn (3.7) den eðnai kal c orismènh diìti h eswterik seirˆ pou dðnei touc suntelestèc Taylor endèqetai na mh sugklðnei. An ìmwc upojèsoume ìti f D α gia kˆpoio < α <, tìte me th bo jeia thc anisìthtac Hardy([DU, sel. 48]) k=n a n n + π f H, (3.8) 5

h opoða isqôei gia sunart seic f H, kai apì thn parat rhsh ìti < α < èqoume D α H 2 H, sumperaðnoume ìti h akoloujða twn suntelest n pou emfanðzontai sth seirˆ (3.7) eðnai fragmènh, epomènwc h (3.7) orðzei analutik sunˆrthsh sto dðsko. O metasqhmatismìc A mporeð na grafeð wc olokl rwma A(f)(z) = z z f(ζ)dζ (3.9) upì thn proôpìjesh ìti h sunˆrthsh f eðnai arketˆ kal ste na eðnai oloklhr simh pˆnw se kˆje eujôgrammo tm ma [, z], z D. Tètoiec sunart seic eðnai ìlec oi sunart seic tou q rou Hardy H, lìgw thc anisìthtac Fejer-Riesz [DU, Theorem 3.3]. Gia tètoiec sunart seic, epilègontac san kampôlh olokl rwshc thn γ(t) = tz + ( t), t, kai qrhsimopoiìntac to je rhma kuriarqoômenhc sôgklishc gia seirèc, gia thn dikaiolìghsh thc enallag c tou ajroðsmatoc me to olokl rwma, èqome thn (3.9). O telest c A eðnai fragmènoc stouc q rouc H p, < p < [Si2], kai Bergman A p, 2 < p < [Si3], kai h melèth tou stic parapˆnw ergasðec basðsthke sth sqèsh tou A me mða hmiomˆda telest n sônjeshc. Ja qrhsimopoi soume ed thn Ðdia mèjodo gia na melet soume ton A stouc D α, < α <. JewroÔme thn hmiomˆda ψ t (z) = e t z + e t, h opoða èqei antðstoiqh amfimonìtimh sunˆrthsh h (z) = log 52 z,

apeirostikì genn tora G(z) = z kai DW shmeðo b =. JewroÔme epðshc thn hmiomˆda telest n sônjeshc T t (f) = f ψ t. Apì thn (2.4) èpetai ìti, gia kˆje t T t Dα D α Kα 2 e α 2 t, (3.2) ìpou K > stajerˆ. EpÐshc apì to je rhma (2.2.) h {T t } eðnai isqurˆ suneq c ston D α, < α <, kai èqei apeirostikì genn tora Γ α (f)(z) = ( z)f (z). Apì thn (3.2) èpetai ìti h {T t } èqei frˆgma auxhtikìthtac ω α 2. Epomènwc {λ : Re(λ) > α 2 } ρ(γ α). (3.2) Ex ˆllou gia to shmeiakì fˆsma tou Γ α, sômfwna me thn (2.22), isqôei σ π (Γ a ) = {λg() C : e λh(z) D α } = {λ C : ( z) λ D α } = {λ C : Re(λ) < α 2 }. Se sundiasmì me thn (3.2) sumperaðnoume σ(γ α ) = σ π (Γ α ) = {λ C : Re(λ) α }. (3.22) 2 53

JewroÔme t ra th stajmismènh hmiomˆda S t (f)(z) = e t f(ψ t (z)), f D α. 'Amesa prokôptei ìti, gia kˆje t, S t Dα D α Kα 2 e ( α 2 )t. (3.23) kai ìti h {S t } eðnai isqurˆ suneq c ston D α. O apeirostikìc genn torˆc eðnai α (f)(z) = ( z)f (z) f(z). ParathroÔme ìti giˆ λ C isqôei 'Ara apì thn (3.22) brðskoume λ α = (λ + ) Γ α. σ( α ) = {λ C : Re(λ) α 2 }. (3.24) 'Epetai ìti ρ( α ) kai me èna aplì upologismì brðskoume R(, α ) = A. Epomènwc o A eðnai fragmènoc telest c ston D α, < α <. H epìmenh prìtash dðnei ektðmhsh thc nìrmac kai epðshc to fˆsma tou A. Prìtash 3.3.. 'Estw < α < kai A α o telest c A ston D α. Tìte 2 2 α A Λ α α 2(2 α) ìpou Λ > stajerˆ anexˆrhth tou α. EpÐshc σ(a α ) = {λ C : λ 2 α 2 α }. 54

