Maximal cluster sets of L-analytic functions along arbitrary curves

Maximal cluster sets of L-analytic functions along arbitrary curves L. Bernal-Gonz´alez, A. Bonilla, M.C. Calder´on-Moreno and J. A. Prado-Bassas∗ Abstract Let Ω be a domain in the N-dimensional real space, Lbe an elliptic differential operator, and (Tn) be a sequence whose members belong to a certain class of operators defined on the space of L-analytic functions on Ω. It is proved in this paper the existence of a dense linear manifold of L-analytic functions all of whose nonzero members have maximal cluster sets under the action of every Tnalong any curve ending at the boundary of Ω such that its ω-limit does not contain any component of the boundary. The above class contains all partial differentiation operators ∂α, hence the statement extends earlier results due to Boivin, Gauthier and Paramonov, and to the first, third and fourth authors. 2000 Mathematics Subject Classification: Primary 30D40. Secondary 30E10, 31B35, 41A30, 47F05. Key words and phrases: maximal cluster set, L-analytic function, dense linear manifold, admissible path, elliptic operator, internally controlled operator. ∗The first, third and fourth authors have been partially supported by the Plan Andaluz de Investigaci´on de la Junta de Andaluc´ıa FQM-127 and by MCyT Grants BFM2003- 03893-C02-01 and MTM2004-21420-E. The second author has been partially supported by MCYT-FEDER Project no. BFM2002-02098. 1 1 Introduction The behavior of the complex-valued functions near the boundary of the domain where they are defined has attracted the attention of many mathematicians. Such a behavior can be considered either globally or restricted to certain subsets (mainly, adequate curves) near the boundary, and is defined by the so-called cluster sets, see below. For background about classical results on cluster sets, we refer to [5] and [9]. Next, we fix some standard notation that will be used throughout this paper. The symbols R,C,N,N0will stand for the real line, the complex plane, the set of positive integers, and the set N∪ {0}, respectively. If N∈N then RNis the N-dimensional real space; specially, C=R2. If A⊂RN then A0(∂A, resp.) represents its interior (its boundary, resp.) in RN. In addition, we set Ac:= RN\A, as usual. The open ball with center a∈RNand radius r > 0 –with respect to the euclidean distance don RN– is B(a, r). By Ω we denote a domain in RN, that is, a nonempty connected open subset of RN. Moreover, Ω∗will denote the one-point compactification of Ω. A Jordan domain is a domain Ω ⊂Cwhose boundary in C∗is a topological image of the unit circle {z∈C:|z|= 1}. If A⊂RNand fis a complex-valued function defined on Athen kfkA:= supA|f|. For a multiindex α= (α1, . . . , αN)∈NN 0, we let |α|=α1+· · · +αN,α! = α1!· · · αN!, xα=xα1 1· · · xαN Nfor x= (x1, . . . , xN)∈RNand ∂α= ( ∂ ∂x1)α1· · · (∂ ∂xN)αN. Let r, N ∈Nwith N≥2 and let L(ξ) = P|α|=raαξα(ξ∈RN) be a fixed homogeneous polynomial of degree rwith complex constant coefficients and which satisfies the ellipticity condition L(ξ)6= 0 (ξ∈RN\{0}). We associate to Lthe homogeneous elliptic operator of order rgiven by L=L(∂) = X |α|=r aα∂α. The symbol LN rstands for the class of all homogeneous elliptic operators of order rin RNwith constant complex coefficients. If Ω ⊂RNis a domain and