BOUNDED POINT EVALUATIONS ON

THE BOUNDARIES OF L REGIONS

John E. McCarthy and Liming Yang

Abstract. An L region is a region G obtained by deleting from the open unit disc D a sequence of disjoint closed discs \overline{Δ(a_{n}, r_{n})}, whose centers a_{n} lie on the positive real axis and decrease monotonically to zero. We consider under what conditions lim_{x→0^{−}} f(x) exists for every function f in some space of analytic functions on G. We obtain the following results.

1. Every function in R^{p}(|z|^{t}dm(z)), where m denotes arc-length measure and 1 < p < ∞, has a limit from the left at zero if and only if

∑_{n=1}^{∞} \frac{r_{n}}{a_{n}^{q+t(q−1)}} < ∞, \frac{1}{p} + \frac{1}{q} = 1.

2. If a_{n} = a^{n}, and ω is harmonic measure for G, then every function in H^{p}(G), 1 < p < ∞, has a limit from the left at zero if and only if

∑_{n=1}^{∞} \frac{r_{n}^{q}}{a^{qn}ω(Γ_{n})^{q−1}} < ∞.

3. Under certain conditions on a_{n}, (e.g. a_{n} = a^{n} or a_{n} = \frac{1}{n^{k}}, for k > \frac{p−1}{2} ) every function in L_{a}^{p}(G) has a limit from the left at zero if and only if

∑_{n=0}^{∞} \frac{1}{a_{n}^{2}log \frac{1}{r_{n}}} < ∞, p = 2 or ∑_{n=0}^{∞} \frac{r_{n}^{2−q}}{a_{n}^{q}} < ∞, p > 2

The first author was partially supported by NSF grant DMS 9296099 1989 Mathematics Subject Classification Primary: 30E25,46E15,47B20 Secondary: 30D55,46E20

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0. Introduction

An L region is a region G obtained by deleting from the open unit disc D a sequence of disjoint closed discs \overline{Δ(a_{n}, r_{n})}, whose centers a_{n} lie on the positive real axis and decrease monotonically to zero. These regions were first studied by L. Zalcman in [15], where he proved that every bounded analytic function f on G has a left limit at zero (i.e. lim_{x→0^{−}} f(x) exists) if and only if ∑_{n=1}^{∞} \frac{r_{n}}{a_{n}} < ∞. In this article we consider under what conditions every element of other spaces of analytic functions on G, such as Bergman and Hardy spaces (see below for definitions), has a left limit at zero. The problem is related to finding finite codimensional invariant subspaces of those spaces (see, for example, [1] and [2] ).

For μ a finite positive Borel measure supported on \overline{G}, let R^{p}(\overline{G}, μ) denote the L^{p}(μ) closure of rational functions with poles off \overline{G}. The Hardy space H^{p}(G) consists of all analytic functions f : G → ℂ such that |f|^{p} has a harmonic majorant on G (see [3, p.203], [5, Chapter 3] and [13]). The Bergman space L_{a}^{p}(G) is the space of analytic functions on G whose p^{th} powers are integrable with respect to area measure on G.

If X is one of the above spaces, R^{p}(\overline{G}, μ), H^{p}(G) or L_{a}^{p}(G), we shall say zero is a bounded point evaluation (bpe) if the closure of zX does not equal X. Clearly, if every function in X has a left limit at zero, then zero is a bpe; the converse is valid if functions analytic in a neighborhood of zero are dense in X. We shall say zero is a bounded point derivation of order k if clos(z^{k+1}X) ≠ clos(z^{k}X).

R. Olin [11] (also see [12] ) showed that there is an L region G and a measure μ on the boundary of G such that R^{p}(\overline{G}, μ) is irreducible and zero is a bpe for R^{p}(\overline{G}, μ). S. Axler and P. Bourdon [2] gave an example of an L region G such that zero is a bpe for the Bergman space L_{a}^{p}(G). In both these cases, every function in the space has a left limit at zero.

Throughout this paper, we shall use A to denote area measure, m to denote arc-length measure, q to denote the conjugate index of p (q = \frac{p}{p−1}), and Γ_{n} to denote the boundary of Δ(a_{n}, r_{n}). We shall let c denote a positive constant, that may change from one appearance

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 3

to the next.

The paper will be organized as follows. In section 1, we assume that the measure μ is a weighted arclength measure on the boundary of G, of the form |z|^{t}dm(z). We prove (Theorem 1.5) that the following are equivalent for 1 < p < ∞: (a1) Zero is a bpe for R^{p}(\overline{G}, μ); (b1) Every function in R^{p}(\overline{G}, μ) has a left limit at zero; (c1)

∫_{∂G} \frac{dm(z)}{|z|^{q+t(q−1)}} < ∞

(d1)

∑_{n=1}^{∞} \frac{r_{n}}{a_{n}^{q+t(q−1)}} < ∞.

Section 2 obtains some results for the Hardy spaces H^{p}(G), under the assumption that a_{n} = a^{n}, for some a between 0 and 1. We prove (Theorem 2.10) that the following conditions are equivalent ( 1 < p < ∞ ): (b2) Every function in H^{p}(G) has a left limit at zero; (c2)

∫_{∂G} \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} < ∞

(d2)

∑_{n=1}^{∞} \frac{r_{n}^{q}}{a^{qn}ω(Γ_{n})^{q−1}} < ∞.

We discuss the same problem for Bergman spaces. In section 3 we give necessary and sufficient conditions for a point to be a bpe for the Bergman space in terms of the Bessel capacity, which are very similar to the results of Fernström and Polking for R^{p}(\overline{G}, A) [4]. In Section 4, we look for geometric conditions in terms of a_{n} and r_{n} for zero to be a bpe. We must again impose some assumptions on a_{n}, though they are less strict than those needed in section 2; for p = 2, the restriction is that

∑_{j=1}^{∞}(a_{j}^{2} + a_{j+1}^{2}) log \frac{c_{j}^{2} − c_{j+2}^{2}}{c_{j+1}^{2} − c_{j+2}^{2}} < ∞

4 JOHN E. MCCARTHY AND LIMING YANG

with an analogous condition for p > 2. The condition is satisfied if e.g. a_{n} = a^{n} or a_{n} = \frac{1}{n^{k}}, for k > \frac{p−1}{2}. For these a_{n}, we are able to prove (Theorems 4.5 and 5.1) that the following statements are equivalent for p ≥ 2: (a3) Zero is a bpe for L_{a}^{p}(G); (b3) Every function in L_{a}^{p}(G) has a left limit at zero; (d3) If p = 2, then

∑_{n=1}^{∞} \frac{1}{a_{n}^{2}log \frac{1}{r_{n}}} < ∞.

If p > 2, then

∑_{n=1}^{∞} \frac{r_{n}^{2−q}}{a_{n}^{q}} < ∞.

Because \frac{1}{z} is in L_{loc}^{p}(A) for p < 2, it is clear that for these p zero is never a bpe for L_{a}^{p}(G).