Apìdeixh. Apì thn (3.24) kai to fasmatikì je rhma [DS] 'Epetai ˆmesa ìti σ(a α ) = { z : z σ( α)} = { z : Re(z) α 2 } = {λ C : λ 2 α 2 α }. A α 2 2 α. Gia thn eôresh tou ˆnw frˆgmatoc thc nìrmac grˆfoume A(f)(z) = S t (f)(z)dt = e t f(ϕ t (z))dt. kai èqoume A α (f) 2 D α = A α (f)() 2 + A α (f) (z) 2 ( z 2 ) α dm(z) D 2 = e t f( e t )dt 2 + S t (f) (z) dt ( z 2 ) α dm(z). D 55

'Omwc, lìgw thc (2.4), e t f( e t )dt 2 ( K2 α = K2 α = ) 2 e t f( e t ) dt ( ( α 2 K α(2 α) 2 f 2 D α. ) 2 e t e α 2 t dt f 2 D α ) 2 f 2 D α Ex ˆllou, lìgw thc genikeumènhc anisìthtac Minkowski kai thc (3.2) 2 S t (f) (z) dt ( z 2 ) α dm(z) D [ ( ) ] 2 S t (f) (z) 2 ( z 2 ) α 2 dm(z) dt D ( S t (f) Dα dt K α ( K α(2 α) 2 f 2 D α. Apì ta parapˆnw prokôptei [ A α (f) 2 D α kai to zhtoômeno èpetai. ) 2 ) 2 e ( α 2 )t dt f 2 D α K α(2 α) 2 + K α(2 α) 2 ] f 2 D α = Λ α(2 α) 2 f 2 D α 56

Kefˆlaio 4 PÐnakec Hausdorff kai telestèc sônjeshc 4. Eisagwg 'Estw {µ n } akoloujða migadik n arijm n kai o telest c diafor n µ n = µ n µ n+, n =,, 2,... Gia k =,, 2,... orðzoume tic µ n = µ n kai k µ n = ( k µ n ). 'Enac pðnakac Hausdorff H = H(µ n ), me genn tria akoloujða {µ n }, eðnai ènac kˆtw trigwnikìc pðnakac me stoiqeða h n,k = h,.. h, h,.. h 2, h 2, h 2,2..... ( ) n n k µ k, k gia k n 57

kai h n,k =, gia k > n. Arqikˆ autoð oi pðnakec melet jhkan sthn jewrða ajroisimìthtac [H]. Melet jhkan epðshc wc telestèc se q rouc akolouji n [Rh],[De], [Le], kai ta suneq analogˆ touc se q rouc oloklhr simwn sunart sewn [BM]. MÐa shmantik kathgorða pinˆkwn Hausdorff prokôptei an h {µ n } eðnai akoloujða rop n, dhl. µ n = t n dµ(t), ìpou µ èna peperasmèno jetikì mètro Borel sto (, ]. SumbolÐzoume touc pðnakec autoôc me H µ. 'Opwc prokôptei apì èna sôntomo upologismì, ta stoiqeða touc eðnai thc morf c ( ) n h n,k = t k ( t) n k dµ(t), k n. k SumbolÐzoume epðshc me A µ ton anˆstrofo tou H µ. 'Estw p <. 'Eqei deiqjeð [H2], ìti ìtan to mètro ikanopoieð t p dµ(t) <, tìte o antðstoiqoc pðnakac Hausdorff orðzei, sto q ro twn akolouji n l p, èna fragmèno telest H µ : l p l p H µ (a n ) = { n h n,k a k }, {a n } l p. k= 'Otan mˆlista to µ eðnai mètro pijanìthtac, tìte h nìrma tou eisagìmenou telest eðnai [Rh] H µ = 58 t p dµ(t).

Parakˆtw anafèroume merikˆ gnwstˆ paradeðgmata pinˆkwn Hausdorff, oi opoðoi orðzoun fragmènouc telestèc stouc q rouc l p. Parˆdeigma. Gia a >, oi pðnakec a-cesàro prokôptoun apì to mètro dµ(t) = a( t) a dt kai èqoun genn tria akoloujða µ n = Γ(a + )Γ(n + ) Γ(n + a + ) 'Otan a = paðrnoume ton pðnaka tou telest Cesàro. Parˆdeigma 2. Gia a >, q > p, ìpou p, q >, oi genikeumènoi Cesàro prokôptoun apì to mètro dµ(t) = me genn tria akoloujða Γ(q + a) Γ(q)Γ(a) tq ( t) a dt, a >, q > p, µ n = Γ(a + q)γ(n + q) Γ(n + a + q)γ(q). Parˆdeigma 3. Gia a >, oi pðnakec Hölder (H a ), prokôptoun gia dµ(t) = Γ(a) (log t )a dt, a >, me genn tria akoloujða µ n = (n + ) a.. 59

