GROWTH OF THE BERGMAN KERNEL ON PLANAR REGIONS

John E. M^{c}Carthy and Liming Yang Washington University Department of Mathematics One Brookings Drive Saint Louis, Missouri, 63130-4899

Statement of Results

Let Ω be a bounded open set in the complex plane. The Bergman space L_{a}^{2}(Ω) is the Hilbert space of holomorphic functions on Ω that are square-integrable with respect to area measure A. Evaluation at each point λ of Ω is continuous, so there is a corresponding kernel function k_{λ} in L_{a}^{2} (Ω) such that

∫_{Ω} f(z)\overline{k_{λ}(z)}dA(z) = f(λ)

for every function f in L_{a}^{2} (Ω). In this paper, we are interested in estimating the growth of ∥k_{λ}∥ as λ tends to the boundary of Ω.

If Ω is smoothly bounded, ∥k_{λ}∥ will grow like the reciprocal of the distance to the boundary; but if the boundary point is somehow “buried” deep inside Ω, the growth of ∥k_{λ}∥ can be slower (to aid the reader’s intuition: the phenomenon is not caused by cusps, which cause the complement to be too thick, but by little holes accumulating at some point). In [M^{c}CY] the authors found geometric conditions for ∥k_{λ}∥ to remain bounded as a boundary point is approached (so that evaluation at this boundary point is a bounded point evaluation) for certain special domains (L-regions). Also, using results of Fernstrom and

Typeset by AMS-TEX 2 JOHN E. M^{c}CARTHY AND LIMING YANG

Polking [FP], necessary and sufficient conditions were found, in terms of Bessel capacity, for a boundary point of an arbitrary domain to be a bounded point evaluation. We extend this last result to estimating the growth of ∥k_{λ}∥ when it does not remain bounded.

Let Λ_{Θ,Γ} be the sector in the left half-plane bounded by y = x tan Θ, y = −x tan Θ, and x^{2} + y^{2} = Γ^{2}. We shall always assume that 0 is in the boundary of Ω and is the point of interest, and that for some Θ in (0, \frac{π}{2} ) and some Γ > 0,

Λ_{Θ,Γ} ⊂ Ω.

We shall look at the growth rate of ∥k_{λ}∥ as λ tends to 0 along the negative real axis.

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 Bessel capacity is defined as

C(E) = inf_{f} ∫ |f(x, y)|^{2} dxdy

where f ∈ L^{2}(ℝ^{2}), f ≥ 0, and ∫ G(t − x, s − y)f(t, s) dtds ≥ 1 for (x, y) ∈ E. Let A_{k} = {z = x + iy : 2^{−k−1} ≤ |z| < 2^{−k}} and A′_{k} = {z : 2^{−k−2} ≤ |z| ≤ 2^{−k+1}}.

Our results are the following:

Theorem 1. Suppose

∑_{k=1}^{∞} kC(A_{k}\Ω) < ∞.

Then there are constants F_{1} and F_{2} so that, as λ → 0^{−}, k_{λ} satisfies the growth condition

∥k_{λ}∥^{2} ≤ F_{1} ∑_{k=1}^{∞} 2^{2 min(k,log_{2} \frac{1}{|λ|})}C(A_{k} \ Ω);
∥k_{λ}∥^{2} ≥ F_{2} ∑_{k=1}^{∞} 2^{2 min(k,log_{2} \frac{1}{|λ|})}C(A_{k}\Ω).

GROWTH OF THE BERGMAN KERNEL ON PLANAR REGIONS 3

Corollary 2. Let 0 < α < 1. A necessary and sufficient condition for ∥k_{λ}∥ to be O(|λ|^{−α}) as λ → 0^{−} is that

lim sup_{k→∞} 2^{2k(1−α)}C(A_{k} \ Ω) < ∞.

Our techniques also work for the Bergman space L_{a}^{p}(Ω) for 2 < p < ∞. Letting q = \frac{p}{p−1} be the conjugate index of p, define the q-capacity by

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

where f ∈ L^{q}(ℝ^{2}), f ≥ 0, and ∫ G(t − x, s − y)f(t, s) dtds ≥ 1 for (x, y) ∈ E (so our previous definition of capacity is C_{2}, though we shall continue to write it as C with no subscript).