Our techniques also apply to the question of when the k^{th} derivative of every function in a space has a left limit at zero, and when zero is a bounded point derivation of order k. As the proofs are very similar, we state the theorems to encompass bounded point derivations, though we give proofs only for k = 0.

The general condition for bpe’s to exist in terms of capacity does not solve the problem in geometric terms. It does simplify a bare hands approach to getting a sufficient condition because of capacity’s sub-additivity; but it doesn’t seem to help in getting necessary conditions. It would be interesting to know if there were an analogous necessary and sufficient condition for Hardy spaces, which are much harder to localize.

Before closing this section, we state and prove a result that will be used in this article. Let X be a Banach space and let T be a bounded linear operator on X. Let Ω be a

bounded domain and let π be a representation from H^{∞}(Ω) to the set of all bounded linear operators on X such that π(1) = I and π(z) = T.

Theorem 0. Suppose π is weak-* continuous and λ ∈ ∂Ω is a peak point for H^{∞}(Ω); then clos(T − λ)X = X.

Proof: As in the proof of Theorem 3.1 in [7], one finds a sequence of functions {f_{n}} ⊂ H^{∞}(Ω) such that (z − λ)f_{n}(z) goes to 1 pointwise in Ω and ∥(z − λ)f_{n}(z)∥_{∞} ≤ M < ∞.

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 5

So (z − λ)f_{n}(z) converges to 1 in the weak-* topology. Thus, π((z − λ)f_{n}) converges to I weakly. But

π((z − λ)f_{n}) = (T − λI)π(f_{n}),

so clos(T − λ)X = X.

In our cases, the operator T is multiplication by z; so we may always assume that zero is not a peak point for H^{∞}(G), that is,

∑_{n=1}^{∞} \frac{r_{n}}{a_{n}} < ∞

(see [15]). So, throughout the paper we shall always make the following assumption: Assumption (*): \frac{r_{n}}{a_{n}} → 0 as n → ∞.

1. Rational approximation with respect to weighted arclength

In this section, we shall assume that the measure μ is supported on the boundary of G, and that R^{p}(\overline{G}, μ) is irreducible, i.e. there is no partition of ∂G as E_{1} ∪ E_{2} such that

R^{p}(\overline{G}, μ) = R^{p}(\overline{G}, μ|_{E_{1}}) ⊕ R^{p}(\overline{G}, μ|_{E_{2}}).

Therefore, the measures μ and m are mutually absolutely continuous except at zero. We assume for convenience that μ{0} = 0. Let

w(z) = \frac{dμ}{dm}(z).

Theorem 1.1. Suppose that 1 < p < ∞ and

∫ \frac{dm(z)}{|z|^{q}w(z)^{q−1}} < ∞;

then every function in R^{p}(\overline{G}, μ) has a left limit at zero, and therefore 0 is a bounded point evaluation for R^{p}(\overline{G}, μ).

6 JOHN E. MCCARTHY AND LIMING YANG

Proof: Let r be a rational function with poles off \overline{G}, λ < 0 and |λ| < \frac{1}{2}. Then using the Cauchy integral formula, we have

r(λ) = \frac{1}{2πi} ∫_{∂G} \frac{r(z)}{z − λ}dz

Hence,

|r(λ)| ≤ \frac{1}{2π} ∫_{∂G} \frac{|r(z)|}{|z − λ|}dm(z)

≤ \frac{1}{2π} (∫_{∂G\∂D} \frac{|r(z)|}{|z − λ|}dm(z) + ∫_{∂D} \frac{|r(z)|}{|z − λ|}dm(z))

≤ \frac{1}{2π} (∫_{∂G\∂D} \frac{|r(z)|}{|z|}dm(z) + 2 ∫_{∂D} |r(z)|dm(z))

≤ \frac{1}{2π} ((∫ \frac{dm(z)}{|z|^{q}w(z)^{q−1}} )^{\frac{1}{q}} ∥r∥_{p} + 2(∫ \frac{dm(z)}{w(z)^{q−1}} )^{\frac{1}{q}} ∥r∥_{p})

≤ c (∫ \frac{dm(z)}{|z|^{q}w(z)^{q−1}} )^{\frac{1}{q}} ∥r∥_{p}

Thus, there is a constant M > 0 such that for every function f ∈ R^{p}(\overline{G}, μ), we have

|f(λ)| ≤ M∥f∥_{p}

Therefore the functionals of evaluation at λ remain bounded in norm as λ tends to zero, ℜ(λ) < 0. So some sequence converges weakly to a functional, which can be represented by a vector k ∈ L^{q}(μ). Because the rational functions are dense, any other sequence must have the same weak-limit. Therefore

lim_{λ→0−} f(λ) = ∫ f\overline{k}dμ.

Hence, every function has a left limit at zero and zero is a bpe. □

In the rest of this section, we are interested in getting necessary conditions. Let K be a compact subset of ℂ; a function f is called admissible for K if it is analytic on the component of ℂ_{∞} \ 𝕂 containing ∞, has modulus at most one there, and vanishes at ∞. The analytic capacity of K is defined to be:

γ(K) = sup{|f′(∞)| : f is admissible for K}

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 7

Let L be a compact subset of [0, 1],

h^{L}(z) = \frac{1}{2} ∫_{L} \frac{1}{z − x}dx

and

f^{L}(z) = \frac{e^{h^{L}} − 1}{e^{h^{L}} + 1}.

It is proved in [14, p. 15] that

|f^{L}(z)| ≤ 1, for z ∉ L

and

(f^{L})′(∞) = \frac{1}{4}m(L) = γ(L)

Lemma 1.2. For z ≠ 0, we have

lim_{L→0} \frac{zf^{L}(z)}{(f^{L})′(∞)} = 1

where L → 0 means that max{y : y ∈ L} → 0. Proof: Since lim_{L→0} h^{L}(z) = 0, we have

lim_{L→0} \frac{zf^{L}(z)}{(f^{L})′(∞)} = lim_{L→0} \frac{z}{e^{h^{L}} + 1} \frac{e^{h^{L}} − 1}{(f^{L})′(∞)}
= \frac{1}{2} lim_{L→0} \frac{e^{h^{L}} − 1}{h^{L}(z)} \frac{2zh^{L}(z)}{(h^{L})′(∞)}
= 1.□

Let 0 < a < 1, and let A_{j} be the annulus given by:

A_{j} = {z : a^{j+1} ≤ |z| < a^{j}}.

Let dμ(z) = |z|^{t}dm(z), and let

C_{j} = ∪_{a_{n}∈A_{j}}Γ_{n}.

8 JOHN E. MCCARTHY AND LIMING YANG

Lemma 1.3. If 1 < p < ∞ and the integrals

∫_{C_{j}} \frac{dm(z)}{|z|^{q}|z|^{t(q−1)}}

do not converge to zero as j tends to ∞, then clos(zR^{p}(\overline{G}, μ)) = R^{p}(\overline{G}, μ).

