Universality of holomorphic functions bounded on closed sets

Universality of holomorphic functions bounded on closed sets L. Bernal-Gonz´alez and A. Bonilla ∗ Abstract In this note, the existence of translation-universal entire functions which are bounded on certain closed subsets is characterized in terms of topological and geometrical properties of such subsets. Corresponding results are also stated in the space of holomorphic functions on the unit disk and in the space of harmonic functions on the plane. Moreover, it is shown the existence of entire functions which are bounded on many rays and, simultaneously, are universal with respect to a prescribed infinite-order differential operator. 1 Introduction and notation Throughout this paper we will use the following notations, most of them being standard: Nis the set of positive integers, Cis the complex plane, Ris the real line, D:= {z∈C:|z|<1}is the open unit disk, B(a, r) (B(a, r)) is the euclidean open (closed, respectively) ball with center a∈Cand radius r > 0. By Awe ∗The first author has been partially supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127 and by DGES Grant BFM2003-03893-C02-01. The second author has been partially supported by MCYT-FEDER Project no. BFM 2002-02098. 2000 Mathematics Subject Classification: Primary 30E10. Secondary 30C99, 31B05, 47A16, 47B38. Key words and phrases: universal function, Arakelian set, bounded holomorphic function, inscribed radius, infinite-order differential operator. Authors addresses: L. Bernal, Departamento de An´alisis Matem´atico, Avda. Reina Mercedes, Apdo. 1160. 41080-Sevilla, Spain. A. Bonilla, Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, C/Astrof´ısico Fco. S´anchez, s/n, 38271-La Laguna, Tenerife, Canary Islands, Spain. E-mails: [email protected],[email protected]. 1 mean the closure in Cof a subset A⊂C. The symbol Tstands for the unit circle {z∈C:|z|= 1}. Moreover, if Gis a domain (:= connected, nonempty open subset) of C(of RN, where N∈N), then H(G) (h(G)) denotes the space of all holomorphic functions f:G→C(of all harmonic functions u:G→R, respectively). It becomes a separable completely metrizable space (hence a Baire space) when it is endowed with the compact open topology (see [27, pages 238–239]). In particular, H(C) is the space of all entire functions. If Fis a closed subset of C(of RN) then A(F) (h(F)) denotes the class of functions f:F→C(u:F→R) which are continuous on F and holomorphic in the interior of F(which are harmonic in some neighbourhood V=Vuof F, respectively). If Gis a domain of Cand F⊂G, then Fis said to be an Arakelian subset of Gif and only if Fis a nonempty, proper, (relatively) closed subset of G, and G∞\Fis connected and locally connected at the ∞-point of G∞:= the one-point compactification of G. We define an Arakelian subset of RNin an analogous way. Finally, a subset Fof a topological space Xis called of first category whenever Fis the union of countable many nowhere dense subsets of X. In 1929 Birkhoff [11] constructed an entire function which is ‘universal’ for translations. In fact, he proved essentially that given b∈C\ {0}there exists a function f∈H(C) such that its sequence of translates {f(·+nb) : n∈N}is dense in H(C). Birkhoff’s theorem can be observed under the point of view of the operator theory as a universality result; namely, if ϕ:C→Cdenotes the translation z7→ z+b then the composition operator Cϕ:f∈H(C)7→ f◦ϕ∈H(C) is universal. In general, if Xis a (necessarily separable) topological vector space and (Tn) is a sequence of operators (= continuous