Theorem 3. Suppose 2 < p < ∞, and

∑_{k→∞} 2^{k(2−q)}C_{q}(A_{k} \ Ω) < ∞.

Then as λ → 0^{−}, the norm of the functional of evaluation at λ in the Bergman space L_{a}^{p}(Ω) is comparable to

[ ∑_{k=1}^{∞} 2^{q min(k,log_{2} \frac{1}{|λ|})}C_{q}(A_{k} \ Ω) ]^{\frac{1}{q}} .

We note that a disk Δ of radius δ has

C(Δ) ∼ [log(\frac{1}{δ})]^{−1}

and, for q < 2,

C_{q}(Δ) ∼ δ^{2−q},

so it is easy to construct examples of regions of the form 𝔻(0, 1) \ ∪_{n=1}^{∞}𝔻(2^{−n}, r_{n}) that have kernels growing at a desired rate.

Proofs

We shall let F denote a generic constant, that may change from one line to the next. We shall let K_{ε} = {z : dist(z, K) < ε}. Lemma 4. (a) For each Borel set E,

C(E) = inf_{E⊂U}C(U)

where U ranges over the open sets.

(b) For any Borel sets E_{1} and E_{2} C(E_{1} ∪ E_{2}) ≤ C(E_{1}) + C(E_{2}).

Proof. See [Me].

Lemma 5. Let K be a compact subset of ℂ. Suppose that

lim sup kC(A_{k}\K) = 0.

Then there exists a constant M > 0 such that, for each ε > 0 and each k ≥ 0, there is a function ψ_{k} ∈ C^{∞} satisfying

(i) ψ_{k}(z) = 1 for z in a neighborhood of A′_{k}\K_{ε} (ii) ∫_{|z|≤2^{−k+1}} |¯∂ψ_{k}|^{2}dA ≤ M · C(A′_{k}\K) (iii) ∫_{|z|≤2^{−k+1}} |ψ_{k}|^{2}dA ≤ M · 2^{−2k}C(A′_{k}\K).

Proof. The proof follows from the proof of Lemma 10 in [FP]. Their hypotheses are much stronger - namely that ∑ 2^{2k}C(A_{k}\K) < ∞ - but their proof works in our special case provided just kC(A_{k}\K) tends to zero. To use their proof, we only have to show that there exists k_{0} such that if k ≥ k_{0} then

∫_{\sqrt{t^{2}+s^{2}}>2^{−k+2}} G(x − t, y − s)g_{k}(t, s) dtds < \frac{1}{2}  (1)

GROWTH OF THE BERGMAN KERNEL ON PLANAR REGIONS 5

for x^{2} + y^{2} < 2^{−2k+3}, where g_{k} is defined as in their proof. We have

∫\sqrt{t^{2}+s^{2}}>2^{−k+2}G(x − t, y − s)g_{k}(t, s) dtds

≤ \{ ∫_{\sqrt{t^{2}+s^{2}}≥2^{−k}} G(t, s)^{2} dtds\}^{\frac{1}{2}} · ∥g_{k}∥

≤ F ( log \frac{1}{2^{−k}} )^{\frac{1}{2}} · [C(A′_{k}\K)]^{\frac{1}{2}} (2)

where the second inequality follows from Lemma 4 in [FP] and the fact that ∥g_{k}∥ ≤ 2[C(A′_{k}\K)]^{\frac{1}{2}} . As the expression in (2) tends to 0, we get (1) as desired. □

For a positive Borel measure μ, let

U^{μ}(z) = ∫ |z − w|^{−1} dμ(w)

and

c(E, V) = sup_{μ}μ(E)

where the supremum is taken over all positive Borel measures with spt μ ⊆ E such that

∫_{V} |U^{μ}(z)|^{2} dA ≤ 1

(Throughout the paper, V will be a fixed bounded open set containing Ω or K. It is only necessary to introduce it as |U^{μ}|^{2} is not integrable in a neighborhood of infinity). Lemma 6. Let E be a Borel set, then

C(E)^{\frac{1}{2}} ≤ c(E, V) ≤ F C(E)^{\frac{1}{2}}

(For a discussion of why this is true, and for other equivalent notions of analytic 2capacities, see [He]).

