A lot of “counterexamples” to Liouville's theorem

TITLE: A LOT OF “COUNTEREXAMPLES” TO LIOUVILLE’S THEOREM. AUTHOR: LUIS BERNAL-GONZ´ ALEZ. AFFILIATION: DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO. FACUL- TAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email p o ec ed]. FOOTNOTES TO THE TITLE: *This wo k is suppo ed in pa by DGICYT g an PB93-0926. A.M.S. Subjec Classi ica ion: P ima y 30D15. Seconda y 30E10, 44A05. Key wo ds and ph ases: Liou ille’s heo em, en i e unc ions, dense linea mani old, Radon ans o m, A akelian se , s ips and sec o s, g ow h index. 1 ABREVIATED TITLE: LIOUVILLE’S THEOREM. NAME AND MAILING ADDRESS OF THE AUTHOR TO WHOM PROOFS SHOULD BE SENT: LUIS BERNAL-GONZ´ ALEZ. DEPARTA- MENTO DE AN´ ALISIS MATEM´ ATICO. FACULTAD DE MATEM´ ATICAS. AVENIDA REINA MERCEDES. APARTADO 1160. 41080 SEVILLA, SPAIN. E-MAIL: lb[email p o ec ed]. 2 A LOT OF “COUNTEREXAMPLES” TO LIOUVILLE’S THEOREM by LUIS BERNAL–GONZ´ ALEZ* Abs ac . We p o e in his pape ha , gi en α∈(0,1/2), he e exis s a linea mani old Mo en i e unc ions sa is ying ha Mis dense in he space o all en i e unc ions and, in addi ion, limz→∞ exp(|z|α) (j)(z) = 0 on any plane s ip o e e y ∈Mand o e e y de i a ion index j. Mo eo e , i is shown he exis ence o an en i e unc ion wi h in ini e g ow h index sa is ying he la e p ope y. 1. INTRODUCTION AND NOTATION One o he mos elemen a y, su p ising and beau i ul esul s in Complex Analy- sis is Liou ille’s heo em: each bounded en i e unc ion is cons an . Ne e heless, i boundedness condi ion is sligh ly weakened ( o ins ance, by allowing boundedness *This wo k is suppo ed in pa by DGICYT g an PB93-0926. 1991 Ma hema ics Subjec Classi ica ion: P ima y 30D15. Seconda y 30E10, 44A05. Key wo ds and ph ases: Liou ille’s heo em, en i e unc ions, dense linea ma- ni old, Radon ans o m, A akelian se , s ips and sec o s, g ow h index. 3 on e e y line), hen noncons an en i e unc ions can be ob ained. Fo his, see o ins ance [7, pp. 9-10], whe e i is e en shown a noncons an en i e unc ion ha ing limi ze o on any line. This is achie ed by using a esul o A. Ro h abou app oxima ion on closed se s. Bu se e al sha pe esul s ha e been ob ained. Recen ly, D.H. A mi age [3] has cons uc ed a noncons an en i e unc ion such ha each de i a i e (n)is in eg able on e e y line lwi h espec o leng h measu e sand ∫l (n)ds = 0 o e e y n∈N0, whe e N0=N∪ {0}and Nis he se o posi i e in ege s. Mo eo e , sa is ies lim z→∞ z∈l (n)(z) = 0 o e e y n∈N0and e e y line l( he e o e each de i a i e is bounded on each line). This is ob ained by an elemen a y pole-pushing echnique. In he same pape i is no ed ha i is a con inuous unc ion on he complex plane C, in eg able on 4 Cwi h espec o plane Lebesgue measu e, and i ∫l ds = 0 (1) o e e y line l, hen is iden ically ze o (see [2] o ins ance). We ema k he e ha he mapping ˆ (l) = ∫l ds (l∈L) is known as he Radon ans o m o . We ha e deno ed by L he se o all s aigh lines in he plane. Thus, we a e dealing wi h he injec i i y o he linea ans o m 7→ ˆ in R2. The co esponding p ob- lem o hype planes in Rnis ex ensi ely s udied in [8] o ce ain unc ion spaces. L. Zalcman [16] in 1982 had indica ed ha i ac ually suffices o (1) o hold only o almos e e y line belonging o a dense se o di ec ions in o de ha ≡0. In he same pape , he cons uc ed –by using a esul (Theo em 1 below) due o A ake- lian conce ning angen ial app oxima ion (see [1, p. 1189]and [6, pp. 160-162])– a noncons an en i e unc ion sa is ying (1) on e e y line l. Nonnull en i e unc- ions ending o ze o on e e y line had al eady been cons uc ed by Mi ag-Leffle [10, pp. 290-294], and D.J. Newman [11] p o ided an explici noncons an en i e unc ion which is bounded on e e y line h ough he o igin. Zalcman also poin ed ou ha , i nea ly no hing is assumed o , hen he e exis nonnull unc ions o 5 which (1) holds: ake =χE, whe e Eis Sie pinski’s nonmeasu able se wi h he p ope y ha any line in e sec s Ein a mos wo poin s (see [15]). All hese esul s can be ca ied in o he se ing o ha monic unc ions on Rn(n≥2) (see [4] and [5]). In his pape we imp o e all abo e asse ions abou en i e unc ions and p o ide a dense linea mani old o en i e unc ions “ iola ing” Liou ille’s heo em. All nonnull unc ions in his mani old ha e in ini e g ow h o de . We can ge