aa r X i v : . [ m a t h . C V ] S e p INTERPOLATION IN MODEL SPACES
PAMELA GORKIN AND BRETT D. WICK
Abstract.
In this paper we consider interpolation in model spaces, H ⊖ BH with B a Blaschke product. We study unions of interpolating sequences for two sequencesthat are far from each other in the pseudohyperbolic metric as well as two sequencesthat are close to each other in the pseudohyperbolic metric. The paper concludes witha discussion of the behavior of Frostman sequences under perturbations. Introduction
Let H ∞ denote the space of bounded analytic functions and let H denote the Hardyspace of functions on the unit circle T satisfying sup
BM O denotes the space of functions of bounded mean oscillation on the unit circle.For a sequence ( a j ) of points in the open unit disk D satisfying the Blaschke condition P j (1 − | a j | ) < ∞ , we consider Blaschke products, or functions of the form B ( z ) = λ ∞ Y j =1 | a j |− a j (cid:18) z − a j − a j z (cid:19) where λ ∈ T . (Here, as in the future, we interpret | a j | /a j = 1 if a j = 0 .) We are particularly interestedin Blaschke products for which the zero sequence ( a j ) is an interpolating sequence for H ∞ . Mathematics Subject Classification.
Primary: 30H05; Secondary: 30J10, 46J15.
Key words and phrases.
Hardy space, model space, Blaschke product, interpolation.
In [5], Dyakonov proved the following:
Theorem 1.1 ([5]) . Suppose that ( α j ) is an ℓ ∞ sequence and B is an interpolatingBlaschke product with zeros ( a j ) . In order that there exist a function f ∈ K ∞ B for which f ( a j ) = α j for all j , it is necessary and sufficient that (1) sup k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j α j B ′ ( a j )(1 − a j a k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ . Note that Theorem 1.1 assumes only that ( a j ) can be interpolated to a particular sequence ( α j ) ; in particular, one satisfying the conditions of equation (1).In this paper, we combine Dyakonov’s techniques with those of Kenneth Hoffman toobtain further results about interpolation in K ∞ θ .To discuss these results, we need a measure of separation of points in the open unitdisk D . The natural metrics are the hyperbolic or pseudohyperbolic distances. We beginwith the latter. Let ρ ( a, z ) = (cid:12)(cid:12)(cid:12)(cid:12) z − a − az (cid:12)(cid:12)(cid:12)(cid:12) denote the pseudohyperbolic distance between two points a and z in D .If ( a j ) and ( z j ) are two sequences of points in D and we assume that we can interpolate ( a j ) to any ℓ ∞ sequence ( α j ) and ( z j ) to any ℓ ∞ sequence ( β j ) then, using Hoffman’sresults, it is not difficult to show that ρ -separation of ( a j ) and ( z j ) implies that we caninterpolate an (ordered) union of the sequences to any ℓ ∞ sequence. For ease of notation,we will primarily consider the union defined by alternating points of the sequences.In this paper, we first consider the case when the sequences ( z j ) and ( a j ) are “far fromeach other”: We show (Theorem 3.1) that if ( a j ) can be interpolated to ( α j ) in K ∞ B and ( z j ) can be interpolated to ( β j ) in K ∞ C , then the union of the two sequences canbe interpolated to the union of ( α j ) and ( β j ) (in the appropriate order) in K ∞ BC if thesequences ( a j ) and ( z j ) are ρ -separated; that is, there exists a constant λ > such that ρ ( a j , z k ) ≥ λ for all j and k . Using Theorem 1.1 allows us to rephrase this as a statementabout a series like the one appearing in equation (1).We then consider two ρ -separated sequences ( a j ) and ( z j ) that are “near each other”;that is, with the property that there exists λ < with ρ ( a j , z j ) < λ < for all j . Inthis case, we show that the modified statement of Proposition 2.1 is true for sequences inmodel spaces (Theorem 4.2); that is, if ( a n ) is interpolating for K ∞ B and the two sequencesare near each other, then ( z n ) is interpolating for K ∞ C .From this result, we obtain some information about (uniform) Frostman Blaschke prod-ucts . Recall that a sequence ( a j ) in D satisfies the Frostman condition if and only if(2) sup (X j − | a j || ζ − a j | : ζ ∈ T ) < ∞ . As a consequence of Vinogradov’s work [14], it follows that an H ∞ -interpolating sequence ( a j ) is Frostman if and only if it is interpolating for K ∞ B . This can also be seen byconsidering Theorem 1.1 and using the following: In [4, Section 3], Cohn shows that aninterpolating sequence ( a k ) is a Frostman sequence if and only if(3) sup n X k − | a k || − a k a n | < ∞ . NTERPOLATION IN MODEL SPACES 3