Parˆdeigma 4. Gia a >, c >, oi pðnakec Gamma (Γ a c), prokôptoun me dµ(t) = me genn tria akoloujða ca Γ(a) tc (log t )a dt, a >, µ n = ( ) a c. n + c Oi pðnakec Hausdorff mporoôn na jewrhjoôn san metasqhmatismoð epð analutik n sunart sewn tou dðskou, oi opoðoi prokôptoun apì ton pollaplasiasmì tou pðnaka me thn akoloujða twn suntelest n Taylor. Sugkekrimèna, an µ èna mètro Borel sto (, ], H µ = (h n,k ) o antðstoiqoc pðnakac Hausdorff, A µ o anˆstrofìc tou kai f(z) = a nz n A(D), orðzoume th dunamoseirˆ n H µ (f)(z) = ( h n,k a k )z n, (4.) kaj c kai thn akìloujh dunamoseirˆ (ìtan aut eðnai dunatì na orisjeð) A µ (f) = ( h k,n a k )z n. (4.2) k=n ParathroÔme ìti an µ eðnai to mètro Lebesque tìte apì tic parapˆnw sqèseic paðrnoume ton telest Cesàro kai ton telest A antðstoiqa, stouc opoðouc anaferj kame sto prohgoômeno kefˆlaio. Sthn perðptwsh pou dµ(t) = a( t) a, Re(a) >, eðnai gnwstì ìti prokôptoun fragmènoi telestèc stouc q rouc Hardy, kaj c kai se ˆllouc q rouc analutik n sunart sewn k= 6

Stic epìmenec enìthtec ja melet soume touc pðnakec Hausdorff, wc telestèc stouc q rouc Hardy, gia thn perðptwsh pou to µ eðnai èna tuqaðo peperasmèno mètro Borel sto diˆsthma (, ]. Sugkekrimèna ja doôme poiec sunj kec prèpei na ikanopoieð to µ ste oi sqèseic (4.) kai (4.2) na orðzoun fragmènouc telestèc stouc H p, p [, ). Sunj kec epð tou µ ste oi H µ kai oi anˆstrofoð touc na eðnai fragmènoi telestèc stouc q rouc Hardy H p melet jhkan epðshc apì ton Oliver Rudolf [RO]. Ta apotelèsmata thc [RO] den eðnai pl rh diìti oi sunj kec pou dðdontai den eðnai oi fusiologikèc gia kˆpoiec timèc tou p. Sugkekrimèna gia thn perðptwsh twn H µ s- touc q rouc H p, h sunj kh pou dðdetai sthn [RO] eðnai aut thc paroôshc ergasðac, en gia p < 2 dðdetai miac diaforetik c fôsewc sunj kh h opoða den eðnai bèltisth. Epi pleìn oi mèjodoi thc paroôshc ergasðac eðnai amesìterec. Prèpei ìmwc na anafèroume ìti sthn [RO] melet ntai epi plèon se èktash ta fˆsmata twn telest n. 4.2 PÐnakec Hausdorff se q rouc Hardy Sthn enìthta aut ja melet soume touc pðnakec Hausdorff stouc q rouc H p. Ja deðxoume pr ta ìti gia f H, h dunamoseirˆ (4.) orðzei mða analutik sunˆrthsh ston D. Prˆgmati an f(z) = a nz n H, tìte h akoloujða {a n } eðnai mhdenik, giatð h sunoriak sunˆrthsh thc f eðnai oloklhr simh epð thc perifèreiac tou dðskou. 6

An M = sup n a n, tìte n n n ( ) n h n,k a k h n,k a k t k ( t) n k dµ(t) a k k k= k= n ( ) n M t k ( t) n k dµ(t) k k= ( n ( ) n = M )t k ( t) n k dµ(t) k k= = M k= (t + ( t)) n dµ(t) = Mµ((, ]) Epomènwc oi suntelestèc thc dunamoseirˆc (4.) apoteloôn fragmènh akoloujða, ˆra h aktðna sôgklishc thc eðnai. Sth sunèqeia, ìpwc kai sthn perðptwsh tou telest Cesàro, deðqnoume ìti h H µ (f) mporeð na grafeð wc mèsoc ìroc orismènwn stajmismènwn telest n sônjeshc. Shmei noume ìti h anaparˆstash aut twn pinˆkwn Hausdorff emfanðzetai gia pr th forˆ, sthn paroôsa melèth, kaj c epðshc kai sthn [RO]. 'Estw p <. Gia t (, ] jewroôme th sunˆrthsh ϕ t (z) = tz ( t)z, z D h opoða apeikonðzei ton D ston eautì tou. 'Ara, apì thn parˆgrafo (.2), o telest c sônjeshc f f ϕ t eðnai fragmènoc ston H p. EpÐshc gia kˆje t (, ], h w t (z) = ( t)z eðnai fragmènh sunˆrthsh twn z D. 'Etsi o telest c T t (f)(z) = w t (z)f(ϕ t (z)) (4.3) 62