linear selfmappings) on Xthen (Tn) is said to be universal (or hypercyclic) provided that there exists some vector x∈X –called universal for (Tn)– for which the orbit {Tnx:n∈N}of xunder (Tn) is dense in Y. And a single operator is called universal whenever the sequence of iterates (Tn) (that is, T1=T,T2=T◦T, and so on) is universal; in this case it is easy to see that the set of universal vectors is dense. If Xis Baire and metrizable and Tis universal then the set of universal vectors for Tis residual, that is, its complement is of first category. See [26] for a good account about these concepts and their history. 2 Since 1929 many papers have dealt with the subject of universality through translations in one complex variable. Let us make a brief report, now in the language of the universality of operators; see also the survey [26] –specially its Section 4a– which contains a rather complete list of references including domains G6=C,Dand spaces X6=H(G). In 1941 Seidel and Walsh [32] were able to construct a function f∈H(D) which is universal in H(D) with respect to the sequence of composition operators generated by the noneuclidean translates z7→ z+an 1+anz(n∈N), where |an| → 1. In 1976 Luh [29] proved that for a prescribed unbounded sequence (bn)⊂Cthe sequence (Cϕn) is universal on H(C), where ϕnis the translation z7→ z+bn. In 1984 DuyosRuis [18] showed by functional analysis methods that Cϕ(ϕ(z) = z+b, b ∈C\ {0}) is universal on H(C) (hence there is a residual subset of universal functions), while the residuality of the (Cϕn)-universal entire functions –where the ϕnare the above translations– was observed by Grosse-Erdmann [25] and Gethner and Shapiro [23]. In 1988 Zappa [34] considered the universality of functions of H(C\ {0}) with respect to ‘multiplicative’ translations. In 1995 Bernal and Montes [9] characterized the sequences of automorphisms of Cor Dgenerating universal sequences of composition operators. Corresponding results in several complex variables can be seen in a number of papers by Godefroy, Shapiro, Abe, Zappa, Le´on, Prado and the first author (see [24], [1], [2], [28], [10] and [8]). As for translation-universality in the space h(RN), see [4] by Armitage and Gauthier. The universality of the differentiation operator D:f∈H(C)7→ f0∈H(C) was stated by MacLane [30] in 1952. Godefroy and Shapiro [24] (see also [26] and [10] for generalizations and improvements) unified both theorems of Birkhoff and MacLane by showing that any infinite-order differential operator Φ(D) (see Section 2) on H(C) which is not a scalar multiple of the identity is universal. We want to bring here a question concerning the existence of translation-universal functions when boundedness conditions on certain subsets are added. In this setting, the second author [13] defined a universal harmonic function as a function f∈h(RN) 3 such that to each g∈h(RN) corresponds a sequence (an)⊂RNdepending on gsatisfying limn→∞ f(x+an) = g(x) uniformly on compact sets. Among other properties, he proved in [13, Theorem 1] the existence of a universal harmonic function which has strong decay (in particular, it is bounded) on any hyperplane strip. M.C. Calder´on [14, Theorem 2.1] gave an analogous concept for the space of entire functions. She showed the existence of a universal entire function decaying very fast on every strip and on every sector {z: 0 ≤arg z≤α}with α∈(0,2π) (as a matter of fact, she considered the action of certain operators Ton H(C), including the identity operator). Very recently, Costakis and Sambarino [15, Theorem 5] proved