f: Ω →Cis a C∞-function, then fis called L-analytic (or L-holomorphic) on Ω if it satisfies the equation L(∂)f= 0 on Ω. We denote by L(Ω) the linear space of L-holomorphic functions on Ω endowed with the compact-open topology. For instance, if N= 2 then the space of 2 ∂-holomorphic functions is the space H(Ω) of holomorphic functions in the usual sense, and for arbitrary N, ∆-holomorphic is the same as harmonic in RNif ∆ denotes the Laplace operator. In general, L(Ω) turns out to be a Fr´echet space (see [12]), which is separable because it is a subspace of C(Ω), the space of complex continuous functions on Ω, also endowed with the compact-open topology. By an operator on L(Ω) we mean a continuous linear selfmapping T:L(Ω) →L(Ω). Let Ω be a domain in RN,N≥2, Ω 6=RNand γ: [0,1) →Ω be a curve in Ω tending to ∂Ω, that is, γis continuous and, given a compact set K⊂Ω, there is t0=t0(K)∈(0,1) for which K∩γ([t0,1)) = ∅. Then the ω-limit (or oscillation set) of γis the set Osc (γ) of points in ∂Ω which are in the closure of γ([0,1)). Let b∈∂Ω. Following [3, Section 6], we shall say that a continuous path γ: [0,1] →RNis admissible for Ω with end point bif γ([0,1)) ⊂Ω and γ(1) = b. Note that, in particular, if ∂Ω has no connected component consisting of a single point then for any admissible path γwe have that Osc (γ) contains no component of ∂Ω. By abuse of language, we sometimes identify γ=γ([0,1)). Given a continuous function f: Ω →Cand a curve γin Ω tending to its boundary, we denote by Cγ(f) the cluster set of falong γ, that is, Cγ(f) = {w∈C∗: there exists a sequence {tn} ⊂ [0,1) such that tn→ 1 and f(γ(tn)) →was n→ ∞}. In [3, Theorem 5], Boivin, Gauthier and Paramonov proved that if L∈ LN r and Ω ⊂RNis a domain with Ω 6=RNsuch that ∂Ω has no component consisting of a single point, then there is at least one function g∈L(Ω) with the property that for each b∈∂Ω, each admissible path γending at band each α∈NN 0, one has Cγ(∂αg) = C∗, that is, each such cluster set is maximal. On the other hand, in [1, Theorem 2.1] it is shown that if Ω is a Jordan domain in Cthen there is a dense linear manifold D ⊂ H(Ω) all of whose nonzero members fsatisfy that the cluster set of falong γis maximal for every curve γin Ω tending to the boundary with Osc (γ)6=∂Ω. In this paper, we state a theorem (see Section 4) that unifies and extends largely both results above; our theorem will be valid for a large class of operators on L(Ω) which will be introduced in Section 2. Section 3 is devoted to present some additional terminology together with a preparatory result about approximation. 3 2 A new class of operators We collect in the following definition the adequate class of operators to be used later. Definition 2.1. Let r, N ∈Nwith N≥2 and Ω ⊂RNbe a domain. Assume that L∈ LN rand that Tis an operator on L(Ω). We say that Tis internally controlled if given ε > 0 and a pair of compact sets F, G ⊂Ω with F⊂G0, there exists δ=δ(ε, F, G)>0 such that [f∈L(Ω) and kfkG< δ] implies kTfkF< ε. Note that the notion of “internal control” is stronger than the mere continuity. In fact, the former requires in particular the continuity of every mapping T1,2:X1→X2, where Xi(i= 1,2) is the normed space consisting of all functions of L(Ω) under the respective norms k · kF,k·kGand (F, G) is any pair of compact subsets of Ω with F⊂G0, while the continuity of T only requires to fix a compact subset Fand find some compact set Gin Ω such that the corresponding mapping T1,2is continuous. Another property to be imposed in Section 3 on an operator Ton L(Ω) will be that ran T(:= T(L(Ω))) contains the constant functions. Of course, this is equivalent to 1 ∈ran T. Let us provide some examples of “classical” operators that satisfy some of the above conditions. Examples 2.2. 