Our paper concludes with a fact about (uniform) Frostman Blaschke products that wehave not seen in the literature. Recall that a Frostman Blaschke product is a Blaschkeproduct with zeros ( a n ) that satisfy the Frostman condition (2). An example of such aBlaschke product appears in [9] (or [2, p. 130]) and is given by a n = (cid:18) − n (cid:19) exp (cid:18) i n n (cid:19) . In general, it is not easy to check that something is a Frostman Blaschke product.Vasyunin has shown that if B is a uniform Frostman Blaschke product with zeros ( a n ) ,then P ∞ n =1 (1 − | a n | ) log(1 / (1 − | a n | )) < ∞ , but this is not a characterization. For gen-eralizations of this as well as more discussion see [1]. Here, we show that if you startwith a uniform Frostman Blaschke product and move the zeros, but not too far pseudo-hyperbolically speaking, then the resulting Blaschke product is also a uniform Blaschkeproduct. In view of the difficulty of proving something is a Frostman Blaschke product,this result could be useful. We accomplish this by using Dyakonov’s methods and resultto conclude that as long as we move the zeros of a Frostman Blaschke product within afixed pseudohyperbolic radius r < of the original zeros, the resulting Blaschke productwill remain a Frostman Blaschke product.2. Preliminaries
In this section we collect all the necessary background and estimates that play a rolein the proofs in later sections. We first recall the fact that if points are close to aninterpolating sequence, then they are interpolating as well.
Proposition 2.1. [6, p. 305]
Let ( a j ) be an interpolating sequence for H ∞ and ( z j ) a ρ -separated sequence with ρ ( a j , z j ) < λ < , for all j , then ( z j ) is an interpolating sequence for H ∞ . This proposition is an exercise in [6]. For a proof, see [10, Theorem 27.33]. Using thesame notation as above, we will need the following estimate that appears in the proof:(4) − ρ ( a j , a k ) ≤ (cid:18) λ − λ (cid:19) (1 − ρ ( z j , z k )) . Recall that for two points z and w in D , the pseudohyperbolic distance is ρ ( z, w ) = (cid:12)(cid:12) z − w − wz (cid:12)(cid:12) and the hyperbolic metric is given by β ( z, w ) = 12 log 1 + ρ ( z, w )1 − ρ ( z, w ) . In what follows, we will consider two interpolating sequences ( a j ) and ( z j ) that are ρ -separated or far from each other ; that is, with the property that there exists ε > with(5) inf j,k ρ ( a j , z k ) ≥ ε. We then consider sequences that are near each other in the sense that there exists ε < with ρ ( a j , z j ) < ε < for all j . In this case, we have the following estimates that we willrefer to later. Let ε be chosen with < ε < . Suppose that ρ ( a j , z j ) ≤ ε for all j . Then PAMELA GORKIN AND BRETT D. WICK r := sup j,k β ( a j , z k ) ≤ log − ε < ∞ . Let s = tanh r ∈ (0 , and apply ([15, Proposition4.5]) to obtain for each j and k ,(6) − s ≤ − s | z k | − | z k | ≤ | − a j z k | and − s | a j | − | a j | ≤ | − a j z k | . Thus, − | z k | | − a j z k | ≥ − s | z k | ≥ − tanh r, and a similar inequality holds with z j replaced by a j .Our work relies on Dyakonov’s proof techniques, which rely on the following two resultsof W. Cohn. The convergence below is taken in the weak- ∗ topology of BM OA := BM O ∩ H , and it also converges in H . Thus, the convergence also holds on compactsubsets of D . Lemma 2.2. [3, Lemma 3.1]
Given an interpolating Blaschke product B with zeros ( a j ) ,the general form of a function g ∈ K ∗ B is g ( z ) = X j c j − | a j | − a j z , where ( c j ) ∈ ℓ ∞ . Lemma 2.3. [4, Corollary 3.2]
Let B be an interpolating Blaschke product with zeros ( a j ) and let g ∈ K ∗ B . Then ( g ( a j )) ∈ ℓ ∞ if and only if g ∈ H ∞ . Another key ingredient in our proofs are the following three theorems from KennethHoffman’s seminal paper, which we recall here.
