Continuity and convergence properties of extremal-interpolating disks

Publicacions Matem`atiques, Vol 39 (1995), 335–347. CONTINUITY AND CONVERGENCE PROPERTIES OF EXTREMAL-INTERPOLATING DISKS Pascal J. Thomas Abstract Let abe a sequence of points in the unit ball of Cn. Eric Amar and the author have introduced the nonnegative quantity ρ(a)= infαinfkj:j=kdG(αj,α k), where dGis the Gleason distance in the unit disk and the first infimum is taken over all sequences αin the unit disk which map to aby a map from the disk to the ball. The value of ρ(a) is related to whether ais an interpolating sequence with respect to analytic disks passing through it, and if ais an interpolating sequence in the ball, then ρ(a)>0. In this work, we show that ρ(a) can be obtained as the limit of the same quantity for the truncated finite sequences, and that ρ(a) depends continuously on awhen ais finite. Furthermore, we describe some of the behavior of the minimizing sequences of maps involved in the extremal problem used to define ρ. 0. Introduction. This article is being written to provide some additional properties of the notion of extremal disks introduced in the joint paper [A-T]. It would not have arisen without the stimulating discussions I had with Eric Amar, for which I wish to thank him. Some of the genesis of this work took place while I was enjoying the hospitality of the University of California at Los Angeles and the University of Wisconsin-Madison, and my thanks go to them as well. First we recap and simplify a few notations from [A-T]. As usual, Ddenotes the unit disk in the complex plane, and Bnthe unit ball in Cn:Bn:= {z∈Cnst z·¯z=n 1|zj|2<1}. Let a={ak,k∈Z∗ +}⊂Bn,α={αk,k∈Z∗ +}⊂D, possibly finite sequences. Given φa holomorphic map from Dto Bn, we shall employ the abbreviated notation φ(α)=ato mean that for all k∈Z∗ +,φ(αk)=ak. 336 P. J. Thomas When αis such that a φas above does exist, we say that αmaps to a, and write α→ afor short. It means that the map φdescribes an analytic disk passing through the points of the sequence awithout “slopping over” the boundary of the ball. Definition [A-T]. For a⊂Bn,n≥1, k∈Z∗ +, ρk(a) := inf{δk(α):α→ a}; and ρ(a) := inf{δ(α):α→ a}, where δk(α):=  j:j=k dD G(αk,α j), δ(α) := inf kδk(α), and, for zand win a domain Ω, dΩ Gdenotes the Gleason distance: dΩ G(z,w) := sup{|f(w)|:f∈H∞(Ω) st f(z) = 0 and f∞≤1}. Note that, for any nonnegative integer n, 1−dBn G(z,w)2=(1 −|z|2)(1 −|w|2) |1−z·¯w|2, in particular dD G(z,w)= |z−w| |1−z¯w|. (see [Ga], [Ru]). Definition [A-T]. We say that φis an extremal-interpolating disk iff there exists an α⊂Dsuch that φ(α)=aand δ(α)=ρ(a). Such disks are those for which the pre-image sequence αis, in a sense, as close together as it can within the unit disk (the Schwarz Lemma is preventing its points from being arbitrarily close to each other). Results about existence of extremal-interpolating disks and some of their first properties were given in [A-T] (where they were simply called “extremal disks”), as well as motivations for the study of this notion. Essentially, ρmeasures whether ais an interpolating sequence with respect to holomorphic functions bounded on the analytic disks passing through it (as opposed as being interpolating with respect to functions bounded on the whole ball). Extremal-interpolating disks 337 1. Convergence along finite subsequences and semi-continuity. Theorem 1. Let a={ak,k∈Z∗ +}⊂Bn, then ρ(a) = lim N→∞ ρ({ak,1≤k≤N}) = inf N∈Z∗ + ρ({ak,1≤k≤N}). Corollary 1. The function a→ ρ(a)is upper semi-continuous with respect