1. If L∈ LN rand Ω is a domain in RNthen from known estimations about the norms of the solutions of a homogeneous elliptic partial differential equation (see for instance [10, pages 188–189, Lemma 1]) it is deduced that each operator ∂αis internally controlled on L(Ω). On the other hand, in [3, Lemma 3] it is proved that 1 ∈ran L. 2. In particular, by Example 1 –or by using Cauchy’s integral formula for derivatives– we obtain for N= 2 and L=∂that each derivative operator Dn(n∈N0) is internally controlled on H(Ω). Here D0f=fand Dn+1f= (Dnf)0for every n∈N0and every f∈H(Ω). More, if Φ(z) = P∞ n=0 anzn is an entire function of subexponential type (that is, given ε > 0 there is a constant A∈(0,+∞) such that |Φ(z)| ≤ Aexp(ε|z|) for all z∈C), then its associated (in general, infinite order) differential operator Φ(D) = P∞ n=0 anDnis well defined and internally controlled on H(Ω). This is easy to see just by taking into account that Φ is of subexponential type if and only if 4 limn→∞(n|an|1/n) = 0 (see for instance [2]). Moreover, 1 ∈ran Φ(D) as soon as Φ 6≡ 0. Indeed, let m= min{n∈N0: Φ(n)(0) 6= 0}; then Φ(D)(h) = 1 if h(z) := zm/Φ(m)(0). 3. Again in the classical holomorphic case, if α∈H(Ω) then its associated multiplication operator Mα:f7→ αf is always internally controlled on H(Ω). And 1 ∈ran Mαif and only if αhas no zeros in Ω. On the contrary, if ϕ: Ω →Ω is a holomorphic selfmapping, then the composition operator Cϕ:f∈H(Ω) 7→ f◦ϕ∈H(Ω) is internally controlled only if ϕis the identity. We have that 1 ∈ran Cϕfor any holomorphic selfmapping ϕ. 4. We know that internal control implies continuity. In fact, the former property is strictly stronger. For instance, the Volterra operator Von H(C) given as (V f)(z) = Rz 0f(t)dt is easily seen to be not internally controlled. Observe that, in addition, 1 6∈ ran Vbecause (V f)(0) = 0 for every f∈ H(C). 5. The family of internally controlled operators is a vector algebra in the space of all operators on L(Ω), that is, if α, β are scalars and T, S are internally controlled operators on L(Ω), then the operators αT +βS and T◦Sare internally controlled too. Indeed, this is evident for αT +βS. As for the composition T◦S, fix a pair (F, G) = (F1, F2) of compact sets as well as a number ε > 0 as in Definition 2.1. Then choose a compact set F3with F1⊂F0 3⊂F3⊂F0 2, for instance, F3:= Sx∈F1B(x, α/2), where α=d(F1, Fc 2). Now apply that Tis internally controlled to the pair (F3, F2) and, finally, apply that Sis internally controlled to (F1, F3). 