Lemma 2.4 (Hoffman’s Lemma) . [7] , [6, p. 395] Suppose that B is an interpolatingBlaschke product with zeros ( z n ) and inf n (1 − | z n | ) | B ′ ( z n ) | ≥ δ > . Then there exist λ := λ ( δ ) with < λ < and r := r ( δ ) with < r < satisfying lim δ → λ ( δ ) = 1 and lim δ → r ( δ ) = 1 such that { z : | B ( z ) | < r } is the union of pairwise disjoint domains V n with z n ∈ V n and V n ⊂ { z : ρ ( z, z n ) < λ } . Let M ( H ∞ ) denote the maximal ideal space of H ∞ or the set of non-zero multiplicativelinear functionals on H ∞ . Identifying points of D with point evaluation, we may thinkof D as contained in M ( H ∞ ) . Carleson’s Corona Theorem tells us that D is dense in thespace in the weak- ∗ topology. The maximal ideal space breaks down into analytic diskscalled Gleason parts. These may be a single point, in which case we call them trivial, orthey may be true analytic disks, in which case we call them nontrivial. It is a consequenceof Hoffman’s work that points in the closure of an interpolating sequence are nontrivial.(See [7, Theorem 4.3].) NTERPOLATION IN MODEL SPACES 5
Theorem 2.5. [7, Theorem 5.3]
Let B be a Blaschke product and let m be a point of M ( H ∞ ) \ D for which B ( m ) = 0 . Then either B has a zero of infinite order at m or else m lies in the closure of an interpolating subsequence of the zero sequence of B . In the same paper of Hoffman, [7, Theorem 5.4], shows that an interpolating Blaschkeproduct cannot have a zero of infinite order. Therefore, if B is an interpolating Blaschkeproduct and B ( m ) = 0 , then m must lie in the closure of the zero sequence of B . Theorem 2.6 (Hoffman’s Theorem) . A necessary and sufficient condition that a point m of the maximal ideal space lie in a nontrivial part is the following: If S and T are subsetsof the disk D and if m belongs to the closure of each set, then the hyperbolic distance from S to T is zero. As a result of Hoffman’s theorem we show that, if ( a j ) is interpolating for K ∞ B and ( z j ) is interpolating for K ∞ C and the ρ distance between the two sequences is positive, then(see Corollary 2.7) B is bounded below on { z j } and C is bounded below on { a j } . This isknown, but for future use we isolate this as a corollary to Theorem 2.6. Corollary 2.7.
Let ( a j ) and ( z j ) be two interpolating sequences for H ∞ with correspond-ing Blaschke products B and C , respectively. Suppose further that the ρ -distance betweenthe two sequences satisfies inf j,k ρ ( a j , z k ) ≥ ε > . Then there exists η > such that inf j | C ( a j ) | ≥ η and inf j | B ( z j ) | ≥ η. Proof.
If not, we may suppose that inf j | B ( z j ) | = 0 . Therefore, there exists a subsequence ( z j k ) of ( z j ) with B ( z j k ) → . Let m ∈ M ( H ∞ ) \ D be a point in the closure of the set { z j k } . Then B ( m ) = 0 . By the aforementioned work of Hoffman, m lies in the closure ofthe zeros of B , namely the closure of { a j } . On the other hand, m lies in the closure of { z j } , by the choice of m . By Theorem 2.6 the hyperbolic distance between the two setsmust be zero. But since the pseudohyperbolic distance between the two is bounded awayfrom zero, this is impossible. (cid:3) Sequences that are far from each other
In this section, we will consider unions of finitely many interpolating sequences definedin the following manner: Let ( α j ) and ( β j ) be sequences. Define ( α j ) ∪ ( β j ) to be thesequence ( γ j ) where(7) γ j = (cid:26) α j if j is odd β j if is even . For simplicity of presentation, we have defined the sequence ( γ j ) via this simple “every-other” interlacing. It is clear that from the proof techniques that one could interlacethe sequences ( α j ) and ( β j ) in other ways. Interlacing in other more exotic ways wouldnecessitate the introduction of additional more complicated notation and to present theideas most clearly we have chosen to use only these simple process described here.In what follows, for a Blaschke product B with zeros ( a j ) , let b j ( z ) = | a j |− a j ( z − a j )(1 − a j z ) PAMELA GORKIN AND BRETT D. WICK and let B j ( z ) = B ( z ) /b j ( z ) . (We interpret | a j |− a j = 1 if a j = 0 .)If we wish to interpolate ( a j ) ∪ ( z j ) (as defined in equation (7)) to the sequence ( α j ) ∪ ( β j ) and we know that ( a j ) is interpolating for K ∞ B and ( z j ) is interpolating for K ∞ C ,and both B ( z j ) and C ( a j ) are bounded below over all j , then we can interpolate to ( α ′ j ) := ( α j /C ( a j )) and ( β ′ j ) := ( β j /B ( z j )) with g ∈ K ∞ B and g ∈ K ∞ C , respectively. So G := Cg + Bg ∈ K ∞ BC will do the interpolation. However, if we don’t know that we cando the interpolation to every bounded sequence, then we need to combine Dyakonov andHoffman’s work to obtain a result. Theorem 3.1.