to the topology given by the distance d(a, b) := supkdG(ak,b k). Proof of Corollary 1: Lemma 2 in [A-T] proved that the function ρis u.s.c. over the set of finite sequences with a given number of points, thus a→ ρ({ak,1≤k≤ N}) is an u.s.c. function, and the greatest lower bound of a family of u.s.c. functions is also u.s.c. Corollary 2. Any sequence a⊂Bnverifying ρ(a)>0is separated, i.e. there exists δ>0such that for any j=k,dG(aj,a k)≥δ. This Corollary is interesting in the context of interpolating sequences for bounded holomorphic functions (see [C], [Ga] for definitions). Separatedness is an easy necessary condition for a sequence of points to be an interpolating sequence. One also easily sees that ρ(a)>0 is another necessary condition for ato be an interpolating sequence [A-T]. Thus we see that this new necessary condition implies a better-known one. Proof of Corollary 2: By renumbering the sequence, take j=1,k= 2. It is easy to see (cf. [A-T]) that ρ({a1,a 2})=dG(a1,a 2). Apply Theorem 1 for N=2: dG(a1,a 2)≥ρ(a)>0. Proof of Theorem 1: Given any >0, let α⊂Dbe such that δ(α)<ρ(a)+. Then there exists kand Nsuch that  j:j=k, 1≤j≤N dG(αk,α j)<ρ(a)+2. Thus by definition ρ({ak,1≤k≤N})≤ρ(a)+2, and we have ρ(a)≥inf N∈Z∗ + ρ({ak,1≤k≤N}). The rest of the Theorem will follow from the 338 P. J. Thomas Proposition 3. Suppose ais a sequence in Bnand φa holomorphic map from Dto Bn, and, for 1≤k≤N,αk∈Dsuch that φ(αk)=ak. Then, given any >0, there exists a holomorphic map ψfrom Dto Bnand a sequence β⊂Dsuch that ψ(β)=aand δ(β)≤δ({αk,1≤k≤N})+. In particular, for any N<M,ρ({ak,1≤k≤N})≥ρ({ak,1≤k≤ M}). The proof of Proposition 3 will be given in Section 4. End of Proof of Theorem 1: The last clause of the proposition shows that the limit in the theorem exists and equals the infimum. Furthermore, by choosing a sequence α such that δ(α)≤ρ({ak,1≤k≤N})+, we have ρ(a)≤δ(β)≤ρ({ak,1≤k≤N})+2, which proves the required inequality. 2. Convergence of mappings. The first section lends some validation to our approach (in [A-T]) of studying the behavior of ρ(a) mostly when ais a finite sequence. In that case, since ρis defined as an infimum, it is legitimate to wonder what happens when we take a sequence of mappings φpand sequences αp such that, for any p,φp(αp)=a, and limp→∞ δ(αp)=ρ(a). By Montel’s theorem, a subsequence of {φp}pwill converge uniformly on compact subsets of D, but no such convergence is guaranteed for the points in the sequences αp, so the question arises of what the relationship between the limit of the mappings and the original sequence. Normalizations. Since we are dealing with a finite sequence a, we may always assume (after re-numbering) that ρ(a)=ρ1(a), and shall do so for the remainder of this section. Likewise, when have a sequence αwhich maps to a, by applying an automorphism of the disk, we reduce ourselves to the case where α1=0. We introduce a class of special holomorphic mappings from the disk to the ball: Extremal-interpolating disks 339 Definition. We say that φ, a holomorphic map from Dto Bn, is a ball-valued Blaschke product of degree Niff for ζ∈D, φ(ζ)=P1(ζ) Q(ζ),... ,Pn(ζ) Q(ζ), where P1,... ,P nand Qare polynomials with max(deg Pj,1≤j≤ n, deg Q)=N,Qhas no zeros in Dand for any ζsuch that |ζ|=1, 1=|φ(ζ)|2= n  1 |Pj(ζ)|2 |Q(ζ)|2. The following was essentially proved in [A-T, Theorem 1]: Theorem. If a={ak,1≤k≤N}⊂Bn,α⊂D, with φa holomorphic map from Dto Bn, such that φ(α)=a, and δ1(α)=ρ1(a)(in particular if φgives an extremal-interpolating disk for a) then φis a ball-valued Blaschke product of