Proof: For simplicity, assume t ≥ 0 (for t < 0, the proof is almost identical). By passing to a subsequence, if necessary, we may assume that there is an ϵ > 0 such that

∫_{C_{j}} \frac{dm(z)}{|z|^{q}|z|^{t(q−1)}} ≥ ϵ.

Hence,

\frac{m(C_{j})}{a^{q(j+1)}a^{t(q−1)(j+1)}} ≥ ϵ.

Let L_{j} = ℝ ∩ (∪_{⅁_{⋉}∈𝔸_{ℷ}}Δ(⅁_{⋉}, ∖_{⋉})) and

f_{j}(z) = f^{L_{j}}(z).

We shall show that \frac{zf_{j}(z)}{f′_{j}(∞)} remain bounded in norm, so have a weak-cluster point, which by Lemma 1.2 is the function 1. Therefore 1 is in

clos(zR^{p}(\overline{G}, μ)),

so clos(zR^{p}(\overline{G}, μ)) = R^{p}(\overline{G}, μ), and 0 is not a bpe.

We first calculate the integral of |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} over those C_{n} such that |n − j| > 1.

Note that if f(z) is an admissible function for L, then

|f(z)| ≤ \frac{γ(L)}{d(z, L)}

for z ∈ L^{c} (see [6, p. 201] ). Therefore, if n ≠ j − 1, j, j + 1, we get that

∫C_{n} |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} |z|^{t}|dz|
≤ \frac{1}{γ(L_{j})^{p}} a^{pn} ∫_{C_{n}} \frac{γ(L_{j})^{p}}{d(z, A_{j})^{p}} |z|^{t}dm(z)
≤ \frac{2a^{pn}}{a^{p(n+1)}(1 − a)^{p}} μ(C_{n}),

where we used the assumption (∗) that \frac{r_{m}}{a_{m}} is small. Hence,

∫_{∪_{n≠j−1,j,j+1}C_{n}} |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} |z|^{t}dm(z) ≤ \frac{2}{a^{p}(1 − a)^{p}} μ(∂G) < ∞.  (I)

Next, we calculate the integral over C_{j}:

∫_{C_{j}} |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} |z|^{t}dm(z) ≤ \frac{1}{γ(L_{j})^{p}} a^{pj}a^{tj} ∫_{C_{j}} dm(z)
≤ \frac{c}{m(C_{j})^{p}} a^{pj}a^{tj}m(C_{j})
≤ c (\frac{a^{qj}a^{t(q−1)j}}{m(C_{j})} )^{p−1}
≤ c (\frac{1}{a^{q}a^{t(q−1)}ϵ})^{p−1}
≤ c.  (II)

To calculate the integral over C_{j+1}, let

B = {z : d(z, L_{j}) < γ(L_{j})}

and

B_{k} = {z : 2^{k}γ(L_{j}) ≤ d(z, L_{j}) < 2^{k+1}γ(L_{j})},

10 JOHN E. MCCARTHY AND LIMING YANG

where k = 0, 1, 2, ...

∫C_{j+1} |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} |z|^{t}dm(z)
= ∫_{C_{j+1}∩B} |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} |z|^{t}dm(z) + ∑_{k=0}^{∞} ∫_{C_{j+1}∩B_{k}} |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} |z|^{t}dm(z)
≤ \frac{a^{p(j+1)}}{γ(L_{j})^{p}} a^{t(j+1)}2πγ(L_{j}) + ∑_{k=0}^{∞} a^{p(j+1)}a^{t(j+1)} ∫_{C_{j+1}∩B_{k}} (\frac{1}{d(z, L_{j})})^{p} dm(z)
≤ c\frac{a^{p(j+1)}a^{t(j+1)}}{γ(L_{j})^{p−1}} + c ∑_{k=1}^{∞} a^{p(j+1)}a^{t(j+1)} (\frac{1}{2^{k}γ(L_{j})})^{p} 2^{k}γ(L_{j})
≤ c\frac{a^{p(j+1)}a^{t(j+1)}}{γ(L_{j})^{p−1}}
≤ c (\frac{a^{q(j+1)}a^{t(q−1)(j+1)}}{m(C_{j})} )^{p−1}
≤ c(\frac{1}{ϵ})^{p−1}
< ∞.  (III)

Similarly, we get that

∫_{C_{j−1}} |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} |z|^{t}dm(z) ≤ c < ∞.  (IV)

Adding I - IV, we get

∫_{∂G} |\frac{zf_{j}(z)}{f′_{j}(∞)}|^{p} dμ ≤ c < ∞,

where c does not depend on j; so the functions

\frac{zf_{j}(z)}{f′_{j}(∞)}

are uniformly bounded, as desired. □

Theorem 1.4. For 1 < p < ∞,if

∫_{∂G} \frac{dm(z)}{|z|^{q}|z|^{t(q−1)}} = ∞

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 11

then clos(zR^{p}(\overline{G}, μ)) = R^{p}(\overline{G}, μ).

Proof: We keep the notation of Lemma 1.3, and by its result we may assume that

∫_{C_{n}} \frac{dm(z)}{|z|^{q}|z|^{t(q−1)}} → 0.

So for each k, choose N_{k} such that

1 ≤ ∑_{n=k}^{N_{k}} ∫_{C_{n}} \frac{dm(z)}{|z|^{q}|z|^{t(q−1)}} ≤ 2

Let

B_{j} = A_{j−1} ∪ A_{j} ∪ A_{j+1}

and

g_{k}(z) = ∑_{n=k}^{N_{k}} ∫_{C_{n}} \frac{dm(ζ)}{|ζ|^{q}|ζ|^{t(q−1)}} \frac{zf_{n}(z)}{f′_{n}(∞)}

Then

∫C_{j}|g_{k}(z)|^{p}dμ(z)

≤ 2^{p} ∫_{C_{j}} | ∑_{a_{n}∉B_{j}} ∫_{C_{n}} \frac{dm(ζ)}{|ζ|^{q}|ζ|^{t(q−1)}} \frac{zf_{n}(z)}{f′_{n}(∞)}|^{p} dμ(z)

+ 2^{p} ∫_{C_{j}} | ∑_{n=j−1}^{j+1} ∫_{C_{n}} \frac{dm(ζ)}{|ζ|^{q}|ζ|^{t(q−1)}} \frac{zf_{n}(z)}{f′_{n}(∞)}|^{p} dμ(z).

