A Carleson-type condition for interpolation in ... - Semantic Scholar

A CARLESON-TYPE CONDITION FOR INTERPOLATION IN BERGMAN SPACES

Alexander P. Schuster and Kristian Seip

Abstract.

An analogue of the notion of uniformly separated sequences, expressed in terms of canonical divisors, is shown to yield a necessary and sufficient condition for interpolation in the Bergman space Ap , 0 < p < ∞.

A sequence Γ = {zj } of distinct points in the open unit disk D = {z : |z| < 1} of the complex plane is a classical interpolation sequence if for every bounded sequence {aj }, there is a bounded analytic function f such that f (zj ) = aj for all j. A famous theorem of Carleson [3] states that Γ is a classical interpolation sequence if and only if it is uniformly separated, i.e. there exists a δ > 0 such that Y ¯¯ zj − zk ¯¯ ¯ ¯ for all k. (1) ¯ 1 − zj zk ¯ ≥ δ j:j6=k

Carleson’s result can be extended to the Hardy space H p , which, for 0 < p < ∞ consists of the functions f analytic in D with kf kpH p

1 = sup 0 0 such that Gk (0) ≥ δ

for all k,

(4)

where Gk (z) = Gφzk (Γ\{zk }) (z). Note that this theorem may be extended to weighted Bergman spaces with the standard weight (1 − |z|2 )α ; it is of no relevance whether Hedenmalm’s factorization theory applies. (It is proved in [8], for example, that when p = 2 and α > 1, there is no contractive divisor.) However, the theorem does fail for the spaces A−n , as can be seen by the example Γ(e, 2π/n) of [14]. For this particular sequence, the analogue of (4) holds, but the density condition of [13] is violated. It is not difficult to see that the condition (4) is necessary. A standard argument based on the closed graph theorem shows that if Γ is a set of interpolation, then there is a constant M > 0 such that the interpolation problem f (zj ) = aj can be solved by an f ∈ Ap satisfying X kf kpp ≤ M |aj |p (1 − |zj |2 )2 , (5) j

whenever {aj (1 − |zj |2 )2/p } ∈ `p . In particular, consider for each k the sequence ½ aj =

(1 − |zk |2 )−2/p 0

if j = k, if j = 6 k.

Letting fk (z) =

Gk (φzk (z))(φ0zk (z))2/p , Gk (0)

we see that fk is an Ap function with fk (zj ) = aj for all j. This follows from the fact that the map f (z) 7→ f (φζ (z))(φ0ζ (z))2/p acts isometrically in Ap . Moreover, because of the defining property of Gk , we see that if hk is any other solution to the interpolation problem, then khk kp ≥ kfk kp . Therefore, by (5), kfk kpp ≤ M , which implies that Gk (0) ≥ M −1/p . Consider now the sufficiency part of the theorem. Note that for 0 < p ≤ 1, if (4) holds, we have ¯p Z ¯¯X ¯ π X ¯ ¯ ak (1 − |zk |2 )2/p fk (z)¯ dA(z) ≤ p |ak |p (1 − |zk |2 )2 kGk kpp ¯ ¯ ¯ δ D k k π X = p |ak |p (1 − |zk |2 )2 < ∞, δ k

so that f (z) =

X

0 2/p 2 2/p Gk (φzk (z))(φzk (z))

ak (1 − |zk | )

Gk (0)

k

is in Ap and solves the interpolation problem. 3

(6)

It is natural to ask whether (4) implies convergence of (6) in Ap also for 1 < p < ∞. One can show that if Γ is interpolating for A2 , then by the extremal property of the functions Gk , (6) represents the minimal norm solution of the interpolation problem, and thus in particular (6) with p = 2 converges. For p > 3, however, it may happen that (6) diverges, as shown by the following argument. Suppose 0 6∈ Γ. Then, since Gk is a contractive zero divisor, we have Z Z |GΓ (φzk (z))|p |φ0zk (z)|2 |GΓ (z)|p dA(z) = dA(z) ≤ πkGΓ kpp = π. p p |G (φ (z))| |G (z)| k zk k D D It follows that

¯ ¯ ¯ GΓ (0) ¯p ¯ ¯ ¯ Gk (zk ) ¯ ≤ 1.