3 Some auxiliary statements Following [4], a relatively closed subset Fin Ω will be called a Roth- Keldysh-Lavrentiev set, or more simply an Ω-RKL set, if Ω∗\Fis connected and locally connected. The following lemma (see [3] and [6]) will reveal useful in the proof of our main result. Note that Arakelian’s theorem (see [7]) covers the assertion for N= 2 in the holomorphic case. Lemma 3.1. Let Ωbe a domain in RN,N≥2and Fbe a Ω-RKL set. Assume that fbe L-analytic in some neighbourhood of Fin Ω. Then, for each ε > 0, there exists g∈L(Ω) such that kf−gkF< ε. 5 In order to simplify the notation, we denote by Othe family of domains Ω⊂RNsuch that Ω 6=RNand the boundary of Ω in (RN)∗has no connected components that consist of a single point. Moreover, for each domain Ω as before, we denote by Γ(Ω) the family of curves γ⊂Ω tending to the boundary whose ω-limit does not contain any component of the boundary. Then, we establish the following technical lemma. Lemma 3.2. Assume that Ω∈ O. Then there are three sequences (Kj), (Gj),(Fj)of compact subsets of Ωsatisfying the following properties: (i) Ω = S∞ j=1 Kjand Kj⊂K0 j+1 (j∈N). (ii) The Gj’s are pairwise disjoint and Gj⊂Kjfor all j∈N. (iii) For every j∈None has that Kj∩Gj+1 =∅and Kj∪Gj+1 is an RKL-set. (iv) Fj⊂G0 j(j∈N)and every γ∈Γ(Ω) intersects all sets Fj’s except possibly finitely many of them. The content of the last lemma is essentially proved in [3, Proof of Theorem 5], so we omit a detailed proof. Suffice it to say that (i), (iii) and the first part of (ii) and of (iv) are obtained in the cited proof, in which each Gjis constructed as a thin (closed) neighborhood of Fj, which in turn is a subset of Kj. In order to obtain the contention Gj⊂Kjof (ii), it suffices to replace each Kjby a set e Kjslightly larger. The sets Kj, Gj, Fjin [3] are closed, but in addition they are constructed by using the level sets of an adequate real analytic function Ψ that satisfies an inequality [3, inequality (6) on page 960], which produces that the above closed sets are bounded; hence they are compact. Finally, the last part of (iv) is shown in [3] for admissible curves, but the same holds for every γ∈Γ(Ω) just by using the same argument of [3, page 962]. 4 The main result Once the basic terminology and background have been established, we are ready to present our main assertion. Theorem 4.1. Let Ω∈ O. Assume that Tn:L(Ω) →L(Ω) (n∈N)is a sequence of operators satisfying the following conditions: 6 (a) Each Tnis internally controlled. (b) For each n∈N,ran Tncontains the constants. Then there exists a dense linear manifold Din L(Ω) such that for every g∈ D \ {0}, every n∈N, and every curve γ∈Γ(Ω), we have Cγ(Tng) = C∗. Proof. From Lemma 3.2, we can choose sequences (Kj),(Fj),(Gj) satisfying properties (i) to (iv) in such lemma. For every pair (k, j)∈N2, we set the disjoint union Mk,j := Kk+j−1∪Gk+j, which is an RKL-set by (iii). Moreover, we can fix sequences {qj:j∈N},{Qk:k∈N}which are dense in Cand L(Ω), respectively. Let us pick countably many pairwise disjoint infinite sets J(k) (k∈N) such that J(k)⊂ {j∈N:j > k}for all k∈Nand, in addition, the sets J(k)−k:= {j−k:j∈J(k)}(k∈N) are also mutually disjoint. This can be made in many ways; for instance, we may choose J(k) = {3ν−k2:ν > k}. We set N:= Sk∈N(J(k)−k). In turn, we divide each J(k) into infinitely many strictly increasing sequences I(k, n) = {p(k, n, l) : l∈N}(n∈N). Then we have the disjoint union N:= Sk,n∈N(I(k, n)−k). For every pair (k, n)∈N2, the mapping ϕ(k, n, ·) : j7→ l–where lis the unique positive integer with k+j=p(k, n, l)– is a strictly increasing bijection from I(k, n)−k onto N. Therefore, trivially, the set {qϕ(k,n,j):j∈I(k, n)−k}is dense in C. Given j∈ N, there exist a unique k=k(j)∈Nand also a unique n=n(j)∈Nsuch that j∈I(k, n)−k⊂J(k)−k. From (a), there is δj>0 such that kTnhkFk+j<1 2jfor all h∈L(Ω) with khkGk+j< δj.