Blaschke Sets for Bergman Spaces

by Boris Korenblum

\underline{Abstract}.: We characterize subsets S of the open unit disk D such that every zero sequence for a Bergman space A^{p}, p > 0, with elements in S is Blaschke.

1. \underline{Introduction}.

The following definition is an extension of the notion of a Blaschke set introduced by Krzysztof Bogdan [B]. \underline{Definition}: We call S ⊂ D a Blaschke set for a class X of analytic functions on D = {z ∈ C : |z| < 1} if

(i) whenever 0 ≢ f ∈ X, and {z_{n}}_{n} are the zeros of f (counting multiplicities), with z_{n} ∈ S, the Blaschke condition holds:

∑_{n}(1 − |z_{n}|) < ∞ ;  (1)

(ii) whenever Z = {z_{n}}_{n} is a Blaschke sequence (i.e. (1) holds), with z_{n} ∈ S, there is an

f ∈ X whose zero sequence is Z. \underline{Remark}: If X is made up of functions of bounded Nevanlinna characteric then this definition reduces to (ii). If H^{∞} ⊂ X, it reduces to (i). \underline{Examples}:

1. Every subset of D is a Blaschke set for H^{p}, 0 < p < ∞.

2. For analytic Lipschitz classes Lip_{α}(D), α > 0, as well as for A^{∞} = {f : f^{(n)} ∈ H^{∞}, ∀n ≥ 0}, Blaschke sets are characterized by

∫_{0}^{2π} log dist(e^{it}, S)dt > −∞  (2)

where dist denotes the Euclidean distance. Note that for Lip_{α}(D) and A^{∞} the zero sequences Z are characterized by (1) and (2), with S replaced by Z.

3. The Blaschke sets S for the class D of analytic functions with finite Dirichlet integral are characterized by (2) (see [B]). Note that D-zero sequences cannot be described this way because there are f ∈ D whose zeros come arbitrarily close to every point of ∂D (see [C] and [SS]).

1991 Mathematics Subject Classification, Primary 30C15

The purpose of this paper is to obtain a description of the Blaschke sets for Bergman spaces A^{p}(p > 0) and growth spaces A^{−α}(α > 0). Recall that A^{p} consists of functions f analytic on D such that

∥f∥_{p}^{p} = ∫_{D} |f(z)|^{p} \frac{dxdy}{π} < ∞ ;

A^{−α} consists of analytic functions f with

∥f∥_{−α} = sup{(1 − |z|)^{α}|f(z)| : z ∈ D} < ∞ ;

We also consider the space A^{−∞} = ⋃_{α>0} A^{−α} = ⋃_{p>0} A^{p}.

We establish the following

\underline{Theorem}. A set S ⊂ D is a Blaschke set for any of the spaces A^{p}, A^{−α}, A^{−∞} if and only if (2) holds.

To prove this theorem we first reduce condition (2) to a form involving a collection of disjoint “tents” tightly surrounding S. The sufficiency of (2) then follows from the fact that “Stolz stars” S_{F} are A^{−∞}-Blaschke sets if the entropy κ̂(F) is finite (see (3) and [HKZ]). To prove the necessity of (2) we use some density concepts first introduced in [K1] and later refined in [S] and [HKZ]. \underline{Acknowledgement}: The suthor thanks Stefan Richter and Carl Sundberg for useful discussions. Special thanks are due to Carl Sundberg for bringing K. Bogdan’s work [B] to the author’s attention.

\underline{2. An equivalent form of (2)}.

We assume that S contains a disk centered at 0 of radius 1/\sqrt{2}.

We need some terminology.

A \underline{tent} is an open subset h of D bounded by an arc I ⊂ ∂D of length less than π/2 and two straight lines through the endpoints of I forming with I an angle of π/4. The closed arc \overline{I} will be called the \underline{base} of the tent h = h_{I}. A tent h is said to \underline{support} S if h ∩ S = ϕ but \overline{h} ∩ \overline{S} ≠ ϕ. A finite or countable collection of tents \{h_{n}\}_{n} is a \underline{belt} if h_{n} are pairwise disjoint and ⋃_{n} \overline{h}_{n} ⊃ ∂D. A collection of tents \{h_{n}\}_{n} is an \underline{S-belt} if h_{n} are

