An indestructible Blaschke product in the little Bloch space

Publicacions Ma emá iques, Vol 37 (1993), 95-109 . AN INDESTRUCTIBLE BLASCHKE PRODUCT IN THE LITTLE BLOCH SPACE A bs ac CHRISTOPHER J . BISHOP The li le Bloch space, 13 0 , is he space o all holomo phic unc- ions on he uni disk such ha lim1 z 1- l ( '(z)j(1 - Iz12) = 0 . Fini e Blaschke p oduc s a e clea ly in 130, bu examples o in- ini e p oduc s in 80 a e mo e di icul o ob ain ( he e a e now se e al cons uc ions due o Sa ason, S ephenson and he au ho , among o he s) . S ephenson has asked whe he 130 con ains an in ini e, indes uc ible Blaschke p oduc , Le ., a Blaschke p od- uc B so ha (B(z) - a)/(1 - QB(z)), is also a Blaschke p od- uc o e e y a E D . In his pape we gi e an a i ma i e an- swe o his ques ion by cons uc ing such a Blaschke p oduc . We also answe a ques ion o Ca mona and Cu í by cons uc - ing a VMO unc ion, , so ha Il J¡ . = 1 and whose ange se , R( , a) = {w : he e exis s z n , ~ a, (z~) = w}, equals he open uni disk o e e y a E T . 1 . In oduc ion Le D = iz1 < 1} deno e he uni disk . The li le Bloch space, B O , is he space o holomo phic unc ions on D such ha lim  1 '(z)1(1 - Iz12) = 0 . Iz1-1 Basic ac s abou B O can be ound in [21 . A Blaschke p oduc is a holo- mo phic unc ion o he o m This wo k was suppo ed by NSF G an DMS 91-00671 96  C . J . BISHOP B(z) = 11 ~ z n - z Iznl n 1-znz zn , whe e E(1 - Izn1) G oo . Fini e Blaschke p oduc s a e clea ly in B O , bu examples o in ini e p oduc s in BO a e no so ob ious . Such unc ions do exis , as is shown in [1], [8], [10] . A mo e explici example, as well as a cha ac e iza ion o such p oduc s in e ms o he ze o sequence {z n }, has been gi en in [3] . Tha esul answe s se e al ques ions abou 130, bu does no esol e he ollowing ques ion om [10] : does B o con ain an in ini e, indes uc ible Blaschke p oduc , Le ., a Blaschke p oduc B so ha _ B(z) - a Ba(z)  1 - úB(z)' is also a Blaschke p oduc o e e y a ED? The ques ion a ises because o F os man's heo em . An inne unc ion F is a holomo phic unc ion on D wi h bounda y alues o absolu e alue 1 a .e . on T . Any such unc ion can be w i en as a p oduc _ II  ¡o F(z) = B(z)S(z) = ( l 1 n zn z ~)(exp( - e ie ± zdp(e)), o a Blaschke p oduc and a singula inne unc ion (M is a ini e, posi i e measu e singula o dB) . F os man's heo em s a es ha o any inne unc ion F on D, Fa  F(z) - a (z) = 1 - úF(z)' is a Blaschke p oduc o e e y a E D E whe e E is an excep ional se o ze o loga i hmic capaci y [5, Theo em 11 .6 .4] . The cons uc ions o Blaschke p oduc s in [8], [10] i s build an inne unc ion in B o and hen apply F os man's heo em . S ephenson asked i his was una oidable, e .g ., does 13 o con ain any indes uc ible Blaschke p oduc s? The example in [3] is buil by cons uc iog he ze o se , so does no use F os man's heo em . Fu he mo e, a a ian o S ephenson's cons uc ion gi es a Blaschke p oduc wi hou using F os man's heo em (see nex sec ion) . In his no e we expand on his obse a ion o gi e a "cu and pas e" cons uc ion o an indes uc ible Blaschke p oduc in 130 . INDESTRUCTIBLE BLASCHKE PRODUCT IN BO  9 7 One could also y o p oduce such an example by inding a su icien condi ion on he ze os o he p oduc o be indes uc ible and which includes some sequences sa is ying he 13 o condi ion om [3] . One such su icien condi ion o indes uc ibili y is ha he sequence