The next lemma is the key to proving the upper bound estimate.

6 JOHN E. M^{c}CARTHY AND LIMING YANG

Lemma 7. Let K be a compact subset of ℂ, 0 ∈ ∂K, and Λ_{Θ,Γ} ⊆ K for some 0 < Θ < \frac{π}{2} and Γ > 0. Suppose that

lim sup_{k→∞} kC(A_{k}\K) = 0.

Then there is a constant M such that for each rational function r with poles off K, and each λ in ℝ ∩ Λ_{Θ,Γ},

|r(λ)| ≤ M · ( ∫_{K} |r|^{2} dA)^{\frac{1}{2}} ( ∑_{k=1}^{∞} 2^{2 min(k,log_{2} \frac{1}{|λ|})}C(A_{k} \ K))^{\frac{1}{2}} .

Proof. Without loss of generality, we can assume K is contained in the disk 𝔻(0, \frac{1}{4}), and so Γ < \frac{1}{4}. It follows from Lemma 4 that

lim sup_{k→∞} kC(A′_{k}\K) < ∞.

Construct a smooth function φ ∈ C_{o}^{∞}(ℝ^{2}) such that

φ(z) = \{ \begin{matrix}0 & if z ∈ \{z : |z| ≥ 2\}or |z| ≤ \frac{1}{4}\} \\ 1 & if z ∈ \{z : \frac{1}{2} ≤ |z| ≤ 1\} ∩ (Λ_{Θ,1})^{c} \\ 0 & if z ∈ Λ_{\frac{Θ}{2} ,1} \end{matrix}

For each integer k set

φ_{k}(z) = \{ \begin{matrix}φ(2^{k}z)/ ∑_{j=−∞}^{∞} φ(2^{j}z) & if φ(2^{j}z) ≠ 0 for some j \\ 0 & else \end{matrix}

Fix a rational function r with poles off K. Let g be a smooth function which is 1 in a neighborhood of K, zero in a neighborhood of the poles of r and zero off 𝔻(0, \frac{1}{4}). Let λ be between −Γ and 0. From Green’s theorem,

r(λ) = \frac{1}{π} ∫_{𝔻(0, \frac{1}{4} )} \frac{1}{λ − z} ¯∂(gr) dA(z).

GROWTH OF THE BERGMAN KERNEL ON PLANAR REGIONS 7

Let ε be small enough for K_{ε} to be contained in the set where g is 1. Then ¯∂(gr) = 0 on K_{ε}. Let ψ_{k} be as in Lemma 5; then ∑ ψ_{k}φ_{k} = 1 near 𝔻\K_{ε}, because if z is in A_{k}, φ_{j}(z) is non-zero only for j = k − 1, k, k + 1, and ψ_{j}(z) = 1 for these values of j. Therefore

r(λ) = \frac{1}{π} ∫ \frac{1}{λ − z} ∑_{k=1}^{∞} ψ_{k}φ_{k} ¯∂(gr) dA

= \frac{1}{π} ∑_{k} ∫_{A_{k}\Λ_{\frac{Θ}{2} ,Γ}} \frac{1}{z − λ} ¯∂(ψ_{k}φ_{k}) · gr dA.

Hence,

|r(λ)| ≤ F ∑_{k} ∫_{A_{k}\Λ_{Θ,Γ}} \frac{1}{|z − λ|}|¯∂(ψ_{k}φ_{k})||gr| dA.  (3)

Let −2^{−N} ≤ λ < −2^{−N−1} and z ∈ A_{k}\Λ_{Θ,Γ}. If k ≥ N,, then

\frac{1}{|z − λ|} ≤ F · \frac{1}{2^{−N}};

if k < N, then

\frac{1}{|z − λ|} ≤ F · \frac{1}{2^{−k}} .