degree no greater than N−1, uniquely determined by α. Only the precise form of the rational map was not explicitly given in [A-T], but it is easy to obtain by following the induction performed there, observing that at each step we only perform composition by M¨obius automorphisms of the disc or ball, and multiplication by ζ. We can now state: Theorem 2. Let a={ak,1≤k≤N}⊂Bn, and {φp}pa sequence of mappings from the disk to the ball such that for each p,φp(αp k)=ak,1≤k≤N, where αp k∈D,αp 1=0, and lim p→∞ δ1(αp) = lim p→∞ N  k=2 |αp k|=ρ(a). Then there exist subsequences, denoted again by {φp}pand {αp}p,a ball-valued Blaschke product φ, and a sequence α={αk,1≤k≤N}⊂ Dsuch that (i) limp→∞ φp=φ, uniformly on compact sets of D, (ii) limp→∞ αp k=αk∈D, and (iii) φ(αk)=akiff |αk|<1. 340 P. J. Thomas Let S:= {k∈{1,... ,N}st |αk|<1}. Then ρ({ak,k∈S})=ρ(a) and φis of degree no greater than #S−1. Remarks. This theorem says that we do have convergence towards some extremal-interpolating disk, but passing only through a subsequence of a. In the case where N= 3 and aitself does not admit an extremalinterpolating disk, we see that the theorem implies that a subsequence of the minimizing sequence of mappings has to converge to an affine disk through two of the points of the original sequence. Proof of Theorem 2: By Montel’s Theorem and compactness of D, it is easy to extract subsequences having properties (i) and (ii). The “if” part of (iii) follows by equicontinuity of the converging subsequence of maps. Now ρ(a)=ρ1(a) = lim p→∞ δ1(αp) = lim p→∞   k≥2,k∈S |αp k| k/∈S |αp k|  =δ1({αk,k∈S})≥ρ1({ak,k∈S})≥ρ({ak,k∈S}), which itself is no less than ρ(a) by Proposition 3, so we actually have equality throughout. Since φ({αk,k ∈S})={ak,k ∈S}, and δ1({αk,k∈S})=ρ1({ak,k∈S}), φis a ball-valued Blaschke product of degree no greater than #S−1, so that for k/∈S,|φ(αk)|=1,so that φ(αk)=ak, which finally proves the “only if” part of (iii). It would be nicer to be able to describe the subsequence through which an extremal-interpolating disk passes, {ak,k∈S}, in terms of the sequence a. Observe that {ak,k∈S}is a sequence with the same ρas the original sequence. If such a subset Sis given, we have the: Theorem 3. Let S⊂{1,... ,N}, such that 1∈S, be minimal for the property that ρ1({ak,k∈S})=ρ1(a)=ρ(a). Then there exists a sequence of finite sequences in the disk, {αp}pand a sequence of mappings from the disk to the ball {φp}psuch that (i) φp(αp)=a, (ii) ρ1(a) = limp→∞ δ1(αp), Extremal-interpolating disks 341 (iii) limp→∞ αp k=αk∈Dand S={k∈{1,... ,N}st |αk|<1}, (iv) limp→∞ φp=φ, a ball-valued Blaschke product of degree no greater than #S−1, and (v) φ({αk,k∈S})={ak,k∈S},δ1({αk,k∈S})=ρ1({ak,k∈ S})=ρ1(a)=ρ(a). Proof of Theorem 3: The sufficient condition for ρto be attained given in [A-T, Lemma 3] covered the special case where the only such set Sis the whole {1,... ,N}. The present proof will use the same idea. For a given p, pick first, using the definition of ρ1, a sequence {βp k,k∈ S}so that δ1({βp k,k∈S})≤ρ1({ak,k∈S})+1 p; then modify and complete this sequence according to Proposition 3 to get αp={αp k,1≤k≤N}so that ρ1(a)≤δ1({αp})≤ρ1({ak,k∈S})+2 p=ρ1(a)+2 p. This forces limp→∞ |αp k|= 1 for k/∈S. On the other hand, for k∈S, by the minimality property of S, ρ1({aj,j∈S\{k}})>ρ(a). Then |αp k|=j∈S|αp j| j∈S, j=k|αp j|≤j∈S|αp j| ρ1({aj,j∈S\{k}})≤γk<1, for plarge enough. Taking subsequences as before, we get the convergence of the points αp kto limits within the open disk when k∈S, and of the mappings to a mapping φ. We obtain (v) as