Denote the first integral by F_{j} and the second integral by S_{j}. Using Minkowski’s inequality, we have

F_{j} ≤ ( ∑_{a_{n}∉B_{j}} ∫_{C_{n}} \frac{dm(ζ)}{|ζ|^{q}|ζ|^{t(q−1)}} (∫_{C_{j}} |\frac{zf_{n}(z)}{f′_{n}(∞)}|^{p} dμ)^{\frac{1}{p}} )^{p}

≤ ( ∑_{a_{n}∉B_{j}} ∫_{C_{n}} \frac{dm(ζ)}{|ζ|^{q}|ζ|^{t(q−1)}} (∫_{C_{j}} |\frac{z}{d(z, L_{n})}|^{p} dμ)^{\frac{1}{p}} )^{p}

≤ ( ∑_{a_{n}∉B_{j}} ∫_{C_{n}} \frac{dm(ζ)}{|ζ|^{q}|ζ|^{t(q−1)}} )^{p} \frac{1}{(1 − a)^{p}} μ(C_{j})

≤ 2^{p} \frac{1}{(1 − a)^{p}} μ(C_{j})

12 JOHN E. MCCARTHY AND LIMING YANG

To calculate S_{j}:

(∫_{C_{j+1}} \frac{dm(ζ)}{|ζ|^{q}|ζ|^{t(q−1)}} )^{p} ∫_{C_{j}} |\frac{zf_{j+1}(z)}{f′_{j+1}(∞)}|^{p} dμ

≤ c\frac{γ(L_{j+1})^{p}}{a^{pq(j+2)}a^{t(q−1)p(j+2)}} \frac{a^{pj}a^{tj}}{γ(L_{j+1})^{p}} γ(L_{j})
≤ c\frac{1}{a^{qj}a^{t(q−1)j}} γ(L_{j})
≤ c ∫_{C_{j}} \frac{dm(z)}{|z|^{q}|z|^{t(q−1)}}

A similar computation yields that

(∫_{C_{n}} \frac{dm(ζ)}{|ζ|^{q}|ζ|^{t(q−1)}} )^{p} ∫_{C_{j}} |\frac{zf_{n}(z)}{f′_{n}(∞)}|^{p} dμ ≤ c ∫_{C_{j}} \frac{dm(z)}{|z|^{q}|z|^{t(q−1)}}

for n = j, j − 1. Hence,

S_{j} ≤ 3 × 3^{p}c ∫_{C_{j}} \frac{dm(z)}{|z|^{q}|z|^{t(q−1)}} ,

and so ∫_{∂G} |g_{k}(z)|^{p}dμ is bounded by some constant independent of k. It follows from Lemma 1.2 that there exists a subsequence {g_{k_{n}}(z)} such that

g_{k_{n}}(z) → d

pointwise except at 0, where d is a constant between 1 and 2. Therefore

g_{k_{n}}(z) → d

weakly, and as g_{k_{n}}(z) is divisible by z in R^{p}(\overline{G}, μ), 1 is in clos(zR^{p}(\overline{G}, μ)). Therefore 0 is not a bpe. □

All the above proofs work mutatis mutandis if one uses the Cauchy integral formula for r^{(k)}(λ); doing this, and combining Theorems 1.1 and 1.4, we get:

Theorem 1.5. Let dμ(z) = |z|^{t}dm(z), and 1 < p < ∞. The following statements are equivalent:

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 13

(a1) Zero is a bounded point derivation of order k for R^{p}(\overline{G}, μ). (b1) The k^{th} derivative of every function in R^{p}(\overline{G}, μ) has a left limit at zero. (c1)

∫ \frac{dm(z)}{|z|^{(k+1)q}|z|^{t(q−1)}} < ∞.

(d1)

∑_{n=1}^{∞} \frac{r_{n}}{a_{n}^{(k+1)q+(q−1)t}} < ∞.

Example 1.6. It follows that, for any k ≤ ∞, one can choose a_{n} and r_{n} so that closz^{j+1}R^{p}(\overline{G}, μ) ⊊ closz^{j}R^{p}(\overline{G}, μ) for j < k, and, if ∞ > j ≥ k, closz^{j}R^{p}(\overline{G}, μ) = closz^{k}R^{p}(\overline{G}, μ).

Example 1.7. Suppose that

∑ \frac{r_{n}}{a_{n}} < ∞

and

∑ \frac{r_{n}}{a_{n}^{2}} = ∞.

Then 0 is a bpe for R^{2}(\overline{G}, \frac{dm(z)}{|z|}) and 0 is not a bpe for R^{2}(\overline{G}, dm(z)). Hence, the operator of multiplication by z on R^{2}(\overline{G}, \frac{dm(z)}{|z|}) is not quasisimilar to the operator of multiplication by z on R^{2}(\overline{G}, dm(z)). Therefore, in [9], the hypo-Dirichlet assumption cannot be weakened even to such relatively nice infinitely connected domains as these.

2. Rational approximation with respect to harmonic measure and H^{p}(G)

In this section, we assume that a_{n} = a^{n}, n = 1, 2, 3, ... for some a between 0 and 1. Let us first note that there is an equivalent way of defining the Hardy spaces H^{p}(G) (see

[3, p. 205]). Fix a point b ∈ G, let ω(G, ., b) be the harmonic measure for G evaluated at b and let {G_{n}} be a sequence of regions contained in G such that b ∈ G_{1}, clos(G_{n}) ⊂ G_{n+1} and G = ∪_{n=1}^{∞}G_{n}. Let ω_{n} = ω(G_{n}, ., b). The Hardy space H^{p}(G) consists of all analytic functions f : G → ℂ such that

sup_{n} ∫_{∂G_{n}} |f|^{p}dω_{n} < ∞.

Since ∫_{∂G_{n}} |f|^{p}dω_{n} is an increasing sequence, the norm can be defined by

∥f∥ = sup_{n} ∫_{∂G_{n}} |f|^{p}dω_{n} = lim_{n} ∫_{∂G_{n}} |f|^{p}dω_{n}.

Let A(a; R_{1}, R_{2}) be the annulus centered at a with radii R_{1} and R_{2}.

Lemma 2.1. Let Γ_{1} = {z : |z| = 1}, Γ_{2} = {z : |z| = R_{1}} and Γ_{3} = {z : |z| = R_{2}}, where 1 < R_{1} < R_{2}. Let Ω = A(0; 1, R_{2}). Then there are two constants M_{0}, M_{1} > 0 (depending only on R_{1}, R_{2}) such that, for any arc I on Γ_{1}, and any point z in Γ_{2}

M_{0}m(I) ≤ ω(Ω, I, z) ≤ M_{1}m(I).

Proof: By Harnack’s inequality, there are two constants M′_{0}, M′_{1} > 0 such that

M′_{0}ω(Ω, I, z′) ≤ ω(Ω, I, z) ≤ M′_{1}ω(Ω, I, z′)

for all z, z′ ∈ Γ_{2} (see [5, p. 19] ). Let I and J be two arcs of equal length on Γ_{1}. By symmetry, one sees that

M′_{0}ω(Ω, J, z) ≤ ω(Ω, I, z) ≤ M′_{1}ω(Ω, J, z)

for all z ∈ Γ_{2}. Therefore, there are M′′_{0}, M′′_{1} > 0 such that

M′′_{0}m(I)ω(Ω, Γ_{1}, z) ≤ ω(Ω, I, z) ≤ M′′_{1}m(I)ω(Ω, Γ_{1}, z)

We know that

ω(Ω, Γ_{1}, z) = \frac{log \frac{R_{2}}{R_{1}}}{logR_{2}}

since log \frac{R_{2}}{|z|} is a harmonic function on Ω. Let M_{0} = M′′_{0} \frac{log \frac{R_{2}}{R_{1}}}{logR_{2}} and M_{1} = M′′_{1} \frac{log \frac{R_{2}}{R_{1}}}{logR_{2}} . The lemma is proved. □

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 15

The proof of the following lemma was shown to us by Albert Baernstein II.

