The closed convex hull of the interpolating Blaschke products

Publicacions Matem`atiques, Vol 41 (1997), 659–669. THE CLOSED CONVEX HULL OF THE INTERPOLATING BLASCHKE PRODUCTS Knut Øyma Abstract The closed convex hull of the interpolating Blaschke products contains any bounded analytic function of sufficiently small norm. Let H∞be the set of bounded analytic functions on the open unit disc D. Each f∈H∞has a non-tangential limit f(eiθ) at almost every point eiθ ∈T=∂D. An inner function fis a bounded analytic function satisfying |f(eiθ)|= 1, a.e. eiθ ∈T. If {zn}⊂Dsatisfies (1 −|zn|)<∞, then ∞  n=1 −zn |zn| z−zn 1−znz is called the Blaschke product with zeros {zn}.Ifzn= 0, replace −zn |zn| by 1. A Blaschke product is an inner function. These products have a lot of interesting properties: If f∈H∞then there is a Blaschke product B and a non-vanishing g∈H∞such that f=Bg.IfIis an inner function then I−λ I−λI is a Blaschke product for λin a dense subset of D. Therefore the Blaschke products are dense in the inner functions in the uniform norm. The closed convex hull of the Blaschke products is the unit ball of H∞. See [G] for the proof of these and many other properties of the Blaschke products. A square is a set of the form Q={reiθ :1−h<r<1,θ 0<θ≤θ0+h}. 660 K. Øyma The length of Qis |Q|=h. The pseudohyperbolic metric on Dis defined by ρ(z,w)= z−w 1−zw . A sequence {zn}⊂Dis called interpolating if the mapping H∞→ ∞:f(z)→{f(zn)}is onto. Carleson, see [G], proved that {zn}is interpolating if and only if inf m=nρ(zn,z m)>δ>0, and  zn∈Q (1 −|zn|2)<C|Q| for all squares. Given an square Q, its top half is T(Q)=reiθ ∈Q:r<1−1 2|Q|. A Blaschke product whose zero set is an interpolating sequence is called an interpolating Blaschke product. These products have been studied in great detail, and they play a significant role in the theory of H∞, see [G]. An important open problem is whether or not they are dense in the set of inner functions. In this paper we study the closed convex hull Kof the interpolating Blaschke products. Theorem. If f≤10−7, then f∈K. The proof of this modest result is long and technical. The ideas may disappear in the formalism, so I delete details at some points. Let B(z) be a Blaschke product with zeros {zn}. Assume that ρ(z,zn) is close to 1 for all n. Then log 1 |B(z)|2= (1 + 0(1)) (1 −|zn|2)(1 −|z|2) |1−znz|2. Assume that the zeros are contained in disjoint squares Qkand that |Qk|(1 −|z|). Choose ξk∈Qk∩T. By the density of Qw.r.t. Bwe mean 1 |Q| zn∈Q (1 −|zn|2). Convex hull of interpolating Blaschke products 661 Assume that the density of all Qkis smaller than a number d. Then log 1 |B(z)|2= (1 + 0(1))  k zn∈Qk (1 −|z|2)(1 −|zn|2) |1−ξkz|2 ≤(1 + 0(1))  k (1 −|z|2)|Qk|d |1−ξkz|2 ≤(1 + 0(1))d2π 0 1−r2 |1−rei(θ−t)|2dt= (1 + 0(1))d2π. Here z=reiθ. This proves Lemma 0. Under the above assumptions, |B(z)|≥e−πd(1+0(1)). Lemma 1. Given a square Q, and constants κ<1,1/2>δ>0, 1>a1−|Q|, there exist a finite Blaschke product Bwith zeros on (|z|=a)∩Qand a region Rof the unit disc, such that 1. The zeros of Bare uniformly distributed on |z|=a 2. The density of Qw.r.t. B≤(1 + 0(1)) 1 πlog 1 δ 3. If z∈Q,|z|<a, then ρ(z,R)≤A<1,A=A(κ, δ) 4. If z∈R, then |B(z)|≤δκ. Below we consider only the dyadic squares: Q=reiθ :2πk 2n<θ<2π(k+1) 2n,1−2−n≤r<1. Let Ba,N (z)= zN−aN 1−aNzN, where ais close to 1 and aN=δ. The zeros are zk=ae2πi Nk.Now Nlog a= log δand N∼log 1 δ 1−a since ais close to 1. Also 1 2π(1 −|zk|2)<1 π(1 −|zk|)= 1 πN(1 −a) = (1 + 0(1)) 1 πlog 1 δ. If Qis a square such that |Q|1−a, then the density of Qis (1 + 0(1)) 1 πlog 1 δsince {zk}are uniformly distributed. 