pairwise disjoint, S-supporting, and ⋃_{n} \overline{h}_{n} ⊃ ∂D\\overline{S}. Note that an S-belt does not have to

be a belt. If S is such that ∂D\\overline{S} ≠ ϕ, S-belts exist: to obtain one we start at an arbitrary point ζ_{0} ∈ ∂D\\overline{S} and, moving counterclockwise, consecutively find points ζ_{1}, ζ_{2} … such that the arcs between them are the bases of S-supporting tents; then we proceed similarly from ζ_{0} in the opposite direction. We thus obtain a system of tents whose bases cover a component of G = ∂D\\overline{S}. Continuing this process for all the components we obtain an S-belt.

An elementary computation shows that if h = h_{I} is a tent supporting S then

−|I| log \frac{1}{|I|} − c|I| ≤ ∫_{I} log dist(ζ, S)|dζ| ≤ −|I| log \frac{1}{|I|} + c|I|

where c is a numerical constant. We thus obtain

\underline{Lemma 1}. Let S be a subset of D such that ∂D\\overline{S} ≠ ϕ. Let \{h_{I_{n}}\}_{n} be an S-belt. Then (2) holds if and only if (A) the set F_{0} = \overline{S} ∩ ∂D has zero Lebesgue length; (B)

∑_{n} κ(I_{n}) < ∞ where κ(I) = |I| log \frac{2πe}{|I|} .

(κ(I) is called the κ length of I).

Note that (A) and (B) together are equivalent to

κ̂(F) = ∫_{∂D} log \frac{2π}{d(ζ, F)}|dζ| < ∞  (3)

where F = F_{0} ∪ Ξ and Ξ consists of the endpoints of those bases such that \overline{I}_{n} ⊂ G; d denotes the angular distance.

The quantity κ̂(F) is defined for all sets F ⊂ ∂D and is called the \underline{entropy} of F. Closed sets with finite entropy are called \underline{Beurling-Carleson sets}.

\underline{3. Sufficiency of (3)}.

Let Ξ_{1} ⊃ Ξ consist of \underline{all} endpoints of the bases I_{n} (including those that are in F_{0}). Pick an increasing sequence F_{1} ⊂ F_{2} ⊂ … of finite subsets of Ξ_{1} such that ⋃_{n} F_{n} = Ξ_{1}.

Then (3) implies

lim_{n→∞} κ̂(F_{n}) = κ̂(F) .

Each F_{n} determines a belt whose tents are based on complementary arcs of F_{n}. Let H_{n} be the union of these tents. (Note that some of these tents are not S-supporting because they contain ponts from S). The complement D\H_{n} = τ_{n} is a “Stolz Star”, i.e. the union of Stolz angles with vertices in F_{n} and apertures of π/2.

Since κ̂(F_{n}) are bounded, it follows that, whenever 0 ≢ f ∈ A^{−∞}, the Blaschke sums for those zeros of f lying in τ_{n} are bounded (see [HKZ], p. 118, Theorem 4.25). We have ∑_{n} τ_{n} ⊃ S and τ_{1} ⊂ τ_{2} ⊂ …, which implies that the Blaschke sum for the zeros of f lying

in S is finite.

\underline{4. Necessity of (3)}.

Suppose now that κ̂(F) = ∞. Given an arbitrary fixed p > 0 we are going to construct a sequence Z = {z_{n}}_{n}, z_{n} ∈ S, such that Z is an A^{p}-zero sequence but ∑(1 − |z_{n}|) = ∞. In addition to the standard tools of A^{−∞}-theory (density notions, premeasures, etc.) we will use some technical lemmas whose proofs are deferred to section 5.

Recall that F = F_{0} ∪ Ξ where F_{0} = \overline{S} ∩ ∂D and Ξ is a finite or countable set lying in G = ∂D\F_{0}. The cluster points (if any) of Ξ are in F_{0}.