eðnai fragmènoc ston H p. JewroÔme to olokl rwma F (z) = w t (z)f(ϕ t (z))dµ(t), (4.4) to opoðo orðzetai gia kˆje analutik sunˆrthsh f. Prˆgmati autì eðnai fanerì giatð, gia stajerì z D, kai sup w t (z) t (,] z sup f(ϕ t (z)) sup f(ζ) <, t (,] ζ z giatð apì to l mma tou Schwarz èqoume ϕ t (z) z. T ra gia f H p, z D, h dunamoseirˆ (4.) sugklðnei apìluta, lìgw tou ìti oi suntelestèc thc apoteloôn fragmènh akoloujða. 'Etsi h allag sth seirˆ ˆjroishc eðnai dunat kai ˆra H µ (f)(z) = n ( h n,λ a λ )z n = λ= h λ,n a n z λ. λ=n Ex' ˆllou jewroôme thn analutik sunˆrthsh F (z) = = ( t)z f( tz ( t)z )dµ(t) t n z n a n ( ( t)z) n+dµ(t). Epeid h seirˆ mèsa sto olokl rwma sugklðnei omoiìmorfa wc 63

proc t (, ], èqoume F (z) = a n = = = t n ( ( t)z) n+dµ(t)zn ( ) a n t n (k + n)! ( t) k z k dµ(t)z n n!k! k= ( ) n + k a n t n ( t) k z n+k dµ(t) k k= ( ) λ a n t n ( t) λ n z λ dµ(t). n λ=n Pˆli lìgw omoiìmorfhc sôgklishc wc proc t, thc upì olokl rwshc seirˆc, èqoume ìti ( ) λ F (z) = a n t n ( t) λ n dµ(t)z λ λ=n n = h λ,n a n z λ. 'Ara katal goume H µ (f)(z) = λ=n = ( n h n,λ a λ )z n = λ= λ=n h λ,n a n z λ ( t)z f( tz ( t)z )dµ(t). SunoyÐzoume ta parapˆnw sto akìloujo l mma : L mma 4.2.. 'Estw µ èna peperasmèno mètro Borel sto (, ] kai H µ = (h n,k ) o antðstoiqoc pðnakac Hausdorff. 'Estw epðshc p < kai f(z) = a nz n H p. Tìte : 64

(i) h dunamoseirˆ H µ (f)(z) thc sqèshc (4.) orðzei mða analutik sunˆrthsh ston D. (ii) Gia kˆje z D, h H µ (f) mporeð na grafeð me thn akìloujh oloklhrwtik morf H µ (f)(z) = w t (z)f(ϕ t (z))dµ(t). (4.5) Ac epistrèyoume t ra sth dunamoseirˆ (4.2). polu numo, tìte ta ajroðsmata B n = h k,n a k k=n An h f eðnai eðnai peperasmèna kai B n =, n n gia kˆpoio n. Sthn perðptwsh aut h (4.2) eðnai analutik sunˆrthsh ston D, wc poluwnumik. Ja doôme ìti kai aut grˆfetai me anˆlogh oloklhrwtik morf. 'Estw p <. Gia kˆje t (, ] jewroôme thn analutik sunˆrthsh ψ t (z) = tz + t, z D h opoða apeikonðzei ton D ston eautì tou. Aut, sômfwna me thn me thn parˆgrafo (.2) eisˆgei fragmèno telest sônjeshc Q t (f)(z) = f(ψ t (z)) (4.6) ston H p. T ra an h f(z) = a nz n eðnai polu numo (ˆra a n = telikˆ), jewroôme to olokl rwma G(f)(z) = = Q t (f)(z)dµ(t) = a n (tz + t) n dµ(t). f(tz + t)dµ(t) 65

Profan c, gia kˆje z D, to parapˆnw olokl rwma upˆrqei kai epeid ta ajroðsmata eðnai peperasmèna G(f)(z) = = = = a n (tz + t) n dµ(t) n ( ) n a n t k z k ( t) n k dµ(t) k k= n (( ) n ) a n t k ( t) n k dµ(t) z k k k= n a n ( h n,k z k ) = ( h k,n a k )z n k= = A µ (f)(z). En gènei, gia mða f H p, mh poluwnumik, to parapˆnw epiqeðrhma den mporeð na efarmosteð. EÐnai ìmwc gnwstì ìti sthn perðptwsh aut f H p f(z) c p, z D, ( z ) p ìpou c p stajerˆ pou exartˆtai mìno apì to p [DU, sel. 36]. 'Ara gia kˆje z D, G(f)(z) c p f H p c p f H p ( z ) p k=n f(tz + t) dµ(t) dµ(t) ( tz + t ) p dµ(t). t p 66