Let B and C be interpolating Blaschke products with zeros ( a j ) and ( z j ) respectively, satisfying inf j,k ρ ( z j , a k ) ≥ ε > . If ( a j ) can be interpolated to ( α j ) in K ∞ B and ( z j ) can be interpolated to ( β j ) in K ∞ C , then ( x j ) := ( a j ) ∪ ( z j ) can be interpolated to ( γ j ) := ( α j ) ∪ ( β j ) in K ∞ BC .Proof. By Hoffman’s theorem the sequence ( x j ) is interpolating for H ∞ and BC is inter-polating. Note that K ∞ B = H ∞ ∩ BzH ∞ = H ∞ ∩ BCzCH ∞ ⊆ K ∞ BC and, similarly, K ∞ C ⊆ K ∞ BC . Corollary 2.7 also implies that there exists δ > such that inf j (min {| B ( z j ) | , | C ( a j ) |} ) ≥ δ > . We define ˜ γ j for j = 1 , , , . . . , by ˜ γ j − := − a j | a j | γ j − B j ( a j ) and ˜ γ j = − z j | z j | γ j C j ( z j ) . Then (˜ γ j ) ∈ ℓ ∞ . Let g be defined by g ( z ) = ∞ X j =1 ˜ γ j − (1 − | a j | )1 − a j z + ∞ X j =1 ˜ γ j (1 − | z j | )1 − z j z , and use Lemma 2.2, the fact that BC is interpolating, and K B ∪ K C ⊆ K BC to concludethat g ∈ K ∗ BC . In particular, g ∈ H .Now for almost every z ∈ T , we have(8) B ( z ) C ( z ) zg ( z ) = X j ˜ γ j − B ( z ) C ( z ) z (1 − | a j | )1 − a j z + X j ˜ γ j B ( z ) C ( z ) z (1 − | z j | )1 − z j z . For the first summand and almost every z ∈ T , B ( z ) C ( z ) z (1 − | a j | )1 − a j z = −| a j | a j C ( z ) B j ( z ) (cid:18) z − a j − a j z (cid:19) z (1 − | a j | )1 − za j = −| a j | a j C ( z ) B j ( z ) (1 − | a j | )1 − a j z . The summation converges in H and each summand is in H , so the function also liesin H . Therefore BCzg ( z ) ∈ (cid:16) BCzH (cid:17) \ H = K BC . The same computations, with appropriate adjustments, hold for the second summand.Therefore,
NTERPOLATION IN MODEL SPACES 7 (9) G ( z ) := B ( z ) C ( z ) zg ( z ) = ∞ X j =1 ˜ γ j − (cid:18) −| a j | a j B j ( z ) C ( z ) (1 − | a j | )1 − a j z (cid:19)| {z } G + ∞ X j =1 ˜ γ j (cid:18) −| z j | z j B ( z ) C j ( z ) (1 − | z j | )1 − z j z (cid:19)| {z } G ∈ K BC . Note that the equality G ( z ) := ∞ X j =1 −| a j | a j ˜ γ j − B j ( z ) C ( z ) (1 − | a j | )1 − a j z + ∞ X j =1 −| z j | z j ˜ γ j B ( z ) C j ( z ) (1 − | z j | )1 − z j z also holds in D .Now from (8) G ∈ H ∩ BC ( zH ) = K BC , G ∈ CH , and G ( a j ) = α j C ( a j ) . But we assume there exists f ∈ K ∞ B with f ( a j ) = α j for all j , and therefore ( Cf )( a j ) = α j ( C ( a j )) . It follows that G − Cf = Bh for some h ∈ H . But since G ∈ CH and B and C have no common zeros, we see that C must divide h .Thus, we have G − Cf = BCh for some h ∈ H . Thus G − Cf ∈ BCH . Note alsothat f ∈ K ∞ B implies that Cf ∈ K ∞ BC . So, G − Cf ∈ ( BC ) H ∩ K BC = { } . Therefore, G = Cf ∈ H ∞ . The same computations show that G ∈ H ∞ . Therefore G = G + G ∈ H ∞ ∩ K BC , which implies the result. (cid:3) From Theorem 1.1 we have the following:
Corollary 3.2.