e en o a leas one o hese unc ions o ha e ex emely as g ow h. We will need a bi o no a ion and a p elimina y esul . I > 0, B deno es he closed ball {z:|z| ≤ }. A s ip is he plane egion lying be ween wo pa allel s aigh lines. I β∈(0,2π), he sec o sβis he se sβ={z: 0 ≤a g z≤β}(he e a g zis e iden ly being allowed o be ou side he “p incipal in e al” (−π, π] used by some au ho s) . Σ will s and o he amily consis ing o all s ips in Cand all sec o s sβ(β∈(0,2π)). H(C) is he space o all en i e unc ions, endowed wi h he compac -open opology. The ex ended plane C∞is he one-poin compac i ica ion o C. I F⊂Cis a closed se , hen A(F) is he space o all con inuous unc ions 6 on Fwhich a e holomo phic in he in e io o F. A closed subse F⊂Cis said o be an A akelian se [13] whene e C∞ Fis bo h connec ed and locally connec ed a in ini y. We will use he ollowing (abo e men ioned) heo em due o A akelian. THEOREM 1. Assume ha F⊂Cis an A akelian se and ha ε( )is con- inuous and posi i e o ≥0. In addi ion, suppose ha ∫∞ 1 −3/2log ε( )d > −∞.(2) Then o e e y g∈A(F) he e exis s an en i e unc ion such ha | (z)−g(z)|< ε(|z|)∀z∈F. The s a emen does no emain alid o e e y Fi (2) is iola ed. No e ha , o ins ance, ε( ) = exp(− 1/2) does no sa is y (2), bu ε( ) = exp(− α) does o α < 1/2. Finally, le be an en i e unc ion. I > 0, we de ine exp1 = exp , expk+1 = exp(expk ) (k∈N), and µ( ) = max{| (z)|:|z|= }. Fo > 0 7 la ge enough, we deno e log1 = log , logk+1 = log(logk ) (k∈N). The ollow- ing de ini ions can be ound in [12] and [14]. The g ow h k-o de ρk=ρk( ) o is ρk= lim sup →∞ logk+1 µ( ) log . I k= 1, hen ρkis called he o de o , and we shall deno e i by ρ( ). The g ow h index i( ) o is i( ) = min{k∈N:ρk( )<∞}, whe e we se i( ) = ∞when ρk( ) = ∞ o all k. 2. RESULTS Recall ha H(C) is sepa able: he amily {Pn}∞ 1o holomo phic polyno- mials ha ing coefficien s wi h a ional eal and imagina y pa s is an example o a coun able dense subse o H(C). Wi h his in mind, we a e now eady o s a e he ollowing heo em. THEOREM 2. Assume ha α∈(0,1/2). Then he e is a linea mani old M⊂H(C)which is dense in H(C)and sa is ying lim z→∞ z∈S exp(|z|α) (j)(z) = 0 (3) 8 ∀S∈Σ,∀ ∈Mand ∀j∈N0. PROOF. Conside he sequence {Pn}∞ 1men ioned a he beginning o he pa ag aph and ix α∈(0,1/2). Fix also a numbe β∈(α, 1/2). Fo e e y n∈N, he unc ion ε( ) = εn( )≡min{1/n, exp(− β)} is posi i e and con inuous o ≥0 and sa is ies (2). Le Fn=Bn∪En, whe e En={z∈C:|z| ≥ n+ 1 and dis (z, P )≥1} and Pis he pa abolic cu e P={x−ix1/2:x≥0}. Then Fnis closed. In addi ion, a glance o e i s complemen e eals ha Fnis an A akelian se . De ine he unc ion gn:Fn→Cby gn(z) = {Pn(z) i z∈Bn 0 i z∈En. Since Bn∩En=∅,gnis well-de ined. T i ially, gn∈A(En). By Theo em 1, he e exis s an en i e unc ion nsuch ha | n(z)−gn(z)|< ε(|z|)∀z∈Fn, 9 hen he se T(V) = { ∈H(C) : is bounded on V} is o he i s ca ego y, because T(V) = ∞ ∪ 1 Cnwi h Cn={ ∈H(C) : | (z)| ≤ non V}and each Cnis closed and has emp y in e io ( o his, no e ha Cn∩ {noncons an polynomials}=∅and ha he second se in he in e sec ion is dense in H(C)). Finally, obse e ha AL ⊂∩{T( ) : is a ay om he o igin}. So AL is, in his sense, e y small. REFERENCES 1. N.V. ARAKELIAN, Uni o m app oxima ion on closed se s by en i e unc ions, Iz . Akad. Nauk SSSR Se . Ma . 28 (1964), 1187-1206. 2. 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MITTAG-LEFFLER, Su la ep ´esen a ion analy ique d’une b anche uni- o me d’une onc ion monog`ene (Sixi`eme no e), Ac a Ma h. 42 (1920), 285-308. 11. D.J. NEWMAN, An en i e unc ion bounded in e e y di ec ion, Ame . Ma h. Mon hly 83 (1976), 192-193. 12. A.R. REDDY, On en i e Di ichle se ies o in ini e o de , Re . Ma . Hisp.- Ame . 27 (1967), 120-131. 13. J.P. ROSAY and W. RUDIN, A akelian’s app oxima ion heo em, Ame . Ma h. Mon hly 96 (1989), 432-434. 14. D. SATO, On he a e o g ow h o en i e unc ions o as g ow h, Bull. Ame . Ma h. Soc. 69 (1963), 410-414. 15. W. SIERPINSKI, Su un p obl`eme conce nan les ensembles mesu ables supe - 17 iciellemen , Fund. Ma h. 1(1920), 112-115. 16. L. ZALCMAN, Uniqueness and nonuniqueness o he Radon ans o m, Bull. London Ma h. Soc. 14 (1982), 241-245. DEPARTAMENTO DE AN´ ALISIS MATEM´ ATICO FACULTAD DE MATEM´ ATICAS AVENIDA REINA MERCEDES APARTADO 1160 41080 SEVILLA, SPAIN E-MAIL: lb[email p o ec ed] 18