Lemma 2.2. Let G be an L region. Let Γ_{n}^{1} = Γ_{n}, Γ_{n}^{2} = {z : |z − a_{n}| = 2r_{n}} and Γ_{n}^{3} = {z : |z − a_{n}| = 4r_{n}}. Suppose A(a_{n}; r_{n}, 4r_{n}) ⊂ G. Fix a point b ∈ G \ ∪A(a_{n}; r_{n}, 4r_{n}). Then there are two constants M_{0}, M_{1} > 0 such that

M_{0} ≤ \frac{ω(G, I, b)}{ω(G, J, b)} ≤ M_{1}

whenever I and J are arcs of equal length on Γ_{n}^{1}.

Proof: Let ϕ(z) = \frac{z−a_{n}}{r_{n}}. Fix a point z ∈ Γ_{n}^{2}. Using Lemma 2.1 and the fact that harmonic measures are conformally invariant, we conclude that

ω(G, I, z) = ω(ϕ(G), ϕ(I), ϕ(z))

≥ ω(A(0; 1, 4), ϕ(I), ϕ(z)) ≥ M_{0}m(ϕ(I))

≥ M_{0} \frac{m(I)}{r_{n}}

On the other hand,

ω(ϕ(G), ϕ(I), ϕ(z)) ≤ ω((Γ̂_{n}^{1})^{c}, ϕ(I), ϕ(z))

≤ M_{1}m(ϕ(I))

= M_{1} \frac{m(I)}{r_{n}}

where K̂ denote the polynomial convex hull of K. Hence, there are two constants M_{0}, M_{1} > 0 such that

M_{0} ≤ \frac{ω(G, I, z)}{ω(G, J, z)} ≤ M_{1}

for all z ∈ Γ_{n}^{2}. However,

\frac{ω(G, I, b)}{ω(G, J, b)} = \frac{∫_{Γ_{n}^{2}} ω(G, I, z)dω(G \ Γ̂_{n}^{2}, ., b)}{∫_{Γ_{n}^{2}} ω(G, J, z)dω(G \ Γ̂_{n}^{2}, ., b)}

Therefore,

M_{0} ≤ \frac{ω(G, I, b)}{ω(G, J, b)} ≤ M_{1}

16 JOHN E. MCCARTHY AND LIMING YANG

□

Theorem 2.3. Let b ∈ G and ω = ω(G, . b). Suppose that

∫ \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} = ∞

Then

clos(zR^{p}(\overline{G}, ω)) = R^{p}(\overline{G}, ω)

Proof: From Lemma 2.2, it is easy to show that

∫ \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} = ∞

if and only if

∑_{k=0}^{∞} \frac{r_{k}^{q}}{a^{qk}ω(Γ_{k})^{q−1}} = ∞

Let the function h_{j}(z) be \frac{z}{z − a^{j}} instead of \frac{zf_{j}(z)}{f′_{j}(∞)} and mimic the proofs of Lemma 1.3 and Theorem 1.4 to conclude that either h_{j} or some linear combination tends to a non-zero constant, which is therefore in clos(zR^{p}(\overline{G}, ω)) (the calculations are much less messy in this case, because of the separation of the centers a^{k}). □

Corollary 2.4. The following statements are equivalent for 1 < p < ∞:

(a) Zero is a bounded point evaluation for R^{p}(\overline{G}, ω);

(b) Every function in R^{p}(\overline{G}, ω) has a left limit at zero; (c)

∫ \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} < ∞

(d)

∑_{k=0}^{∞} \frac{r_{k}^{q}}{a^{qk}ω(Γ_{k})^{q−1}} < ∞

Corollary 2.5. Suppose that 1 < p < ∞ and

∫ \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} = ∞

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 17

Then not every function in H^{p}(G) has a left limit at zero.

Proof: This follows from the inclusion R^{p}(\overline{G}, ω) ⊆ H^{p}(G). □

Remark: We would like to prove that 0 is a bpe for H^{p}(G) if and only if the preceding integral is finite. We cannot do this, however, because do not know whether the functions analytic at zero are dense in H^{p}(G). The remainder of this section is devoted to proving that if the integral in 2.5 is finite, every function in the space H^{p}(G) does have a left limit at zero.

Let

Γ_{k}^{l} = \{z : |z − a^{k}| < (1 + \frac{1}{l})r_{k}\}

and

G_{l} = D \ (\{z : |z| ≤ \frac{1 + a}{2}a^{l}\} ∪ (∪_{k=0}^{l}Γ̂_{k}^{l}))

For definiteness, fix −\frac{1}{2} as the norming point for ω.

Lemma 2.6. Let C_{k} in G be small circles that surround Γ_{k} and do not intersect, and let C_{0} be a circle centered at 0 with radius slightly less than one. Suppose that

∫ \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} < ∞

Then for −\frac{1}{2} ≤ λ ≤ 0,

L_{λ}(f) = \frac{1}{2πi} ∑_{k=0}^{∞} ∫_{C_{k}} \frac{f(z)}{z − λ}dz

is a bounded linear functional on H^{p}(G). Moreover,

lim_{λ→0−} L_{λ}(f) = L_{0}(f)

for every f ∈ H^{p}(G).

Proof: Let ω_{l} = ω(G_{l}, ., −\frac{1}{2}). We have the following computation:

|∑_{k=n}^{m} \frac{1}{2πi} ∫_{C_{k}} \frac{f(z)}{z − λ}dz|

≤ ∑_{k=n}^{m} \frac{1}{2π} ∫_{Γ_{k}^{l}} |\frac{f(z)}{z − λ}| dm(z)
≤ \frac{1}{2π} (∑_{k=n}^{m} ∫_{Γ_{k}^{l}} \frac{dm(z)}{|z|^{q}(\frac{dω_{l}}{dm(z)})^{q−1}} )^{\frac{1}{q}} ∥f∥

We know that ω_{l}(Γ_{k}^{l}) → ω(Γ_{k}), as l → ∞. Using Lemma 2.2, we see that there exists an absolute constant c > 0 such that for l large enough

∫_{Γ_{k}^{l}} \frac{dm(z)}{|z|^{q}(\frac{dω_{l}}{dm(z)})^{q−1}} ≤ c ∫_{Γ_{k}} \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}}

So

|∑_{k=n}^{m} \frac{1}{2πi} ∫_{C_{k}} \frac{f(z)}{z − λ}dz| ≤ c (∑_{k=n}^{m} ∫_{Γ_{k}} \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} )^{\frac{1}{q}} ∥f∥

Therefore, L_{λ} is a bounded linear functional for λ ≤ 0.