in the previous proof, and the resulting extremality forces to φto be a ball-valued Blaschke product. Questions. Is the set Sof Theorem 2 always “minimal”, i.e. of the type given in Theorem 3? It is clear that it contains a “minimal” set S, and that any “minimal” set that contains it must be equal to it. Also, the set S is included in some set S maximal for the property that an extremalinterpolating disk does pass through {ak,k∈S}; must S,Sand S coincide? More modestly, are there examples of sequences where several different minimal sets Scan be found? 342 P. J. Thomas 3. Continuity in the finite case. Despite their limitations, the ideas of the previous section enable us to prove the following continuity result: Theorem 4. For every N∈Z∗ +, the function {ak,1≤k≤N}→ρ({ak,1≤k≤ N})is continuous from (Bn)Nto R+. Proof of Theorem 4: We shall adopt the same normalisations as those in the previous section. Since ρ(a) = min1≤k≤Nρk(a), it will be enough to prove that ρ1is a continuous function. By Lemma 2 in [A-T], we already know it to be upper semi-continuous. We shall proceed by induction on N. Since ρ({a1,a 2})=dG(a1,a 2), the case N= 2 is clear. Now suppose the property true for all sequences asuch that #a≤ N−1. Given a∈(Bn)N, assume, to get a contradiction, that there is a sequence of sequences ap⊂(Bn)Nsuch that limp→∞ ρ1(ap)<ρ 1(a). Then for any proper subset S⊂{1,... ,N}, by the induction hypothesis, lim p→∞ ρ1({ap k,k∈S})=ρ1({ak,k∈S})≥ρ1(a), by Proposition 3. Therefore for plarge enough, ρ1(ap)<min{ρ1({ap k,k∈S}):S⊂{1,... ,N},#S≤N−1}, so an application of [A-T, Lemma 3] shows that for each such pthere exists a sequence αpin the disk such that δ1(αp)=ρ1(ap) and a mapping φpsuch that φp(αp)=ap. Now, for small enough, plarge enough, and any k∈{2,... ,N}, we have |αp k|=δ1(αp) δ1({αp j,j=k})≤ρ1(ap) ρ1({ap j,j=k})≤limp→∞ ρ1(ap) ρ1(a)+<1. This implies that all the points αp kremain in a relatively compact disk within the unit disk, therefore by extracting a subsequence we may assume that for each k, limp→∞ αp k=αk∈D, and limp→∞ φp=φ, with uniform convergence on compact subsets of the unit disk. This implies that φ(α)=a, but since δ1(α) = lim p→∞ δ1(αp) = lim p→∞ ρ1(ap)<ρ 1(a), we get a contradiction with the definition of ρ1. Extremal-interpolating disks 343 4. Proof of Proposition 3. As usual, we may assume without loss of generality that α1= 0 and δ(α)=δ1(α), where α={αj}1≤j≤N. Also, renumber aso that |aj+1|≥ |aj|for any j≥N+1. We write Mβfor the constant of interpolation of a sequence β, i.e. Mβ:= inf{M>0st∀a⊂Bn,there is φ:D→Bn(0,M)stφ(β)=a}. Pick r0<1 such that α⊂D(0,r 0) and N  j=2  αj r0 <δ 1(α)+, and 1>0 small enough so that 1≤1−|aN+1| Mα+2 and φ(D(0,r 0)) ⊂ Bn(0,1−(Mα+2)1). Lemma 4. There exists a holomorphic function E∈A(D)(i.e. continuous up to the boundary) so that (i) |E(ζ)|≤1, for any ζ∈D; (ii) |E(ζ)−1|≤1, for any ζ∈D(0,r 0); (iii) E(1) = 0, but E(ζ)=0for any ζ∈D. Proof: For a∈(−1,+1) let φa(ζ)= ζ−a 1−aζ , and E(ζ)=(1−φa(ζ))/2. Then E(1) = 0 and E∞≤1. But the disk φa(D(0,r 0)) admits the line segment [−r0−a 1+ar0;r0−a 1−ar0] for its diameter and lima→1−r0−a 1−ar0=−1, so that for aclose enough to 1, φa(D(0,r 0)) ⊂D(−1,21), which yields (ii). Lemma 5. There exists F∈C 0([r0,1),Bn)and {αj}j≥N+1 a strictly increasing sequence such that (i) F(αj)=ajfor j≥N+1; (ii) M{αj}j≥1≤M{αj}1≤j≤N+1; (iii) 1 −|F(x)|≥1(Mα+2)|E(x)|, for any x∈[r0,1); (iv) F(r0)=φ(r0). Proof: Once the αjare given, we will define Fby linear interpolation: F(θr0+(1−θ)αN+1):=θφ(r0)+(1−θ)aN+1 and F(θαj+(1−θ)αj+1):=θaj+(1−θ)aj+1,