We consider separately two cases depending on whether κ̂(F_{0}) is infinite or finite. \underline{Case 1}: κ̂(F_{0}) = ∞. By Lemma 2, s.5, there is a sequence \{ζ_{ν}\}_{1}^{∞} of distinct points in F_{0} such that the corresponding arcs {J_{ν}}_{1}^{∞} between ζ_{ν} and ζ_{ν+1} are pairwise disjoint, cover ∂D, i.e. ⋃_{ν} \overline{J}_{ν} = ∂D, and κ̂(\{ζ_{ν}\}) = ∞, which is equivalent to

∑_{ν=1}^{∞} κ(J_{ν}) = ∑_{ν=1}^{∞} |J_{ν}| log \frac{2πe}{|J_{ν}|} = ∞ .

(Note that lim_{ν→∞} ζ_{ν} = ζ_{1}). Construct a premeasure (see [K1], [K2], [HKZ]) dμ = p|dζ| − dσ whose positive part has the density

p(ζ) = log \frac{2π}{d(ζ, \{ζ_{ν}, ζ_{ν+1}\})}, ζ ∈ J_{ν}, ν ≥ 1 ,

and the negative singular part −dσ puts on every point ζ_{ν} a Dirac mass equal to −κ(J_{ν}). Although both positive and negative parts are infinite on ∂D, dμ is κ-bounded above, which means that there is a constant c > 0 such that for all arcs I ⊂ ∂D

μ(I) ≤ c|I| log \frac{2πe}{|I|} .

This enables us to consider a zero-free analytic function

f_{ε}(z) = exp\{ε ∫_{∂D} \frac{ζ + z}{ζ − z} dμ(ζ)\}

which is in A^{p} (or A^{−α}) provided that ε > 0 is sufficiently small, and p (or α) are arbitary but fixed positive numbers.

Now we use Lemma 3, s.5, to reduce all the singular masses at ζ_{ν} by a factor 1/2 and compensate for that by extra zeros of high multiplicity at z_{ν} ∈ S. We can ensure that the resulting function Φ is in A^{p}. The zeros z_{ν} of ϕ (counting multiplicities) form a non-Blaschke sequence of points from S (see Lemma 3 for details).

\underline{Case 2}: κ̂(F_{0}) < ∞. Then we must have κ̂(Ξ) = ∞. Recall that Ξ includes all the endpoint of the base arcs of the S-belt that are in G = ∂D\\overline{S}. Let \{J_{ν}\}_{ν} be the sequence of these arcs arranged by decreasing lengths. Then κ̂(F_{0}) < ∞ together with κ̂(Ξ) = ∞ yield

∑_{ν=1}^{∞} κ(J_{ν}) = ∑_{ν=1}^{∞} |J_{ν}| log \frac{2πe}{|I_{ν}|} = ∞ .

It is always possible to find a decreasing sequence 1 > λ_{1} > λ_{2} > … → 0 such that

∑_{ν=1}^{∞} λ_{ν}κ(J_{ν}) = ∞ .

Every \overline{J}_{ν} is the base of a tent h_{J_{ν}} that supports S; therefore there is at least one common point, say w_{ν}, in \overline{S}, \overline{h}_{J_{ν}} and D. Take every w_{ν} and repeat it [\frac{λ_{ν}κ(J_{ν})}{1−|w_{ν}|}] times. Let the resulting sequence be Z = {z_{k}}_{k}. \underline{Claim}: Z is a zero sequence for every A^{p}, p > 0. To prove the claim we employ the notion of \underline{upper} \underline{asymptotic} \underline{κ-density} of a sequence in D. There are several equivalent definitions of this density. We will use the definition based on \underline{radial} \underline{stars} (see [HKZ]):

For an arbitrary finite set M ⊂ ∂D let r_{M} denote the union of radii from 0 to points in M. If A = \{a_{k}\}_{k} is any sequence of points in D, we form the \underline{partial} \underline{Blaschke} \underline{sum} for A and r_{M}:

B(A, r_{M}) = ∑_{κ}{1 − |a_{k}| : a_{k} ∈ r_{M}} ,

and define

D^{+}(A) = lim sup_{κ̂(M)→∞} \frac{B(A, r_{M})}{κ̂(M)}  (4)

where lim sup is taken over all finite M ⊂ ∂D.

The following result, although short of a full characterization of A^{p}-zero sets, is sharp enough for our purposes.