For ϵ > 0, there is an N such that

( ∑_{k=N+1}^{∞} ∫_{Γ_{k}} \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} )^{\frac{1}{q}} < ϵ

On the other hand,

|∑_{k=1}^{m} \frac{1}{2πi} ∫_{C_{k}} \frac{f(z)}{z − λ}dz − ∑_{k=1}^{m} \frac{1}{2πi} ∫_{C_{k}} \frac{f(z)}{z}dz|

≤ 2c ( ∑_{k=N+1}^{m} ∫_{Γ_{k}} \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} )^{\frac{1}{q}} ∥f∥

+ |∑_{k=1}^{N} \frac{1}{2πi} ∫_{C_{k}} \frac{f(z)}{z − λ}dz − ∑_{k=1}^{N} \frac{1}{2πi} ∫_{C_{k}} \frac{f(z)}{z}dz|

Hence,

|L_{λ}(f) − L_{0}(f)| ≤ cϵ∥f∥ + |∑_{k=1}^{N} \frac{1}{2πi} ∫_{C_{k}} \frac{f(z)}{z − λ}dz − ∑_{k=1}^{N} \frac{1}{2πi} ∫_{C_{k}} \frac{f(z)}{z}dz|

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 19

Thus we have that

L_{λ}(f) → L_{0}(f) as λ → 0

for all f ∈ H^{p}(G). □

We must show that L_{λ}(f) really is f(λ) for all f in H^{p}(G); first we need a simple estimate of harmonic measure.

Lemma 2.7. There is a constant c > 0 such that

ω(G_{n}, Γ′_{n}, −\frac{1}{2}) ≥ c\frac{1 + a}{2}a^{n}

where Γ′_{n} = \{z : |z| ≤ \frac{1+a}{2}a^{n}\}.

Proof: Denote

Ω_{n} = {z ∈ D : Rez < 0} ∩ G_{n}

Then

ω(G_{n}, Γ′_{n} ∩ ∂Ω_{n}, −\frac{1}{2})
≥ ω(Ω_{n}, Γ′_{n} ∩ ∂Ω_{n}, −\frac{1}{2})
≥ ω(\{z : |z| < 1, Rez < 0\}, Γ̂′_{n} ∩ \{z : Rez = 0\}, −\frac{1}{2})
≥ cm(Γ′_{n} ∩ {z : Rez = 0})
≥ c\frac{1 + a}{2}a^{n}

Therefore,

ω(G_{n}, Γ′_{n}, −\frac{1}{2}) ≥ c\frac{1 + a}{2}a^{n}.□

Lemma 2.8. Suppose that 1 < p < ∞ and

∫ \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} < ∞

Then for λ < 0, we have that

L_{λ}(f) = f(λ)

20 JOHN E. MCCARTHY AND LIMING YANG

for all f ∈ H^{p}(G).

Proof: Let

n ≥ \frac{1 − log|λ|}{log|\frac{1}{a}|}.

Then

f(λ) = \frac{1}{2πi} ∫_{Γ′_{n}} \frac{f(z)}{z − λ}dz + \frac{1}{2πi} ∑_{k=0}^{n−1} ∫_{C_{k}} \frac{f(z)}{z − λ}dz

and

|f(λ) − \frac{1}{2πi} ∑_{k=0}^{n−1} ∫_{C_{k}} \frac{f(z)}{z − λ}dz|

≤ \frac{1}{2π} ∫_{Γ′_{n}} \frac{|f(z)|}{|z − λ|}dm(z)
≤ \frac{c}{|λ|} (∫_{Γ′_{n}} \frac{|dz|}{(\frac{dω(G_{n}, ., − \frac{1}{2} )}{dm(z)})^{q−1}} )^{\frac{1}{q}} ∥f∥

Using Lemma 2.2 and Lemma 2.7, we see that on Γ′_{n}

\frac{dω(G_{n}, ., −\frac{1}{2})}{dm(z)} ≥ c\frac{ω(G_{n}, Γ′_{n}, −\frac{1}{2})}{m(Γ′_{n})} ≥ c,

so

|f(λ) − \frac{1}{2πi} ∑_{k=0}^{n−1} ∫_{C_{k}} \frac{f(z)}{z − λ}dz|

≤ c\frac{1}{|λ|} (\frac{1 + a}{2}a^{n})^{\frac{1}{q}}

Letting n tend to ∞, we get that f(λ) = L_{λ}(f) for all f ∈ H^{p}(G). □ Combining the last three lemmata, we get: Theorem 2.9. If 1 < p < ∞ and

∫ \frac{dm(z)}{|z|^{q}(\frac{dω}{dm(z)})^{q−1}} < ∞,

then every function in H^{p}(G) has a left limit at zero.

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 21

Again, derivatives are handled just as well by the Cauchy integral formula, so we get:

Theorem 2.10. The following statements are equivalent for 1 < p < ∞: (b2) The k^{th} derivative of every function in H^{p}(G) has a left limit at zero. (c2)

∫ \frac{dm(z)}{|z|^{(k+1)q}(\frac{dω}{dm(z)})^{q−1}} < ∞.

(d2)

∑_{n=0}^{∞} \frac{r_{n}^{(k+1)q}}{a^{qn(k+1)}ω(Γ_{n})^{q−1}} < ∞.

3. Bergman spaces : Conditions for bpe’s in terms of Bessel capacities

Let G(x, y) be the Bessel kernel, which is most easily defined as the inverse Fourier transform of (1 + x^{2} + y^{2})^{− \frac{1}{2}} . For each set E in ℂ, the q-Bessel capacity C_{q} is defined by

C_{q}(E) = inf_{f} ∫ |f(x, y)|^{q}dxdy

where f ∈ L^{q}(ℝ^{⊭}), f(x, y) ≥ 0 and ∫ G(t − x, s − y)f(t, s)dtds ≥ 1 for (x, y) ∈ E. The following properties of C_{q} will be used in this section. Lemma 3.1. (a) If q < 2, there are constants F_{1} and F_{2} such that

F_{1}δ^{2−q} ≤ C_{q}(D(λ, δ)) ≤ F_{2}δ^{2−q}

(b) If q = 2, there are constants F_{3} and F_{4} such that

F_{3}\{log(\frac{1}{δ})\}^{−1} ≤ C_{q}(D(λ, δ)) ≤ F_{4}\{log(\frac{1}{δ})\}^{−1}

(c) For each set K,

C_{q}(K) = inf_{K⊂U} C_{q}(U)

where U stands for an open subset.

(d) Let K_{1} and K_{2} be two subsets of ℂ, then

C_{q}(K_{1} ∪ K_{2}) ≤ C_{q}(K_{1}) + C_{q}(K_{2}).

For proofs, see Meyers [10].

The following theorem was proved by Fernstrom and Polking [4] for K a compact set in the plane.