that there exists an entire function fwhose translates z7→ f(z+n) (n∈N) are dense in H(C) such that ftends to zero as z→ ∞ on every sector {z:ε≤arg z≤2π(1 −ε)}with ε∈(0,1). They also show (see [15, Theorem 6]) that for a prescribed nowhere dense set E⊂T, there is a D-universal entire function ftending to zero along every ray from the origin passing trough E, that is, limr→+∞f(rt) = 0 for all t∈E. Note that in the concept of universality of [13] and [14] it is equivalent to state that the sequence (an) exists independently of g: indeed, fix a countable dense subset (gk) in h(RN) (or in H(C)) and consider the sequence (bn,k)nwhich performs the approximation to gk; then the adequate sequence (an) is made by joining all terms bn,k (k, n ∈N) in a single sequence. On the other hand, any euclidean (noneuclidean) translation in C(D) is in fact an internal law z7→ z∗a=z+a(z7→ z∗a=z+a 1+az ) making C(D) a topological group, whenever C(D, respectively) is endowed with the euclidean topology. The last two remarks motivate the following definition. Definition 1.1. (a) A plane topological group (PTG) is a topological group (G, ∗), such that Gis a domain of C,Gcarries the euclidean topology and for each a∈G the “translation selfmapping” τa:z∈G7→ z∗a∈Gis holomorphic in D. (b) If (G, ∗) is a PTG and f∈H(G), then we say that fis τ-universal if and only if there exists a sequence (an)⊂Gsuch that the set {f◦τan:n∈N}is dense in H(G). (c) If u:RN→Ris harmonic, then uis called τ-universal whenever there is a sequence (an)⊂RNsatisfying that the set {u(·+an) : n∈N}is dense in H(RN). Observe that we are considering “right” translations τa. Of course, one could give 4 analogous concepts (and obtain analogous results) by considering “left translations” if these are defined in a suitable way. We refer to, for instance, the book [6] for the fundamentals about topological groups. We recall that if (X, ∗) is a metrizable topological group –as, for instance, a PTG– then there exists a distance don X generating its topology such that dis translation-invariant, that is, d(x∗a, y ∗a) = d(x, y) for all x, y, a ∈X. Turning to the question of the boundedness of universal functions, the aim of this paper is to furnish necessary and sufficient conditions for the existence of τ-universal holomorphic functions on a PTG which in addition are bounded on certain prefixed subsets of the domain G, mainly G=C,D. A corresponding result for harmonic functions will be also provided. This will be performed in Section 3. In Section 4 we will deal with the same question for infinite-order differential operators, so completing the above mentioned result on D-universality due to Costakis and Sambarino. Section 2 is devoted to give several results that will reveal useful later, together with some additional terminology. 2 Some auxiliary results In [7] a geometrical notion was used to characterize the universality of certain sequences of differential operators. Now, we state such notion in a more general setting. Recall that if (X, d) is a metric space then the open (closed) d-ball with center a∈Xand radius r > 0 is Bd(a, r) = {x∈X:d(x, a)< r}(Bd(a, r) = {x∈X:d(x, a)≤r}, respectively). Recall also that if A⊂Xthen its diameter is δ(A) = sup{d(x, y) : x, y ∈A}. Definition 2.1. If (X, d) is a metric space and Ais a nonempty subset of Xthen the inscribed radius (or Tchebychef radius) of Ais defined as the number ρd(A)∈[0,+∞] given by ρd(A) = sup{r > 0 : there exists a closed ball Bof radius rwith B⊂A}. It is clear that if Ais open in Xthen ρd(A)>0. When dis the euclidean