\underline{Proposition}. (See [HKZ], p.130, Theorem 4.37). Let A = \{a_{κ}\}_{k} be a sequence of points in D and D^{+}(A) be the upper asymptotic κ-density of A. If D^{+}(A) < \frac{1}{p} then A is an A^{p}-zero sequence. If D^{+}(A) > \frac{1}{p} then A is not an A^{p}-zero sequence.

\underline{Remark}: This is a sharper version, due to Kristian Seip [S], of a simiilar but weaker result from [K1].

Now we can prove the claim by showing that D^{+}(Z) = 0. Let Q = \{q_{ν} = \frac{w_{ν}}{|w_{ν}|}\}_{ν}. Every arc J_{ν} contains exactly one point from Q, namely q_{ν}. Obviously, for computing D^{+}(Z) we can employ only those M that are finite subsets of Q. For such M we have

B(Z, r_{M}) ≤ ∑_{ν}{λ_{ν}κ(J_{ν}) : q_{ν} ∈ M}

and

κ̂(M) ≥ ∑_{ν}{κ(J_{ν}) : q_{ν} ∈ M}

(see Lemma 4, s.5). Therefore

\frac{B(Z, r_{M})}{κ̂(M)} ≤ ∑_{ν}\{λ_{ν}κ(J_{ν}) : q_{ν} ∈ M\}/ ∑_{ν}\{κ(J_{ν}) : q_{ν} ∈ M\} .

It is convenient to use the following notations:

K(M) = ∑_{ν}{κ(J_{ν}) : q_{ν} ∈ M} ,

K_{λ}(M) = ∑_{ν}{λ_{ν}κ(J_{ν}) : q_{ν} ∈ M} .

Let {M_{n}}_{n} be a sequence of subsets of Q such that κ̂(M_{n}) → ∞. Then we have

\frac{B(Z, r_{M_{n}})}{κ̂(M_{n})} ≤ \frac{K_{λ}(M_{n})}{K(M_{n})} .  (5)

Suppose that K(M_{n}) = O(1)(n → ∞). Then obviously the left-hand side of (5) tends to 0. Also, if K(M_{n}) → ∞, then the right-hand (and, with it, the left-hand) side of (5) tends to 0 because λ_{ν} ↓ 0. Therefore every sequence {M_{n}}, M_{n} ⊂ Q, κ̂(M_{n}) → ∞, contains a subsequence {M_{n_{k}}} = {M′_{k}}, n_{1} < n_{2} …, such that

lim_{k→0} \frac{B(Z, r_{M′_{k}})}{κ̂(M′_{k})} = 0 .

which implies D^{+}(Z) = 0. Thus we have obtained a non-Blaschke A^{p}-zero sequence {z_{k}} whose elements are in \overline{S}. Using how Lemma 5, s.5, we can replace z_{k} by nearby points z̃_{k} from S so that the new sequence {z̃_{k}}_{k} is still an A^{p}-zero sequence and non-Blaschke. This completes the proof of the Theorem.

\underline{5. Technical Lemma}.

We give below the statement of the technical lemmas we used in proving the Theorem, together with a brief outline of their proofs. \underline{Definition}: A sequence \{ζ_{n}\}_{1}^{∞} of distinct points in ∂D is called \underline{T-monotone} if the open arcs I_{n} between ζ_{n} and ζ_{n+1} are pairwise disjoint and ⋃_{n} \overline{I} = ∂D. Note that it follows

from this definition that lim_{n→∞} ζ_{∞} = ζ_{1}.

\underline{Lemma 2}. Every closed set F ⊂ ∂D of infinite entropy contains a T-monotone sequence {ζ_{n}}_{n} ⊂ F of infinite entropy.

\underline{Proof}: We have

κ̂(F) = ∫_{∂D} log \frac{2π}{d(ζ, F)}|dζ| = ∞ .

(d denotes the angular distance). By the Heine-Borel lemma there is a point ζ_{0} ∈ F such that every open arc J containing ζ_{0} has the property

∫_{J} log \frac{2π}{d(ζ, F)}|dζ| = ∞ .

Now we can find a nested system of open arcs such that J_{κ} ⊃ \overline{J}_{n+1}, ⋂_{n} I_{n} = \{ζ_{0}\},

and a finite set M_{k} ⊂ (J_{n}\\overline{J}_{n+1}) ∩ F such that

∫_{J_{n}\\overline{J}_{n+1}} log \frac{2π}{d(ζ, M_{n})}|dζ| ≥ 1, n ≥ 1 .