Theorem 3.2. Assume that p ≥ 2. Then λ is a bounded point derivation of order k for R^{p}(K, A) if and only if

∑_{j=0}^{∞} \frac{C_{q}(A_{j}(λ) \ K)}{a^{(k+1)jq}} < ∞

where 0 < a < 1 and A_{j}(λ) = A(λ, a^{j+1}, a^{j}) = {z : a^{j+1} < |z − λ| ≤ a^{j}}. Applying the above results, we are able to prove a similar result for Bergman spaces. Theorem 3.3. Let G be a bounded open subset of ℂ. Suppose p ≥ 2 and λ ∈ ∂G, then

λ is a bounded point derivation of order k for L_{a}^{p}(G) if and only if

∑_{j=0}^{∞} \frac{C_{q}(A_{j}(λ) \ G)}{a^{(k+1)jq}} < ∞

Proof: The proof of necessity is exactly the same as that for R^{p}(K) (see [4] and [8]). It suffices to show the sufficiency. Suppose

∑_{j=0}^{∞} \frac{C_{q}(A_{j}(λ) \ G)}{a^{(k+1)jq}} < ∞.

Let Γ_{j} = {z : |z − λ| = a^{j}}. As

∑_{j=0}^{∞} \frac{C_{q}(Γ_{j+1} \ G)}{a^{(k+1)jq}} ≤ ∑_{j=0}^{∞} \frac{C_{q}(A_{j+1}(λ) \ G)}{a^{(k+1)jq}} < ∞,

Lemma 3.1 (d) yields that

∑_{j=0}^{∞} \frac{C_{q}(\overline{A_{j}(λ)} \ G)}{a^{(k+1)jq}} < ∞.

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 23

Using Lemma 3.1 (c), we can find an open subset U_{j} such that

\overline{A_{j}(λ)} \ G ⊂ U_{j}

and

∑_{j=0}^{∞} \frac{C_{q}(U_{j})}{a^{(k+1)jq}} < ∞

Let K_{j} = \overline{A_{j}(λ)} \ U_{j}, and K = ∪_{j=1}^{∞}K_{j} ∪ \{λ\}. Then K is a compact subset,

∑_{j=0}^{∞} \frac{C_{q}(A_{j}(λ) \ K)}{a^{(k+1)jq}} < ∞

and

K ∩ A_{j}(λ) = (K_{j} ∩ A_{j}(λ)) ∪ (K_{j−1} ∩ Γ_{j})
⊂ (G ∩ A_{j}(λ)) ∪ (\overline{A_{j−1}(λ)} ∩ G ∩ Γ_{j})
= A_{j}(λ) ∩ G

So

K \ {λ} ⊂ G.

It follows from Theorem 3.1 that λ is a bounded point derivation for R^{p}(K, A). Suppose f ∈ L^{p}(G) is analytic in a neighborhood of λ; then f is in R^{p}(K, A) and

|f^{(k)}(λ)| ≤ c(∫_{K} |f|^{p}dA)^{\frac{1}{p}} ≤ c(∫_{G} |f|^{p}dA)^{\frac{1}{p}}

where c is a constant. Hence, λ is a bounded point derivation for L_{a}^{p}(G). □

Corollary 3.4. Let G be an L-region.

(a) If p = 2 and

∑_{n=0}^{∞} \frac{1}{a_{n}^{2(k+1)}log \frac{1}{r_{n}}} < ∞

then zero is a bounded point derivation of order k for L_{a}^{2}(G).

(b) If p > 2 and

∑_{n=0}^{∞} \frac{r_{n}^{2−q}}{a_{n}^{q}(k + 1)} < ∞

then zero is a bounded point derivation of order k for L_{a}^{p}(G).

Proof of (a): Using Lemma 3.1 (d), we see that

C_{2}(A_{j}(0) \ G) ≤ ∑_{a_{n}∈A_{j}(0)} C_{2}(D(a_{n}, r_{n}))
≤ F_{4} ∑_{a_{n}∈A_{j}(0)} \frac{1}{log \frac{1}{r_{n}}} (Lemma 1 (b))

It follows from Theorem 3.3 that zero is a bounded point derivation of order k. The proof of (b) is similar. □

4. Bergman L_{a}^{2} spaces on L regions : Geometric conditions for bpe’s

Because of possible interference between adjacent Δ_{n}, we cannot give necessary and sufficient conditions, in terms of a_{n} and r_{n}, that 0 be a bpe for the Bergman space of an arbitrary L region. We must impose the requirement that there is a sequence {c_{j}} decreasing monotonically to zero, such that the number of a_{n}’s in A_{j} := {z : c_{j+1} ≤ |z| < c_{j}} is less than some constant N, and with the following property:

(P) ∑_{j=1}^{∞} c_{j}^{2}d_{j} < ∞, where

d_{j} = max (log \frac{c_{j}^{2} − c_{j+2}^{2}}{c_{j+1}^{2} − c_{j+2}^{2}} , log \frac{c_{j−1}^{2} − c_{j+1}^{2}}{c_{j}^{2} − c_{j+1}^{2}} ) .

Example: If c_{j} = a^{j} and the number of a_{n}’s in any A_{j} remains bounded as j tends to ∞, condition (P) is satisfied.

If c_{j} = (\frac{1}{j})^{k}, where k > \frac{1}{2}, and the number of a_{n}’s in A_{j} is no more than N, then condition (P) is satisfied.

We now show that if the series in Corollary 3.4 diverges, and G satisfies condition (P), then 0 can not be a bpe. First, we show that this happens if the terms do not tend to zero, then prove it in general.

Lemma 4.1: If

lim inf_{n→∞} a_{n}^{2}log \frac{1}{r_{n}} < ∞

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 25

then zero is not a bpe.

Proof: Let

f_{n}(z) = \frac{z}{z − a_{n}}

Then

∥f_{n}∥^{2} = ∫_{G} |f_{n}(z)|^{2}dA(z)

≤ ∫_{\{z: r_{n}≤|z−a_{n}|<1+a_{n}\}} |\frac{z}{z − a_{n}} |^{2}dA(z)

≤ 4πa_{n}^{2}log \frac{1}{r_{n}} + 4π.

By hypothesis, there is a subsequence of {f_{n}} whose norms remain bounded; therefore some subsequence , {f_{n_{k}}} say, converges weakly. But as f_{n} converges pointwise to 1, this must be the weak-limit of f_{n_{k}}, so 1 is in clos(zL_{a}^{2}(G)). Because functions analytic in a neighborhood of 0 are dense [1], it follows that

clos(zL_{a}^{2}(G)) = L_{a}^{2}(G).□

Theorem 4.2. Suppose {a_{n}, r_{n}} satisfies the condition (P) and

∑_{n=1}^{∞} \frac{1}{a_{n}^{2}log \frac{1}{r_{n}}} = ∞.  (4.3)

Then zero is not a bpe for L_{a}^{2}(G).

Proof: Let

b_{n} = \frac{1}{a_{n}^{2}log \frac{1}{r_{n}}} .