Let B , C , ( a j ) , and ( z j ) be as in Theorem 3.1 and let ( x j ) = ( a j ) ∪ ( z j ) ,where ( a j ) and ( z j ) are the zeros of B and C , respectively. If sup k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j α j B ′ ( a j )(1 − a j a k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ and sup k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j β j C ′ ( z j )(1 − z j z k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ , then sup k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X j γ j ( BC ) ′ ( α j )(1 − α j α k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ∞ where ( γ j ) = ( α j ) ∪ ( β j ) . PAMELA GORKIN AND BRETT D. WICK Sequences that are near each other
In the introduction to the paper, we mentioned (see Proposition 2.1) that if ( a n ) is aninterpolating sequence for H ∞ and ( z n ) is a ρ -separated sequence with ρ ( a n , z n ) < − ε < for all n , then ( z n ) is interpolating for H ∞ . Here we consider the same result for K ∞ B . Proposition 4.1. If ( a n ) is interpolating for K ∞ B , then there is a constant M such that k f k ∞ ≤ M k ( f ( a n )) k ∞ for every f ∈ K ∞ B .Proof. Define T : K ∞ B → ℓ ∞ by T ( f ) = ( f ( a n )) . Then T is a bounded linear operator thatmaps surjectively onto ℓ ∞ . Note that T is also injective, because T ( f ) = T ( g ) implies f − g ∈ BH ∞ . But f − g ∈ K ∞ B ∩ BH ∞ implies that f = g . The desired result now followsfrom the open mapping theorem (or, more specifically, the bounded inverse theorem). (cid:3) We now prove that when points in an interpolating sequence for K ∞ B can be movedpseudohyperbolically, as long as they are not moved too far, the new sequence will beinterpolating for K ∞ B if the original was. Theorem 4.2.
Let B be a Blaschke product and suppose that its zero sequence, ( a n ) , isan interpolating sequence for K ∞ B . Let M be the constant in Proposition 4.1, and supposethat ( z n ) is a sequence of distinct points with ρ ( a n , z n ) < − ε < / (2 M ) . Then ( z n ) isinterpolating for K ∞ B .Proof. Without loss of generality we may assume
M > . Let ( α n ) ∈ ℓ ∞ . Choose f ∈ K ∞ B with f ( a n ) = α n for all n . If necessary, divide ( α n ) by a constant to assume that we canchoose f with norm at most one. Then, by Schwarz’s lemma, for all n we have ρ ( f ( z n ) , f ( a n )) ≤ ρ ( z n , a n ) . Thus, | f ( z n ) − f ( a n ) | ≤ ρ ( a n , z n ) , for all n .So, using our assumptions, for all n we have | f ( z n ) − f ( a n ) | ≤ − ε ) . Now ( a n ) is interpolating for K ∞ B , so we may choose f ∈ K ∞ B so that f ( a n ) = f ( a n ) − f ( z n ) for all n . By Proposition 4.1, we know that k f k ∞ ≤ M k ( f ( a n )) k ∞ ≤ M (1 − ε ) . Therefore, by Schwarz’s lemma, (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) M (1 − ε ) (cid:19) f ( z n ) − (cid:18) M (1 − ε ) (cid:19) f ( a n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ρ ( a n , z n ) . Consequently, for all n we have | f ( z n ) + f ( z n ) − f ( a n ) | = | f ( z n ) − f ( a n ) | ≤ M (1 − ε ) . Now we choose f ∈ K ∞ B with k f k ≤ M k ( f ( z n ) − f ( a n )) k ∞ and f ( a n ) = − ( f ( z n ) − f ( a n )) = f ( a n ) − ( f ( z n ) + f ( z n )) . NTERPOLATION IN MODEL SPACES 9