distance 5 on Cor RNthen the subscript din ρdand in the d-balls will be dropped. Now, we state the following auxiliary geometrical result to be used later. Lemma 2.1. Let (X, d)be a connected metric space with δ(X) = +∞. If Ais a subset of Xwith ρd(A)=+∞and Bis any closed d-ball, then ρd(A\B)=+∞. Proof. Here we will also delete the subscript d. Hence we have that ρ(A) = +∞and that B=B(a, R) for certain a∈X, R > 0. Since δ(X)=+∞, given x0∈Xwe have that the mapping x∈X7→ d(x, x0)∈[0,+∞) is not bounded. But dis continuous and Xis connected. Therefore, given x0∈Xand r > 0, there exists x∈Xsuch that d(x, x0) = r. Fix M > 0. We are looking for a point c∈Xwith B(c, M)⊂A\B. By hypothesis, there exists b∈Xsatisfying B(b, 3M+ 2R+ 1) ⊂A. At this point we distinguish two cases. If d(a, b)> M +Rand c:= bthen B(c, M)⊂B(b, 3M+2R+1) and, by the triangle inequality, B(c, M)∩B(a, R) = ∅, whence we are done. If, on the contrary, d(a, b)≤M+R, then again by the triangle inequality we obtain B(a, R)⊂B(b, M + 2R). Choose c∈Xwith d(b, c)=2M+ 2R+ 1. We claim that B(c, M)⊂A\B. Indeed, a further use of the triangle inequality gives B(c, M)⊂ B(b, 3M+ 2R+ 1) ⊂A. Finally, by way of contradiction, let us suppose that there exists some point x∈B(c, M)∩B. Then d(x, c)≤Mand d(x, a)≤R, hence d(c, a)≤M+R, which yields d(b, c)≤d(c, a)+d(b, a)≤2M+2R, that is absurd. Next, we enunciate the Arakelyan theorem that can be found in [19, pages 153– 154]. Lemma 2.2. If Fis a relatively closed subset of a domain Gin C, then Fis an Arakelian subset if and only if for every g∈A(F)and ε > 0there is a holomorphic function fin Gsuch that |f(z)−g(z)|< ε for all z∈F. We now establish as lemmas two powerful results about tangential approximation of holomorphic or harmonic functions. The first one is a variant for G=Cof the Arakelian theorem, see [3] or [19, pages 160–162]. The second one is a harmonic version by Armitage and Goldstein of this theorem, see [5, Theorem 1.1] or [20, 6 Corollary 5.10]. If x= (x1, . . . , xN)∈RNthen kxkdenotes its norm kxk= (x2 1+ · · · +x2 N)1/2. Lemma 2.3. If Fis a closed subset of C, then Fis an Arakelian subset if and only if for every g∈A(F)and every continuous function ε: [0,+∞)→(0,+∞)with Z+∞ 1 t−3/2log(1/ε(t)) dt < +∞, there is an entire function fsuch that |f(z)−g(z)|< ε(|z|)for all z∈F. Lemma 2.4. If Fis an Arakelian subset of RN, then for each v∈h(F)and each choice of positive numbers a and ε, there exists u∈h(RN)such that |u(x)−v(x)|< ε(1 + kxk)−afor all x∈F. The following result is purely topological and its content can be found in [21, Section 5, pages 242–243]. Given a relatively closed subset Fof a domain G⊂C, we set b F:= F∪c(F), where c(F) denotes the union of the connected components of G\Fhaving compact closure in G, that is, the union of the ‘holes’ of F. So c(F) = ∅ if Fhas no holes. Lemma 2.5. Let G⊂Cbe a domain and Fbe a relatively closed subset of G. Then we have: (a) Fis Arakelian in Gif and only if c(F) = ∅and c(F∪K)is relatively compact in Gfor every compact subset Kof G. (b) If Fis Arakelian in Gand K⊂Gis compact then \ F∪Kis Arakelian in G. (c) If Fis Arakelian in Gand Kis a compact subset of Gwith connected complement such that F∩K=∅, then F∪Kis Arakelian in G. To finish, recall that if Φ(z) = P∞ k=0 akzkis an entire function of exponential type –that is, there are positive finite constants A, B satisfying |Φ(z)| ≤ AeB|z|(z∈C)– then the formal expression Φ(D) = P∞ k=0 akDk(where D0=I= the identity operator) defines in fact an operator –in general, an “infinite-order differential operator”– on H(C). The entire