be hin, Le ., zk - zn kan 1 - znzk Howe e , his condi ion is incompa ible wi h he BO condi ion . An e en mo e ambi ious p oblem is o cha ac e ize indes uc ibili y in e ms o he ze o-se . In [6], Mo se has cons uc ed a des uc ible Blaschke p oduc which becomes indes uc ible when a single poin is dele ed om i s ze o-se . This indica es any cha ac e iza ion o indes uc ibil- i y in e ms o he ze o se would be e y delica e (and p obably e y di icul ) . A ela ed p oblem has been sol ed, howe e . In some sense, ini e p oduc s o in e pola ing Blaschke p oduc s a e he "con o mally in a ian " class o Blaschke p oduc s . In [7] Nicolau has gi en a ze o- se cha ac e iza ion o hose Blaschke p oduc s B so ha B a is a ini e p oduc o in e pola ing Blaschke p oduc s o e e y a in he disk . Thus he has sol ed he con o mally in a ian e sion o he p oblem o cha - ac e izing indes uc ibili y . Ou cons uc ion gi es a Blaschke p oduc whose singula se ( he accumula ion se o he ze os) has measu e ze o . I we could cons uc an example whose singula se was he en i e ci cle, his unc ion would also ha e he p ope y ha i s ange se , R( , a) = {w : he e exis s z n - a, (z n ) = w}, equals he whose disk o e e y a E T . Ca mona and Cu í had asked in [4] i he e was a unc ion in H°° n BO wi h his p ope y . I belie e he cons uc ion can be modi ied o gi e such an example, bu a he han do his, I will ske ch he cons uc ion o a unc ion E H°° n VMO wi h 11 jj,, = 1 and R( , a) = D o e e y aE T . Since VMO C BO, his is an e en s onge esul (again answe ing a ques ion o Ca mona and Cu í) . I hank A u o Nicolau o b inging his ques ion o my a en ion and ou discussions on i . I also hank he e e ee o his help ul commen s . His sugges ions ha e cla i ied he exposi ion in se e al places . 98  C . J . BISHOP 2 . The B O cons uc ion The idea is qui e simple ; we will build a simply connec ed Riemann su ace by aking copies o he uni disk wi h sli s and "gluing" di e en copies along he sli s . A simple example o his idea is o ake in ini ely many copies o D [2,1), and iden i ying he " op" edge o one copy wi h he "bo om" edge o he nex . See Figu e 1 . . .(D (D (D (D ... Figu e 1 . A single example s~ Le So be he ini ial shee , which we also e e o as he "ze o h shee " . This shee con ains a poin co esponding o ze o in he uni disk and we e e o his poin as "0" on he su ace . Le S  , be n h s age o his cons uc ion ( he union o copies -n o n) and S = UnSn he limi ing su ace . Fo each o he e su aces he poin "0" e e s o he poin 0 on So . In he es o his pape we shall assume ha any Riemann mapping o he uni disk o a cons uc ed su ace like S n o S maps 0 in he disk o he poin 0 on he su ace . Whene e we alk abou ha monic on he su ace i is he push o wa d o no malized Lebesgue measu e on he ci cle unde such a Riemann mapping, Le ., ha monic measu e will always be wi h espec o he poin 0 on he ze o h shee . S is simply connec ed so he e is a Riemann mapping ob : D --> S and he e is an ob ious holomo phic p ojec ion P : S -3 D . We claim ha F = P o D mus be an inne unc ion because all he ha monic measu e o S li es on he pa o he bounda y abo e he he uni ci cle . To p o e his we conside S n and show ha he ha monic measu e o he wo adial sli s in i s bounda y a e O(ñ) . To do his we mapS  , o a hal in ini e s ip W = {(x, y) : -oo < x < 0, -(2n - - 1)7 < y < (2n + 1)7 } by he mapping INDESTRUCTIBLE