Therefore k f k ≤ M (1 − ε ) and f ( a n ) = − ( f ( z n ) + f ( z n ) − f ( a n )) . Now by Schwarz’s lemma we have | f ( z n ) / k f k − f ( a n ) / k f k| ≤ ρ ( a n , z n ) ≤ − ε ) , and consequently | f ( z n ) + f ( z n ) + f ( z n ) − f ( a n ) | = | f ( z n ) − f ( a n ) | ≤ M (1 − ε ) . Continuing in this way, we assume we have chosen f , . . . , f m ∈ K ∞ B with | f m ( z n ) + · · · + f ( z n ) − α n | ≤ m +1 M m (1 − ε ) m +1 for all n and k f m k ≤ m M m (1 − ε ) m . We choose f m +1 ∈ K ∞ B with f m +1 ( a n ) = − ( f m ( z n ) + · · · + f ( z n ) − α n ) and k f m +1 k ≤ m +1 M m +1 (1 − ε ) m +1 . Now we have chosen ε so that (1 − ε ) < / (2 M ) and k f m +1 k ≤ (2 M (1 − ε )) m +1 . Letting f = P ∞ j =0 f j we obtain f ∈ K ∞ B with the property that for each n | f ( z n ) − α n | = lim m | f m ( z n ) + · · · + f ( z n ) + f ( z n ) − f ( a n ) | ≤ lim m m +1 M m (1 − ε ) m +1 = 0 . Thus f ∈ K ∞ B and f does the interpolation. (cid:3) Frostman Blaschke products and sequences that are near each other
Tolokonnikov [13] showed that Frostman Blaschke products are always a finite productof interpolating Blaschke products, [12]. In view of this, if we start with two sequences ( a n ) and ( z n ) with ρ ( a n , z n ) ≤ λ < for all n and ( a n ) a Frostman sequence, then we canwrite ( a n ) as a finite union of interpolating sequences and, as long as ( z n ) is ρ -separated,the corresponding subsequences of ( z n ) will also be interpolating, by Proposition 2.1. Forthis reason, we can reduce our discussion to Frostman sequences that are interpolatingfor H ∞ . Proposition 5.1.
Let ( a n ) n ∈ N be a sequence of points in D . If N is an integer for which ( a n ) n>N is a Frostman sequence, then ( a n ) is a Frostman sequence.Proof. Consider the function F ( ζ ) := P Nj =1 1 −| a j | | a j − ζ | on the unit circle. Then F is continuousand therefore bounded. Thus, sup ζ ∈ T N X j =1 − | a j | | a j − ζ | is finite and the result follows. (cid:3) We turn to the main theorem of this section, which says that if we begin moving pointsof a Frostman sequence, as long as we don’t move the sequence too far pseudohyperbol-ically, the new sequence will be interpolating for K ∞ C , where C is the Blaschke productcorresponding to the new sequence. Theorem 5.2.
Let ε > . Let ( a n ) be an interpolating Frostman sequence and let ( z n ) bea ρ -separated sequence with ρ ( a n , z n ) ≤ − ε for all n . Then ( z n ) is a Frostman sequence.Proof. Using Proposition 2.1 and (4) we know that for all j and k , − ρ ( a j , a k ) ≤ (cid:18) − ε )1 − (1 − ε ) (cid:19) (1 − ρ ( z j , z k )) . Since (1 + ρ ( a j , a k )) ≤ and ≤ ρ ( z j , z k ) , it follows that (1 + ρ ( a j , a k ))(1 − ρ ( a j , a k )) ≤ (cid:18) − ε )1 − (1 − ε ) (cid:19) (1 + ρ ( z j , z k )) (1 − ρ ( z j , z k )) . A computation shows that − ρ ( a j , a k ) = (1 − | a j | )(1 − | a k | ) | − a j a k | . Since all of this also holds with the roles of ( a j ) and ( z j ) interchanged, there are positiveconstants C = C ( ε ) and C = C ( ε ) such that(10) C (1 − | a j | )(1 − | a k | ) | − a j a k | ≤ (1 − | z j | )(1 − | z k | ) | − z j z k | ≤ C (1 − | a j | )(1 − | a k | ) | − a j a k | . Now ρ ( a n , z n ) < − ε := r and we know that every pseudohyperbolic disk is a Euclideandisk (see [6], Chapter 1). If we rotate the disk by α n , where α n := | a n | /a n (interpreting α n = 1 if a n = 0 ), we do not change pseudohyperbolic distances; that is, for a, z ∈ D and α ∈ T , ρ ( αa, αz ) = ρ ( a, z ) . So, α n z n ∈ D ρ ( | a n | , − ε ) . Now we use the fact that the pseudohyperbolic disk D ρ ( | a n | , − ε ) is a Euclidean disk centered at the real number p n = 1 − r − r | a n | | a n | ∈ R with radius R n = 1 − | a n | − r | a n | r. Since | a n | → , there are finitely many a n for which | a n | ≤ − ε and finitely manycorresponding z n . If we show that the Blaschke product with zeros ( z n ) n ≥ N is a FrostmanBlaschke product, then Proposition 5.1 implies that the Blaschke product with zeros ( z n ) n is also a Frostman Blaschke product. Thus, we may assume that, for all n . we have | a n | ≥ − ε = r and | z n | ≥ − ε .The assumption that r = 1 − ε < | a n | = ρ (0 , | a n | ) , implies that is not in D ρ ( | a n | , r ) for all such a n , and therefore is not in the Euclidean disk D ( p n , R n ) . Since α n z n ∈ D ρ ( | a n | , r ) = D ( p n , R n ) , and D ( p n , R n ) is a Euclidean disk with center on the positivereal line, all points in D ( p n , R n ) have modulus greater than p n − R n . A computationshows that p n − R n = | a n | − r | a n | − r + r | a n | − r | a n | = ( | a n | − r )(1 + r | a n | )1 − r | a n | = | a n | − r − r | a n | . NTERPOLATION IN MODEL SPACES 11
Since we assume that | a n | > r we have p n − R n = ρ ( | a n | , r ) .Thus, | z n | = | α n z n | ≥ p n − R n = ρ ( | a n | , r ) . So − | z n | ≤ − ρ ( | a n | , r ) . Consequently,(11) − | z n | ≤ (1 − r )(1 − | a n | )(1 − r | a n | ) ≤ r − r (1 − | a n | ) . Thus, for C r := r − r we have − | z n | ≤ C r (1 − | a n | ) , for all n and we note that C r is a constant depending on r but independent of n . Similarly,since ρ ( a n , z n ) < r , we may interchange the roles of a n and z n above to see that − | a n | ≤ C r (1 − | z n | ) , where C r is a constant depending only on r (and, hence, only on ε ).From the work above, we see that (1 − | a m | ) ≍ (1 − | z m | ) ; that is, there are positiveconstants D and D independent of m with(12) D (1 − | a m | ) ≤ − | z m | ≤ D (1 − | a m | ) for all m. Now, for all z ∈ D and all j (see [6, p. 4]) ρ ( a j , z ) ≤ ρ ( a j , z j ) + ρ ( z j , z )1 + ρ ( a j , z j ) ρ ( z j , z ) . Thus, − ρ ( a j , z ) ≥ − (cid:18) ρ ( a j , z j ) + ρ ( z j , z )1 + ρ ( a j , z j ) ρ ( z j , z ) (cid:19) . Simplifying, we have (1 − | a j | )(1 − | z | ) | − a j z | ≥ (1 − ρ ( a j , z j ))(1 − ρ ( z j , z ))(1 + ρ ( a j , z j ) ρ ( z j , z )) . Thus, − | a j | | − a j z | ≥ − ρ ( a j , z j )(1 + ρ ( a j , z j ) ρ ( z j , z )) ! (cid:18) − | z j | | − z j z | (cid:19) . But by assumption ρ ( a j , z j ) ≤ r < for all j , so − | a j | | − a j z | ≥ (1 − r )4 (cid:18) − | z j | | − z j z | (cid:19) . By equation (12), we have − | z j | | − a j z | ≥ D (1 − r )4 (cid:18) − | z j | | − z j z | (cid:19) . Therefore, for all j | − a j z | ≥ D (1 − r )4 (cid:18) | − z j z | (cid:19) . So there is a positive constant C , independent of j , such that for all z ∈ D | − a j z | ≥ C (cid:18) | − z j z | (cid:19) . Choose ζ ∈ T and let z → ζ . Then(13) | − a j ζ | ≥ C | − z j ζ | . Since this holds for all ζ ∈ T , combining (11) and (13), we see that there is a constant C such that for all j , − | a j | | − a j ζ | ≥ C − | z j | | − z j ζ | . Thus, if ( a j ) is Frostman, so is ( z j ) and since this holds with the roles of a j and z j reversed,we have ( z j ) Frostman if and only if ( a j ) is Frostman. (cid:3) We note that the proof can be slightly shortened by using the characterization of Frost-man sequences due to Cohn that appears in (3). Since we can also obtain it directly, weprefer to do so.Theorem 5.2 should be compared with that of Matheson and Ross [9] who showed thatevery Frostman shift of a Frostman Blaschke product is Frostman; that is, if we start witha Frostman Blaschke product B and we consider ϕ a ◦ B where ϕ a ( z ) = ( a − z ) / (1 − az ) ,then ϕ a ◦ B is still a Frostman Blaschke product. We may think of this as saying that ifwe move the zeros of a Frostman Blaschke product in a systematic way (namely, to theplaces at which the Blaschke product assumes the value a ), the resulting product is stillFrostman. Their proof is based on a result of Tolokonnikov [12] (that is itself based on aresult of Pekarski [11]) and a theorem of Hru˘s˘cëv and Vinogradov, [8]. Corollary 5.3.