function Φ is said to be of subexponential type whenever given ε > 0 there is a constant A=Aε∈(0,+∞) satisfying |Φ(z)| ≤ Aeε|z|(z∈C). In other words, Φ is of exponential (subexponential) type if and only if it has either growth order <1 or growth order 1 and finite growth type (it has either growth 7 order <1 or growth order 1 and growth type 0, respectively). Obviously, if Φ is of subexponential type then it is also of exponential type. The content of the following lemma can be found in [12]. Lemma 2.6. An entire function Φ(z) = P∞ k=0 akzkis of exponential type if and only if limk→∞(k!|ak|)1/k = 0. 3 Bounded τ-universality In this section, our results about the existence of a τ-universal function which is bounded on a certain prescribed subset Fwill be listed and proved. The inscribed radius will play a crucial role. Specifically, a necessary condition for such existence on general PTGs is shown in Theorem 3.1. In Theorem 3.3, a complete geometrical and topological characterization of the possible subsets Fis given for C. If we assume that Fis Arakelian, a similar statement of existence is shown for Din Theorem 3.4. Finally, a corresponding result of ‘bounded’ τ-universality in the setting of harmonic functions on RNis stated in Theorem 3.5. Theorem 3.1. Let (G, ∗)be a PTG. Assume that dis a distance on Gsatisfying the following two properties: (i) dis translation-invariant and generates the topology of G. (ii) Every closed d-ball is compact. Let Fbe a nonempty, proper subset of G. If there exists a τ-universal function f∈H(G)such that fis bounded on Fthen ρd(G\F)=+∞. Proof. Suppose, by way of contradiction, that R:= ρd(G\F)<+∞and that there exists a τ-universal function f∈H(G) which is bounded on F. Then there exist a sequence (an)⊂Cand a constant M∈(0,+∞) such that the sequence (f◦τan) of translates of fis dense in H(G) and |f(z)| ≤ Mfor all z∈F. We have that Bd(an, R + 1) ∩F6=∅for all n∈N. Observe that due to (i) it holds that Bd(a, r) = {z∗a:z∈Bd(e, r)}(a∈C, r > 0), where eis the neutral element of (G, ∗). Therefore we can find a sequence (bn)⊂Bd(e, R + 1) with bn∗an∈Ffor 8 all n∈N. Let us consider the constant function g(z) := M+ 1. Then there exists a sequence {n(1) < n(2) <· · ·} ⊂ Nsuch that f◦τan(j)→gas j→ ∞ uniformly on compacta in G. Thus, (ii) yields that, in particular, lim j→∞ sup z∈Bd(e,R+1) |f(z∗an(j))−M−1|= 0, which is absurd, since |f| ≤ Mon Fand we have for any n∈Nthat sup z∈Bd(e,R+1) |f(z∗an)−M−1| ≥ |f(bn∗an)−M−1| ≥ M+ 1 − |f(bn∗an)| ≥ 1. This concludes the proof. Remarks 3.2. 1. If in particular we set G=C, Theorem 3.1 shows that if there exists some translation-universal entire function that is bounded on a prefixed set F, then ρ(C\F) = +∞; indeed, take as ∗the usual sum and as dthe euclidean distance. With an analogous proof, the last theorem also holds for h(RN) when RNis endowed with the ordinary sum and with the euclidean distance. If G=Dand the disk is endowed with the hyperbolic (or Poincar´e) distance dP(z, w) := tanh−1|z−w 1−zw |(see [17] for a quite complete description) and with the internal law z∗w=z+w 1+zw (z, w ∈D), then Theorem 3.1 also applies (recall that dPis invariant under the automorphisms of Dand generates the usual topology on D, and that the dP-balls are euclidean balls in D; specifically, BdP(a, R) = B((1−tanh2R)a 1−|a|2tanh2R,(1−|a|2) tanh R 1−|a|2tanh2R) for all a∈Dand all R∈(0,+∞)) yielding that if Fis a subset of Dand there exists a translationuniversal function f∈H(D) which is bounded on F, then ρdP(D\F) = +∞. 