Taking the union E = ⋃_{n} M_{n} (or a suitable subset of E) and rearranging it in a sequence

will prove the Lemma.

\underline{Lemma 3}. Let f ∈ A^{p}(p > 0) have an “atomic singularity” at z = 1, i.e.

lim sup_{r→1^{−}}(1 − r) log |f(r)| = −2m < 0 .

If m_{1} < m then

(i) F (z) = e^{m_{1} \frac{1+z}{1−z}} f(z) is in A^{p}; (ii) whenever 0 ≠ α_{n} ∈ D and lim_{n→∞} α_{n} = 1, the function

f_{α_{n}}(z) = (\frac{α_{n} − z}{1 − \overline{α}_{n}z} · \frac{|α_{n}|}{α_{n}})^{N_{n}}F (z), where N_{n} = [\frac{m_{1}}{1 − |α_{n}|}] ,

tends to f in the metric of A^{p}.

\underline{Proof}: (i) For any r ∈ (0, ∞) the equation

\frac{1 − |z|^{2}}{|1 − z|^{2}} = r

defines a circle C_{r} internally tangent to ∂D at the point 1. Such circles are called orocycles. If f is in A^{p} and has atomic singularity m at 1, then the function

g(z) = e^{m \frac{1+z}{1−z}} f(z)

may not be in A^{p}; however, the integral

L(r) = \frac{1}{2π} ∫_{C_{r}} |1 − ζ|^{2}|g(ζ)|^{p}|dζ|

is finite and decreasing on (0, r), and

∫_{D} |f(z)|^{p} \frac{dxdy}{π} = ∫_{0}^{∞} e^{−mr}L(r)dr < ∞ .

This implies

∫_{D} |F (z)|^{p} \frac{dxdy}{π} = ∫_{0}^{∞} e^{−(m−m_{1})r}L(r)dr < ∞ .

(ii) then follows by the dominated convergence theorem \underline{Lemma 4}. If I ⊂ ∂D is an arc, M is an arbitrary subset of ∂D, and if at least one point from M is in \overline{I}, then

∫_{I} log \frac{2π}{d(ζ, M)}|dζ| ≥ κ(I) = |I| log \frac{2πe}{|I|} .

\underline{Proof}: The minimum of the integral on the left for a given arc I is attained when M is a one-point set, and this point is one of the endpoints of I. A direct computation yields the required result. \underline{Lemma 5}. Let f ∈ A^{p} have a zero at some point a ∈ D. For arbitrary α ∈ D define

f_{α}(z) = \frac{B_{α}(z)}{B_{a}(z)}f(z)

where B is a Blaschke factor:

B_{α}(z) = \frac{z − α}{1 − \overline{α}z} , B_{a}(z) = \frac{z − a}{1 − \overline{a}z} .

Then f_{α} tends to f in the metric of A^{p} as α → a.

The proof is immediate and left to the reader.

\underline{References}

[B] \underline{K. Bogdan}. On the zeros of functions with finite Dirichlet integral. Kodai Math. J., 19 (1996), 7-16.

[C] \underline{L. Carleson}. On the zeros of functions with bounded Dirichlet integral. Math. Z. 56 (1952), 289-295.

[HKZ] \underline{H. Hedenmalm}, \underline{B. Korenblum}, \underline{K. Zhu}. Theory of Bergman space. Graduate Texts in Mathematics, 199. Springer, 2000.

[K1] \underline{B. Korenblum}. An extension of the Nevanlinna theory. Acta Math. 135 (1975), 187-219.

[K2] \underline{B. Korenblum}. A Beurling-type theorem. Acta Math. 138 (1977), 265-293.

[S] \underline{K. Seip}. Beurling type density theorems in the unit disk. Invent. Math. 113 (1993), 21-39.

[SS] \underline{H.S. Shapiro} and \underline{A.L. Shields}. On the zeros of functions with finite Dirichlet integral and some related function spaces. Math. Z. 80 (1962), 217-229.

Department of Mathematics and Statistics University at Albany, State University of New York 1400 Washington Avenue, Albany, New York 12222 E-mail: borisko@math.albany.edu