662 K. Øyma Also, |Ba,N (z)|=ρ(zN,a N)≤ρ(−|z|N,a N). Assume that |z|=aM, M>1. Then |Ba,N (z)|≤ρ(−aNM,a N)=ρ(−δM,δ)≤δ+δM. Let R⊂Qbe the region bounded by Γ = {z∈Q:|z|=aM}and a circle intersecting Γ in a small angle γ, as in Figure 1. γR |z|=aT Γ Q Figure 1. Ris bounded by Γ and another circle. The angle of intersection, γ, is small. The corners of Rare far, but not too far, from the endpoints of Γ. The figure does not tell the truth: Most of Q∩{z:|z|<a}is contained in R, that is: All points in Q∩{|z|<a}are “close” to Rin the pseudohyperbolic metric. The argument proving Lemma 0 shows that for z∈Rthe contribution to |Ba,N (z)|coming from the zeros outside Q is close to 1, provided the corners of Rare moved away from the endpoints of Γ. By making γsmall and increasing the distance from the corners of Rto the endpoints of Γ, we obtain |B(z)|≤(δ+δM)κfor z∈R,κclose to 1. It is also easy to see that if z∈Q,|z|<a, then ρ(z,R)<A=A(δ, κ)<1. By increasing M, decreasing κslightly and making γsmall, we finish the proof. Lemma 2. Any Blaschke product Bcan be factored B(z)=B1(z)· B2(z)where lim sup r→1 min |z|=r|Bi(z)|=1 for i=1,2. Proof: See [Ø]. Convex hull of interpolating Blaschke products 663 Lemma 3. Let B(z)be a B-product with zeros {zn}. Assume that there exists constants A<1,η>0such that for all nthere exists z∗ n satisfying ρ(zn,z∗ n)<A,|B(z∗ n)|>η. Then B(z)is a finite product of interpolating B-products. Proof: See [G-N]. Lemma 4. Any finite product of interpolating B-products can be uniformly approximated by an interpolating B-product. Proof: See [M-S]. A consequence of Lemma 4 is that the closed convex hull of the interpolating B-products is closed under multiplication. By Lemmas 4 and 2, we may assume that the conclusion of Lemma 2 holds. Given δ>0 close to 1. Let 0 <δ<β<1 to be chosen later. Choose r1<1, r1close to 1 such that |B(z)|>βif |z|=r1. We may assume that |B(0)|>β. Let Q1 1,Q2 1,... be the maximal dyadic squares such that 1. inf z∈T(Q)|B(z)|<δ. 2. T(Q)∩{z:|z|<r 1}=∅. These are the squares of the first generation. The proof of Lemma 0 shows that  B(zn)=0,zn∈Qk 1 (1 −|zn|2)<C(δ)|Qk 1| where C(δ)→0 when δ→1. Choose r2<1, (1 −r1)/(1 −r2) large, such that |B(z)|>βif |z|=r2. The second generation are the maximal dyadic squares such that 1. inf z∈T(Q)|B(z)|<δ. 2. T(Q)∩{z:|z|<r 2}=∅. 3. T(Q)∩{z:|z|<r 1}=∅. Continue inductively. If βis large then (1)  k:Qk n⊂Qj n−1 |Qk n|<ε|Qj n−1|. (See [G, p. 332].) 664 K. Øyma Denote (Qk n\T(Qk n))∩{z:|z|≤rn}=Sk 1,n ∪Sk 2,n and Γk i,n =Sk i,n ∩{z: |z|=rM n}, where M,Γ k i,n and Rk i,n come from Lemma 1 and its proof. See figure below. T |z|=rn Qk n Sk 2,n Sk 1,n Rk 1,n −→− Figure 2 Decompose B(z)=B1(z)B2(z) where Bi(z) has zeros in  n,k Sk i,n. Construct Bk 1,n as in Lemma 1 with a=rn,Q=Sk 1,n. All Bk 1,n of generation nshare the same a(= rn). Let B1,n = k Bk 1,n,B ∗ 1= n B1,n. The estimate (1) shows B∗ 1is an interpolating B-product and if βis close to 1, then |B∗ 1(z)|≈|B1,n(z)|when zis close to Γk 1,n. By choosing δclose to 1 we obtain  B1B∗ 1(zi)=0 zi∈Qk n (1 −|zi|2)≤[1 + 0(1)] 1 2πlog 1 δ|Qk n|. That is: We have control over the density of Qk nw.r.t. B1B∗ 1.Forz∈ Rk 1,n,|B∗ 1(z)|<δ κ. Choose r<κ;rslightly smaller. Then ris close to 1. By Frostman’s theorem we can choose rsuch that B1B∗ 1−δr 1−δrB1B∗ 1 =C1 is a B-product. Convex hull of interpolating Blaschke products 665 Claim. C1is a finite product of interpolating B-products. The proof uses Lemma 3. Assume that C1(w) = 0. Then B1(w)B∗ 1(w)=δr. Let w∗∈D, Arg w∗= Arg wand 10(1 −|w∗|)= (1 −|w|). Split B1(z)B∗ 1(z)=BQ(z)B∼ Q(z) where BQtakes care of the zeros inside Qw(see Figure 3), and consider the following three different situations. Case 1: ρ(w, zn) is large for all nand every Sk 1,n meeting Qwis small, say, |Sk 1,n| 1−|w|<η. Since |B∼ Q(w)|>δ rone has rlog 1 δ≥log 1 |B∼ Q(w)|= (1 + 0(1))1 2 zn/∈Qw (1 −|w|2)(1 −|zn|2) |1−znw|2=A and log 1 |B∼ Q(w∗)|= (1 + 0(1))1 2 zn/∈Qw (1 −|w∗|2)(1 −|zn|2) |1−znw∗|2 <1 4A<r 4log 1 δ. Therefore |B∼ Q(w∗)|>δr 4. The world looks like this: T w∗ w Sk 1,n −→ − Figure 3. Situation in Case 1. 666 K. Øyma Let us consider all the disjoint dyadic subsquares of Qw, of length 100η(1 −|w|). Then the density of all these small, but not too small, subsquares of Qwis less than (1 + 0(1)) 1 2πlog 1 δ. By Lemma 0, |BQ(w∗)|≥δ1 2(1+0(1)) >δ 0.51. Hence |B1(w∗)B∗ 1(w∗)|>δr 4·δ0.51 >δ4 5rsince ris close to 1. Therefore |C1(w∗)|=ρ(B1(w∗)B∗ 1(w∗),δr)≥ρ(δr,δ4 5r). Lemma 3 does the trick. Case 2: ρ(w,zN) is small for some N. Here small means not close to 1. Then zNplays the same role as w∗ in Case 1. Case 3: One Sk 1,N meeting Qwis “not small”, that is, |Sk 1,n| 1−|w|>η, for some n. Then there exists w∗∈Rk 1,N such that ρ(w, w∗) is “not large”. Lemma 1 gives |B1(w∗)B∗ 1(w∗)|<|B∗ 1(w∗)|<δ κ<δ r. Use Lemma 3 again. Remark. This argument will be repeated later, but we will not repeat the details. The parameter of the Frostman transform, δr, satisfied two conditions: 1. δr>δ κ≥max z∈Rk 1,n |B∗ 1(z)|for all n, k. 2. The density dof Qk nw.r.t. B1B∗ 1satisfied e−πd >δ κ. Therefore the argument with a Frostman parameter δ∗works if max z∈Rk 1,n |B∗ 1(z)|<|δ∗|<e −Πd. It is easy to see that the unimodular constants belong to the closed convex hull Kof the interpolating B-products. We have B1B∗ 1=C1+δr 1+δrC1 = ∞  0 anCn 1. Since δ>0, a computation proves that |an|=1+2δr, hence B1B∗ 1∈ (1+2δr)K. Convex hull of interpolating Blaschke products 667 Alternatively. Repeat the argument with δrreplaced by δreiθ.For almost all θthe function Cθ=B1B∗ 1−δreiθ 1−δre−iθB1B∗ 1 is a finite product of interpolating B-products. Approximating with Riemann sums we see that 1 2ππ −π B1B∗ 1−δreiθ 1−δre−iθB1B∗ 1 dθ ∈K. This leads to B1B∗ 1∈1 1−δ2rK. This estimate is better than the previous one if δris small. We now want to repeat the argument with B∗ 1replaced by B∗∗ 1=B∗ 1−σ 1−σB∗ 1 . The zeros of B∗∗ 1are close to the zeros of B∗ 1if σis not too large. The zeros of B∗ 1are concentrated on arcs |z|=rn, one arc for each generation T |z|=rn=a |z|=b × × × Figure 4. The zeros of B∗ 1and B∗∗ 1are marked by crosses and dots respectively. The zeros of B∗ 1are marked by crosses. Close to these zeros |B∗ 1(z)|≥ (1 + 0(1))|B1,n(z)|. Figure 4 shows that if |σ|<ρ(aN,b N), aN=δ, b=as,s<1 then the zeros of B∗∗ 1are concentrated as indicated by the dots. If ais close to 1 we obtain |σ|<ρ(aN,b N)=ρ(δ, δs)= δ−δs 1−δs+1 .