BLASCHKEPRODUCT IN B o  99 - z --> log( z  2 ) 1-2z which has a well de ined b anch on S  , . The poin 0 on he su ace is mapped o -1/2 and he adial edges a e mapped o he ho izon al edges o he s ip . S anda d es ima es (e .g ., map he s ip o a hal plane ia sin(z/i) and use he Poisson in eg al) show ha he ha monic measu e o he ho izon al edges o he s ip wi h espec o he poin -1/2 a e app oxima ely 1/n . By con o mal in a iance o ha monic measu e he claim abou S n is p o ed . Thus he ci cula pa o OS n has measu e >_ 1 - C/n . Taking n --> oo we see ha he ci cula pa o OS has ull measu e, Le, ¡Fl = 1 a .e . on he uni ci cle . In ac , F mus be Blaschke p oduc . To see his, ecall ha a unc ion in he uni ball o HI(D) is a Blaschke p oduc i he leas ha monic majo an o log l 1 is 0 (e .g ., [5, Theo em II .2 .4]) . This says ha F is a Blaschke p oduc i 0 is he leas ha monic majo an o log ¡P(z) 1 on S . Le u be he leas ha monic majo an o logjP(z)1 on S . Then u es ic ed o S,,, is ha monic and has bounda y alues 0 on P -1 (T) and >_ log 2 on he wo adial sli s in OS, . Thus 0 >_ u(0) >_ 2 log 2 . Since his holds o any n, u(0) = 0 ( ecall ha he "0" in u(0) e e s o he designa ed poin on he ze o h shee ) . Since u is nonposi i e his implies u - 0 and so F is a Blaschke p oduc . Le a E D and Ta(z) = (z - a)/(1 - áz) . An a gumen like he one abo e shows shows ha i a :,¿ 2 hen F a = Ta o F is a Blaschke p oduc . Howe e , since no poin o S co e s he poin {2 }, F2 is ne e ze o, so mus be a singula inne unc ion (in ac , since F is con inuous excep o one bounda y poin , up o o a ions i mus be exp(A i+z ), o some A > 0, Le ., he singula inne unc ion co esponding o a posi i e poin mass) . To build an indes uc ible Blaschke p oduc we will ha e o a y he cons uc ion, adding shee s which co e he omi ed poin s o ea lie gene a ions, and in pa icula , so ha he leas ha monic majo an o log 1To,(P(z))1 on S is 0 o any choice o a E D . This says ha no only is each poin co e ed in ini ely o en, bu he e is some sense in which i is " equen ly" co e ed . To ge F in o he li le Bloch space imposes ano he cons ain : gi en any e > 0 only ini ely many o he shee s we a ach may con ain a disk o adius e . This a ises because o a geome ic cha ac e iza ion o ,Cio due o S egenga and S ephenson [9] . Fo analy ic on D and a E D, > 0 hey de ine SZ a ( ) o be he componen o -1 (D ( (a), )) con aining a, F a ( ) = Og a ( ) n T and (a) = sup{ : F a ( ) = 0} . Then E Bo i (a) = o(1) as ¡al --> 1 . In 10 0  C . J . BISHOP pa icula , i he Riemann su ace o is ob ained by iden i ying copies o D along sli s, hen he endpoin s o any such sli a e in he ideal bounda y o he su ace . The e o e will be in he li le Bloch space i o e e y E > 0, hese endpoin s o pas ed edges a e E-dense in D o all bu ini ely many shee s . We will induc i ely cons uc a sequence o posi i e numbe s {E n ,} ending o ze o, a sequence o ini e poin se s {E ,}, a collec ion o adial line segmen s T  ,, and wo sequences o in ege s {gn}, {h  } ending o in ini y . The se s {Ej, {T  } will sa is y (1) Tn C T a+l, EnC En+1, EnC T  . (2) The endpoin o each segmen in T,, is in E,,, . (3) E  ,/E  ,+1 is an e en in ege . (4) Adjacen poin s o & en a segmen o T,, a e a dis an e E , om each o he . (5) SupzED dis (z, E n ,) G lOE  . See Figu e 2 . An "edge" I o T  deno es a subin e al en T  , connec ing wo adjacen poin