2. We illustrate with an example in Cthat some kind of restriction on the subset F is necessary: The set F:= C\Sn≥3B(2n, n) is a closed subset which is not contained in any Arakelian subset of C, and satisfies ρ(C\F) = +∞; an application of the Maximum Modulus Principle shows that if an entire function fis bounded on Fthen it must be bounded on C, so fis constant and therefore it cannot be τ-universal. 3. Condition (ii) in Theorem 3.1 cannot be derived from (i). For instance, the distance d(z, w) := |z−w| 1+|z−w|is a translation-invariant distance on the PTG (C,+) generating the topology of C, but Bd(0,1) = C, which is not compact in C. 9 Thus, we would be done if we were able to find a function f∈ A which is also universal for T. Note that the null function is in Aand that Acan be written as the intersection of the open sets Ak:= {g∈H(C) : |g(z)|< ε(|z|) for all z∈B(0, k)∩F}(k∈N). Consequently, Ais a nonempty Gδsubset of H(C). Hence, by Alexandroff’s theorem, Aendowed with the topology inherited from H(C) is a completely metrizable space, so a Baire space. Thus, we can apply Baire’s category arguments to prove that there exists a residual (so nonempty) subset in Aconsisting of universal functions for T. For this, observe that A∩{T−universal functions}is the intersection of the sets E(s, j, m) = [ n∈N {g∈ A :|Tng−hj(z)|<1 sfor all z∈B(0, m)}(s, j, m ∈N), where {hj}j∈Nis a countable dense subset of HAand A:= Φ−1({|z|>1}). Therefore, it is enough to see that each set E(s, j, m) is open (this is easy by the continuity of each operator Tn) and dense in A. In order to prove the denseness, fix s, j, m ∈N together with a function g0∈Aand numbers δ, R > 0. Because of the nature of the topology of Awe must find a positive integer nand a function f∈ A so that sup |z|≤R |f(z)−g0(z)|< δ and sup |z|≤m |(Tnf)(z)−hj(z)|<1/s. (7) We may assume without loss of generality that R≥m. Let γbe defined as γ= inf z∈F∩B(0,R+2) {ε(|z|)− |g0(z)|} and choose any constant βsuch that 0 < β < min{δ, γ}; note that this implies that we also have β < infz∈F∩B(0,R+2) ε(|z|). Since HBwhere B:= Φ−1({|z|<1}) is dense in H(C), we may find a function q0∈HBsuch that sup |z|≤R+1 |q0(z)−g0(z)|< β/4.(8) Recall that hj∈HA. From Lemma 4.1 and from the facts that q0∈HBand Tnea= Φn(a)ea→0 (n→ ∞) for all a∈B, it is derived the existence of an n∈Nand of an entire function Hsuch that TnH=hjon C, and |H(z)|< β/4 and |Tnq0(z)|<1/2sfor all z∈B(0, R + 1).(9) 16 Let q:= q0+H. Then, by the triangle inequality, sup |z|≤R+1 |q(z)−g0(z)|< β/2.(10) Since Tn= Φn(D) and Φnis also of subexponential type, by Lemma 2.6 we derive that |ak| ≤ M(1/2)k/k! (k≥0) for some finite constant M > 0, where the ak’s are the Taylor coefficients of Φn. If we use Cauchy’s estimates then for every z∈B(0, R) and every h∈H(C) we obtain that |(Tnh)(z)| ≤ ∞ X k=0 |akh(k)(z)| ≤ M ∞ X k=0 (1/2)k k!k!sup|w−z|=1 |h(w)| 1k≤2Msup B(0,R+1) |h|. Consequently, if his entire and supB(0,R+1) |h|<1/(4Ms) =: β1, then sup |z|≤R |(Tnh)(z)|<1/2s. (11) Finally, set e F:= B(0, R + 1) ∪Fand define the function g:e F→Cby g(z) =    q(z) if z∈B(0, R + 1) 0 if z∈ {|w| ≥ R+ 2} ∩ F (1 −t)q(z) if z∈ {|w|=R+1+t} ∩ F, 0≤t≤1. Observe that e Fis an Arakelian set and that gis continuous on it and holomorphic in its interior. It is elementary to construct a continuous positive function ε1(t) on [0,+∞) such that ε1(t)≤ε(t) for all t≥0, ε1(t)<min{β/2, β1}for 0 ≤t≤R+ 2 and such that ε1(t) still satisfies the integrability condition given in Lemma 2.3. Consequently, there exists an entire function fwith |f(z)−g(z)|< ε1(|z|) for every z∈e F. 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