Since Γ is an Ap zero sequence, X (1 − |zk |2 )1+δ < ∞ k

for every δ > 0 by Jensen’s formula, but the series may diverge for δ = 0 [9]. We may therefore choose interpolation data ak =

Gk (0) (1 − |zk |2 )−1/p+² Gk (zk )

with ² > 0. But if (6) solves the interpolation problem, we have X f (0) = (1 − |zk |2 )3/p+² , k

which in general diverges if ² ≤ 1 − 3/p. The proof to be given below of the sufficiency of (4) makes essential use of the description of interpolation sequences obtained in [13]. We do not know a direct proof that (4) implies convergence of (6) when p = 2. In the general case 1 < p < ∞, we may also ask whether a direct proof can be found if (6) is suitably modifed. It is interesting to note that such a constructive proof can be given for interpolation in p H , 1 < p < ∞. To this end, observe that the direct analogue of (6) is f (z) =

X

0 1/p 2 1/p Bk (φzk (z))(φzk (z))

ak (1 − |zk | )

k

=

X k

µ ak

Bk (0)

1 − zk z 1 − |zk |2

¶1−2/p

BΓ (z) . (z − zk )BΓ0 (zk )

By a similar argument as above, this series may diverge if p > 2. However, consider instead the series X BΓ (z) f (z) = ak . (z − zk )BΓ0 (zk ) k

4

To begin with, assume that {ak } is finite. By duality, ¯ ¯ ¯ ¯ Z 2π ¯ ¯X a ¯ ¯X a 1 h(eiθ ) ¯ ¯ ¯ ¯ k k ˜ kf kH p = sup ¯ dθ¯ = sup ¯ h(zk )¯ , 0 0 iθ ¯ BΓ (zk ) 2π 0 e − zk ¯ khkLq =1 ¯ BΓ (zk ) khkLq =1 ¯ k

k

where q is the conjugate exponent of p (1/p + 1/q = 1) and Z 2π 1 h(eiθ ) ˜ h(z) = dθ. 2π 0 eiθ − z Because (1) implies that |BΓ0 (zk )| ≥

δ , 1 − |zk |2

an application of H¨ older’s inequality yields à !1/p à !1/q X X ˜ k )|q (1 − |zk |2 ) kf kH p ≤ δ −1 sup |ak |p (1 − |zk |2 ) |h(z . khkLq =1

k

k

˜ is a bounded map from Lq to H q , the proof that (1) implies that Γ is Since h 7→ h interpolating for H p is completed by an application of Carleson’s embedding theorem [11] and, in order to pass to infinite sequences {ak }, a normal family argument. The argument given above yields a simple constructive proof that (1) is sufficient for Γ to be interpolating for H p , 1 < p < ∞. A constructive proof of a different nature has previously been given by Amar [1]. In the case 0 < p ≤ 1, a proof by explicit construction of a linear operator of interpolation was found by Kabaˇıla; see [5], pp. 153–154. For bounded analytic functions, the existence of a linear operator of interpolation was proved by P. Beurling [4]. Clearly, the problem is more difficult in Ap , since there is no simple relation between the Gk ’s similar to the one between the Bk ’s. However, note that by introducing a suitable convergence factor ((1 − |zk |2 )/(1 − zk z))s in the k th term of the sum in (6), a calculation (see [14], p. 218) shows that (4) is sufficient for Γ to be interpolating for every space Ap−² with ² > 0. We turn now to the main problem, which consists of removing this ². As mentioned above, we will use the density introduced in [13], to be defined next. The sequence Γ is said to be uniformly discrete if there is an η > 0 such that ρ(zi , zj ) ≥ η for all i 6= j, where ρ(z, ζ) = |φζ (z)| is the pseudo-hyperbolic distance between points ζ, z ∈ D. For Γ uniformly discrete and 21 < r < 1, let P 1 1/2