(1) With no loss of generality, we may assume that the sequence (δj) is strictly decreasing and tends to zero. Then we can define τj:= δj−δj+1 >0 (j≥1). On the other hand, thanks to (b), there exists a sequence (Φn)⊂L(Ω) satisfying TnΦn= 1 for all n∈N.(2) Fix now a number k∈N. Inspired by the approach of [11] (see also [8]), we define gk,0:= Qkand fk,1:Mk,1→Cas fk,1(x) =    gk,0(x) for every x∈Kk qlΦn(x) for every x∈Gk+1 if k+ 1 = p(k, n, l)∈J(k) 0 for every x∈Gk+1 if 1 /∈J(k)−k. 7 Notice that fk,1is well defined because if 1 ∈J(k)−kthen there is a unique n∈Nwith 1 ∈I(k, n)−k; so k+ 1 = p(k, n, l) for a unique l∈N, namely, l=ϕ(k, n, 1). Observe also that fk,1is L-analytic in some neighborhood of Mk,1. By Lemma 3.1, there is a function gk,1∈L(Ω) such that kgk,1−fk,1kMk,1<τ1 2k. From this step, we proceed by induction. Define for j≥2 the function fk,j :Mk,j →Cby fk,j(x) =    gk,j−1(x) for every x∈Kk+j−1 qlΦn(x) for every x∈Gk+jif k+j=p(k, n, l)∈J(k) 0 for every x∈Gk+jif j /∈J(k)−k. Again, fk,j is L-analytic in some neighborhood of the RKL-set Mk,j. Hence Lemma 3.1 guarantees the existence of a function gk,j ∈L(Ω) such that kgk,j −fk,jkMk,j <τj 2k.(3) Let us consider the series gk,0+ ∞ X j=1 (gk,j −gk,j−1).(4) For a prescribed compact subset K⊂Ω, there is m∈Nsuch that K⊂ Kk+m−1. Then for j≥mone has that kgk,j −gk,j−1kK≤ kgk,j −fk,jkKk+j−1≤ kgk,j −fk,jkMk,j < τj, due to (3). But Pj≥1τj=δ1<∞, therefore (4) converges uniformly on compacta in Ω. Thus, its sum defines a function gkbelonging to L(Ω). Notice that, from the shape of (4), we have that gk= lim ν→∞ gk,ν =gk,j + ∞ X ν=j+1 (gk,ν −gk,ν−1) for all j∈N.(5) Finally, we define the desired linear submanifold Din L(Ω) as the linear span D= span {gk:k∈N} 8 Let us fix a function h∈L(Ω), a compact set K⊂Ω and a number ε > 0. Then by the exhaustion property (i) we get an m∈Nwith K⊂Km; so Kk+j−1⊃Kfor all k≥mand all j∈N. Therefore, due to (3), (4) and the definition of fk,j, we obtain for k≥mthat kgk−QkkK=kgk−gk,0kK≤ ∞ X j=1 kgk,j −gk,j−1kKk+j−1 ≤ ∞ X j=1 kgk,j −fk,jkMk,j < ∞ X j=1 τj 2k=δ1 2k. Since {Qk:k∈N}is dense in L(Ω) there exists k≥mwith 2k>2δ1/ε and kQk−hkK< ε/2. Hence, by the triangle inequality, kgk−hkK< ε. Thus, the set {gk:k∈N}is dense in L(Ω). But D ⊃ {gk:k∈N}, so Dis also dense. It remains to show that if a number n∈N, a function g∈ D \ {0} and a curve γ∈Γ(Ω) are prescribed, then Cγ(Tng) = C∗. For this, fix n, g, γ as before. We can write g=c1g1+· · · +ckgkfor certain k∈Nand c1, . . . , ck∈Cwith ck6= 0. Moreover, by the property (iv) we can select a number j0∈Nand points xj∈γ∩Fk+j(j≥j0). Observe that the construction of the set I(k, n)−kguarantees that for any j∈I(k, n)−kthe corresponding k(j), n(j) for which (1) is satified are exactly our prescribed indexes k,n. Consequently, from (2), (5) and the linearity of Tn, we obtain for every j∈I(k, n)−kwith j≥j0that |(Tng)(xj)−ckqϕ(k,n,j)| =¯¯¯¯¯TnÃckgk+ k−1 X i=1 cigi!(xj)−ckqϕ(k,n,j)TnΦn(xj)¯¯¯¯¯ =¯¯¯¯¯ckTn(gk,j −qϕ(k,n,j)Φn)(xj) + ckTnÃ∞ X ν=j+1 (gk,ν −gk,ν−1)!(xj) + k−1 X i=1 ciTn(gi)(xj)¯¯¯¯¯ ≤° ° ° ° °ckTn(gk,j −qϕ(k,n,j)Φn) + ckTnÃ∞ X ν=j+1 (gk,ν −gk,ν−1)!+ k−1 X i=1 ciTngi° ° ° ° °Fk+j 9