s o E,,, Le ., a componen o T  , E n . We le en , deno e he numbe o edges in T  , . In he cons uc ion below each such edge will be ea ed as wo sepa a e pieces o he bounda y o he domain R n = D T,, co esponding o i s wo sides . One side will be pas ed e a shee o p e ious gene a ion, he o he pas ed e one o mo e shee s in he nex highe gene a ion . Figu e 2 . E  , T  ,, R n INDESTRUCTIBLE BLASCHKE PRODUCT IN 13p  10 1 Gi en an edge I in he bounda y o R  , we can ei he a ach ano he copy o R n ,, o di ide he edge in o m = En/En+1 edges in T . .+1 ( since E n C En+1) and a ach m copies o R,+1 . Gi en a sequence o in ege s {gi} we could build a Riemann su ace as ollows . S a wi h one copy o R1 and a ach 2e1 copies o o R1 along (bo h sides o ) each edge o T 1 . Call his 5 1 . Then a ach mo e copies o R1 along each edge in OS, o ob ain S2and con inuing o g 1 gene a ions, ob aining a nes ed sequence o su aces S1 C S 2 C . . . C Sgl . The e m "gene a ions" e e s o he ac ha o connec he poin 0 in he ze o h shee So o any o he unpas ed edges o Sk a pa h mus pass hough a leas k + 1di e en shee s (i .e ., copies o R1) belonging o So, S1 So, ... , Sk Sk_1 . We ha e ob ained Sgl by pas ing oge he iden ical shee s, Le ., copies o Rl . To ge he nex su ace, Sgl+1, we a ach o each unpas ed edge o Sgl El/E2 copies o he shee R2 . We ob ain Sgl+2 by pas ing a copy o R2 o each unpas ed edge o Ssl+1 . We con inue in his way o 92 gene a ions, ob aining a su ace S91+92 . The nex su ace Sg1+g2+1, is cons uc ed by a aching copies o R3 o he unpas ed edges o Sg1+g2 . Thus gi en he sequence o in ege s {gk} (which ells us o how many gene a ions o a ach copies o Rk) and con inuing in he ob ious manne , we ob ain an inc easing, nes ed sequence o simply connec ed su aces, {S n } . Then S = UnSn, is a simply connec ed connec ed Riemann su ace . I -P : D - S is he Riemann map (mapping 0 o 0 on So), and P : S --> D he p ojec ion hen F = P o <P is a holomo phic unc ion on he uni disk which we claim is an in ini e Blaschke p oduc in 130, i he pa ame e s a e chosen co ec ly . This is essen ial S ephenson's cons uc ion in [10] . The ac ha F E BO ollows om he cha ac e iza ion o S egenga and S ephenson men ioned ea lie . I he sequence {gi} g ows quickly enough, S ephen- son shows he mapping F is an inne unc ion . I dis (o,T n ) >_ En hen F is ac ually a Blaschke p oduc (again i gn / oo as enough) . To p o e his, conside he leas ha monic majo an u o log IP(z)j es ic ed o SN = Sgl+ . . .+g" The bounda y b eaks in o wo pieces aS N = 01SN U a2SN co esponding espec i ely o P -1 (T) and he adial edges . Then u has bounda y alues 0 on 8 1 SN and u >_ 109 En on á2SN . The se a2SN can be made o ha e as small ha monic measu e as we wish by aking gn la ge enough, so we may ake 0>- U> - W(ó2SN)109E n >- 1 - n i gn is la ge enough ( ecall ha as be o e, ha monic measu e e e s o he ha monic measu e wi h espec o he poin 0 on he ze o h shee 10 2  C . J . BISHOP So) . Thus F is a Blaschke p oduc , bu i canno be indes uc ible since i only akes alues in each E n ini ely o en . As S ephenson poin s ou , his example shows he excep ional se in F os man's heo em may be dense in he uni disk . To make F indes uc ible, we modi y he cons uc ion sligh ly . Asso- cia ed o each E,, de ine ano he se F n , by eplacing each zE E nby a poin w E En,+1 wi h Iz - w i = á E a and such ha w is