From Proposition 3.5, we can assume that lim_{n→∞} b_{n} = 0, so for each m, we can find N_{m} such that

1 ≤ ∑_{n=m}^{N_{m}} b_{n} ≤ 2.

Let

f_{n}(z) = ∑_{n=m}^{N_{m}} b_{n} \frac{z}{z − a_{n}}

26 JOHN E. MCCARTHY AND LIMING YANG

and

B_{j} = A_{j−1} ∪ A_{j} ∪ A_{j+1}

We want to show ∥f_{m}∥ stays bounded. We have ∫_{A_{j}∩G} |f_{m}(z)|^{2}dA(z)

≤ 2 ∫_{A_{j}∩G} ( ∑_{a_{n}∉B_{j}} b_{n}|\frac{z}{z − a_{n}} |)^{2} dA(z) + 2 ∫_{A_{j}∩G} ( ∑_{a_{n}∈B_{j}} b_{n}|\frac{z}{z − a_{n}} |)^{2} dA(z)

Denote the first integral by F_{j} and the second integral by S_{j}. By Minkowski’s inequality

F_{j} ≤ 2[ ∑_{a_{n}∉B_{j}} b_{n}(∫_{A_{j}∩G} |\frac{z}{z − a_{n}} |^{2}dA(z))^{\frac{1}{2}} ]^{2}.

But

∫_{A_{j}} |\frac{z}{z − a_{n}} |^{2}dA(z) ≤ c_{j}^{2} ∫_{A_{j}} |\frac{1}{z − a_{n}} |^{2}dA(z)
= c_{j}^{2}\{ \begin{matrix}πlog \frac{c_{j}^{2} − a_{n}^{2}}{c_{j+1}^{2} − a_{n}^{2}} & a_{n} < c_{j+2} \\ πlog \frac{a_{n}^{2} − c_{j+1}^{2}}{a_{n}^{2} − c_{j}^{2}} & a_{n} > c_{j−1} \end{matrix}

≤ c_{j}^{2}\{ \begin{matrix}πlog \frac{c_{j}^{2} − c_{j+2}^{2}}{c_{j+1}^{2} − c_{j+2}^{2}} & a_{n} < c_{j+2} \\ πlog \frac{c_{j−1}^{2} − c_{j+1}^{2}}{c_{j−1}^{2} − c_{j}^{2}} & a_{n} > c_{j−1} \end{matrix}
≤ πc_{j}^{2}d_{j},

so

F_{j} ≤ 2π( ∑_{a_{n}∉B_{j}} b_{n})^{2}c_{j}^{2}d_{j} ≤ 8πc_{j}^{2}d_{j}.

For the second integral, we have the following computation:

S_{j} ≤ 2 ∫_{A_{j}∩G}( ∑_{a_{n}∈B_{j}} b_{n}^{2}|\frac{z}{z − a_{n}} |^{2})(3N)dA(z)
≤ 6N ∑_{a_{n}∈B_{j}} b_{n}^{2}(4πb_{n}^{−1} + 4π)
≤ 100N ∑_{a_{n}∈B_{j}} b_{n}.

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 27

So we have

∫_{A_{j}∩G} |f_{m}(z)|^{2}dA(z) ≤ c(c_{j}^{2}d_{j} + ∑_{a_{n}∈B_{j}} b_{n}),

and therefore

∫ |f_{m}(z)|^{2}dA(z) ≤ c(∑_{j=1}^{∞} c_{j}^{2}d_{j} + ∑_{n=m}^{N_{m}} b_{n}).

So the norms of f_{m} are uniformly bounded, and as before we conclude that 1 ∈ clos(zL_{a}^{2}(G)), and hence

clos(zL_{a}^{2}(G)) = L_{a}^{2}(G).□

The divergence of (4.3), without assumption (1), is not enough to prevent 0 from being a bpe, as the following example shows.

Example 4.4. Let a_{n} = 2^{−n}, r_{n} = exp(−8^{n}). Let G = D \ ∪_{n=1}^{∞}\overline{Δ(a_{n}, r_{n})}. Then G satisfies the hypotheses of Theorem 3.3, and 0 is a bpe for L_{a}^{2}(G). Now, replace each disc Δ(a_{n}, r_{n}) by 2^{n} small discs, each of radius \frac{r_{n}}{2^{n+1}} equally spaced along the diameter of Δ(a_{n}, r_{n}). Let G′ be the larger region obtained by removing these discs from D. For G′, the sum (3.4) is greater than

\frac{1}{2} ∑_{n=1}^{∞} ∑_{j=1}^{2^{n}} \frac{1}{a_{n}^{2}log \frac{2^{n+1}}{r_{n}}} = \frac{1}{2} ∑ \frac{2^{n}}{n + 1} \frac{1}{a^{2n}log \frac{2}{r_{n}}} = ∞.

But 0 is still a bpe for L_{a}^{2}(G′) because G ⊂ G′. □

Combining Corollary 3.4 and Theorem 4.2, with the obvious modifications to its proof to handle derivatives, we get:

Theorem 4.5. Suppose that G satisfies condition (P). Then the following statements are equivalent. (a3) Zero is a bounded point derivation of order k for L_{a}^{2}(G). (b3) The k^{th} derivative of every function in L_{a}^{2}(G) has a left limit at zero. (d3)

∑_{n=1}^{∞} \frac{1}{a_{n}^{2(k+1)}log \frac{1}{r_{n}}} < ∞

28 JOHN E. MCCARTHY AND LIMING YANG

5. Bergman spaces on L regions : L_{a}^{p}, p > 2

For p > 2, we can use similar techniques. We shall just state the results obtained. Condition (P) on the sequence c_{j} is replaced by: there is a sequence {c_{j}} decreasing

monotonically to zero, such that the number of a_{n}’s in A_{j} := {z : c_{j+1} ≤ |z| < c_{j}} is less than some constant N, and with the following property:

(P′)  ∑_{n=1}^{∞} c_{j}^{p}d_{j} < ∞,

where

d_{j} = max (∫_{A_{j}} |\frac{1}{z − c_{j+2}} |^{p} dA(z), ∫_{A_{j}} |\frac{1}{z − c_{j−1}} |^{p} dA(z)) .

Theorem 4.5 becomes:

Theorem 5.1. If p > 2 and G satisfies the condition (P’), the following statements are equivalent. (a4) Zero is a bounded point derivation of order k for L_{a}^{p}(G). (b4) The k^{th} derivative of every function in L_{a}^{p}(G) has a left limit at zero. (c4)

∑_{n=1}^{∞} \frac{r_{n}^{2−q}}{a_{n}^{q(k+1)}} < ∞

Acknowledgements

The authors would like to thank Albert Baernstein II for pointing out the classical proof of Lemma 2.2 and other valuable conversations.

BOUNDED POINT EVALUATIONS ON THE BOUNDARIES OF L REGIONS 29

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30 JOHN E. MCCARTHY AND LIMING YANG

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Department of Mathematics Washington University St. Louis MO 63130