Let ( a n ) and ( z n ) be ρ -separated sequences with sup n ρ ( a n , z n ) ≤ λ < .Let B and C be the corresponding Blaschke products. Then ( a n ) is interpolating for K ∞ B if and only if ( z n ) is interpolating for K ∞ C .Proof. Suppose first that ( a n ) is interpolating for K ∞ B . Since ( a n ) is then interpolating for H ∞ and ( z n ) is ρ -separated with sup n ρ ( a n , z n ) ≤ λ < , it follows from Proposition 2.1that ( z n ) is interpolating for H ∞ . Similarly, the same is true if we interchange the rolesof z n and a n . The result now follows from Hru˘s˘cëv and Vingogradov’s work. (See also [5,(1.12)].) (cid:3) Acknowledgments.
Since August 2018, Pamela Gorkin has been serving as a ProgramDirector in the Division of Mathematical Sciences at the National Science Foundation(NSF), USA, and as a component of this position, she received support from NSF forresearch, which included work on this paper.Brett D. Wick’s research supported in part by NSF grants DMS-1800057 and DMS-1560955, as well as ARC DP190100970.Any opinions, findings, and conclusions or recommendations expressed in this materialare those of the authors and do not necessarily reflect the views of the National ScienceFoundation.
References [1] Akeroyd, John R.; Gorkin, Pamela Constructing Frostman-Blaschke products and applications tooperators on weighted Bergman spaces. J. Operator Theory 74 (2015), no. 1, 149–175.[2] Cima, Joseph A.; Matheson, Alec L.; Ross, William T. The Cauchy transform. Mathematical Surveysand Monographs, 125. American Mathematical Society, Providence, RI, 2006.
NTERPOLATION IN MODEL SPACES 13 [3] Cohn, William, Radial limits and star invariant subspaces of bounded mean oscillation, Amer. J.Math 108, (1986), 719–749.[4] Cohn, William, A maximum principle for star invariant subspaces, Houston J. Math 14 (1988), 23 -37.[5] Dyakonov, Konstantin M., A free interpolation problem for a subspace of H ∞ , Bull. Lond. Math.Soc. 50 (2018), no. 3, 477–486.[6] Garnett, John B., Bounded Analytic Functions, Revised first edition. Graduate Texts in Mathemat-ics, 236. Springer, New York, 2007.[7] Hoffman, Kenneth, Bounded analytic functions and Gleason parts. Ann. of Math. (2) 86 (1967),74–111.[8] Hru˘s˘cëv, S. V.; Vinogradov, S. A., Inner functions and multipliers of Cauchy type integrals. Ark.Mat. 19 (1981), no. 1, 23–42.[9] Matheson, Alec L.; Ross, William T., An observation about Frostman shifts. Comput. MethodsFunct. Theory 7 (2007), no. 1, 111–126.[10] Mortini Raymond; Rupp, Rudolf, A Space Odyssey: Extension Problems and Stable Ranks, Accom-panied by introductory chapters on point-set topology and Banach algebras, in preparation.[11] Pekarski˘i, A. A., Estimates of the derivative of a Cauchy-type integral with meromorphic densityand their applications. (Russian) Mat. Zametki 31 (1982), no. 3, 389–402, 474.[12] Tolokonnikov, V. A., Blaschke products with the Carleson-Newman condition, and ideals of thealgebra H ∞ . (Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 149 (1986),Issled. Line˘in. Teor. Funktsi˘i. XV, 93–102, 188; translation in J. Soviet Math. 42 (1988), no. 2, 1603–1610.[13] Tolokonnikov, V. A., Carleson’s Blaschke products and Douglas algebras. (Russian) Algebra i Analiz3 (1991), no. 4, 186–197; translation in St. Petersburg Math. J. 3 (1992), no. 4, 881–892.[14] Vinogradov S. A., Some remarks on free interpolation by bounded and slowly growing analyticfunctions, Zap. Nauchn. Sm. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) (1983), 35 – 46.[15] Zhu, Kehe Operator theory in function spaces. Second edition. Mathematical Surveys and Mono-graphs, 138. American Mathematical Society, Providence, RI, 2007. Pamela Gorkin, Department of Mathematics, Bucknell University, Lewisburg, PA USA17837
E-mail address : [email protected] Brett D. Wick, Department of Mathematics & Statistics, Washington University inSt. Louis, St. Louis, Missouri, USA
E-mail address ::