on same adius as z . The se s F  , sa is y app oxima ely he same densi y condi ions as he E n (wi h E n eplaced by 2c,  ,) . Ou idea is o modi y he cons uc- ion by al e na ing he use o he se s E  , and F  in he cons uc ion . Since E  , 1 F n , = 0 his means ou su ace will co e he whole disk and since max(dis (z, En), dis (z, Fn)) >_ c  ,/4 o e e y poin z in he disk, we should be able o p o e ou unc ion is indes uc ible by es ima ing ha monic measu e ei he on he "E  ,-shee s" o 'T,,-shee s" (depending on whe he z is a om E  , o a om Fn) . Howe e since E", 1 F,, = 0, we need some u he modi ica ions o o able o a ach an "F  ,-shee " o an " En - shee " . This is how we a ach a F  ,-shee o an E n ,-shee . Conside a com- ponen in e al I o T n wi h endpoin s in E  , . Le T n , be he ana- logue o T  , o he se F  , and le R,, = D Tn, . Assume (wi h- ou loss o gene ali y) ha F  , has been chosen so T  , C T  , . Le {ao, al, . . . . an} = I n E  ,+1, lis ed in o de (e .g ., ao, a a a e he end- poin s o I) . Le F  ,j = F n U {aj , a j+ 1} . Along each in e al (a j , aj+1) a ach a copy o R n . To his shee we a ach copies o Ñ,, along all componen in e als o ,, F j . We con inue in his way, a aching copies o R n along in e als o Tn Fn, excep o hose shee s eached by ei he looping a ound a j o a ound aj+1, in which case we a e o ced o a ach copies o R n along in e als o he o m T,, F n U{a j } (o Tn F  ,U{aj+1}) . Some o hese iden i ica ions a e illus a ed in Figu e 3 . Mo e p ecisely, Figu e 3 shows egions on ou shee s, labeled I, II, III, IV . Shee Iis pas ed o shee II along he edge [aj, aj + 1] . Shee II is pas ed o shee III along edge [q, aj ] and o shee IV along he edge [p, q], whe e p, q a e poin s o F,,, adjacen o aj . The solid and do ed cu es illus a e pa hs om shee I o shee s III and IV espec i ely which (mus ) pass h ough shee III . No ice ha he poin A E E n in he ideal bounda y o shee I is co e ed when shee s II and IV a e pas ed along [p, q] . Simila ly ao E E n is co e ed when II is pas ed o III along [q, aj] (assuming j 0 0 ; o he wise i would be co e ed by some shee a ached o shee IV) . INDESTRUCTIBLE BLASCHKE PRODUCT IN B O  103 4 4 1 4 aj Figu e 3 . Modi ica ions o co e E n Suppose we ha e al eady cons uc ed a su ace S  , whose bounda y consis s o a cs co e ing T o edges o T,, . To each componen in e al I o T , ,n En we a ach copies o R  , as desc ibed abo e . Do his o g,+1 gene a ions . The esul ing shee s co e E n , ( he only shee s which do no co e e e y poin o E n , a e hose a ached along subin e als o he o m (ao, al) o (an_I, an) in he cons uc ion abo e) . We call he esul ing su ace S n , . To he bounda y o S, z a ach copies o R n+ 1 = D Tn+1 along componen in e als o Tn+1 En+1 o hn+1 gene a ions ( his poses no di icul ies since E n , F n and all poin s o he o m a j in he p e ious s age o cons uc ion we e in En+1 ; hus e e y adial in e al in he bounda y o S n has endpoin s in En+l) . The esul ing su ace is called Sn+I and sa is ies he induc ion hypo hesis . The union o e n o hose (nes ed) su aces is deno ed S and we ob ain he desi ed unc ion by mapping he disk o S and hen p ojec ing back o he disk . All ha emains is o choose he sequences {En}, {gn} and {hn} so ha he ha monic measu e es ima es hold . We will i s choose gn , hen En+1 and hen hn+1 Le u be he leas ha monic majo an o log ITa o P(z) 1 .  We wan