Therefore (3) gives

|r(λ)| ≤ F ∑_{k=1}^{N−1} 2^{k}( ∫_{A_{k}} |¯∂(ψ_{k}φ_{k})|^{2} dA)^{\frac{1}{2}} ( ∫_{A_{k}} |gr|^{2} dA)^{\frac{1}{2}}

+ F ∑_{k=N}^{∞} 2^{N}( ∫_{A_{k}} |¯∂(ψ_{k}φ_{k})|^{2} dA)^{\frac{1}{2}} ( ∫_{A_{k}} |gr|^{2} dA)^{\frac{1}{2}}

≤ F ∥gr∥_{L_{a}^{2}(𝔻)}( ∑_{k=1}^{N−1} 2^{2k} ∫_{A_{k}} |¯∂(ψ_{k}φ_{k})|^{2} dA

+ ∑_{k=N}^{∞} 2^{2N} ∫_{A_{k}} |¯∂(ψ_{k}φ_{k})|^{2} dA)^{\frac{1}{2}} .

Using Lemma 5 and the fact that |¯∂φ_{k}| ≤ F · 2^{k}, we have

∫_{A_{k}} |¯∂(ψ_{k}φ_{k})|^{2} dA ≤ F · C(A′_{k}\K).

8 JOHN E. M^{c}CARTHY AND LIMING YANG

Hence,

|r(λ)| ≤ F ∥gr∥_{L_{a}^{2}(𝔻)}( ∑_{k=1}^{N−1} 2^{2k}C(A′_{k}\K) + ∑_{k=N}^{∞} 2^{2N}C(A′_{k}\K))^{\frac{1}{2}} .

Since g is arbitrary, subject to being 1 on K, we conclude that

|r(λ)| ≤ M( ∑_{k=1}^{∞} 2^{2 min(k,log_{2} \frac{1}{|λ|})}C(A_{k} \ K))^{\frac{1}{2}} ∥r∥_{L^{2}(K,A)}.

□

Proof of Theorem 1. (Upper Bound) Suppose that

lim sup_{k→∞} kC(A_{k}\Ω) = 0,

so by Lemma 4 (b),

lim sup_{k→∞} kC(Ā_{k}\Ω) = 0.

Using Lemma 4 (a), we can find open subsets U_{k} such that

Ā_{k}\Ω ⊂ U_{k}

and

C(U_{k}) < 2C(A_{k}\Ω).

Let K_{k} = Ā_{k} \ U_{k}, and K = ∪_{k=1}^{∞}K_{k} ∪ {0}. Then K is a compact subset,

lim sup_{k→∞} kC(A_{k}\K) = 0,

and

K ∩ A_{k} ⊆ Ω ∩ A_{k},

so

K\{0} ⊆ Ω.

Let f ∈ L_{a}^{2} (Ω) be analytic in a neighborhood of zero. By Runge’s theorem, there exists a sequence of rational functions r_{n} converging to f uniformly on K; so by Lemma 7 we have

|f(λ)| ≤M( ∑_{k=1}^{∞} 2^{2 min(k,log_{2} \frac{1}{|λ|})}C(A_{k} \ K))^{\frac{1}{2}} ( ∫_{K} |f|^{2} dA)^{\frac{1}{2}}
≤M( ∑_{k=1}^{∞} 2^{2 min(k,log_{2} \frac{1}{|λ|})}C(A_{k} \ Ω))^{\frac{1}{2}} ( ∫_{Ω} |f|^{2} dA)^{\frac{1}{2}} .  (4)

Since such functions f are dense in L_{a}^{2} (Ω) (see [Al]), (4) holds for all f in L_{a}^{2} (Ω). Therefore, for λ ∈ ℝ ∩ Λ_{Θ,Γ},

∥k_{λ}∥ = sup_{∥f∥_{L^{2}_{a} (Ω)}=1} |f(λ)| ≤ M( ∑_{k=1}^{∞} 2^{2 min(k,log_{2} \frac{1}{|λ|})}C(A_{k} \ Ω))^{\frac{1}{2}} .

(Lower Bound) By Lemma 6, we can choose positive measures μ_{k} carried by A_{k} \ Ω with

∫_{Ω} |U_{k}^{μ}(z)|^{2} dA ≤ 1,

and ∥μ_{k}∥ comparable to C(A_{k} \ Ω)^{\frac{1}{2}} . Moreover, we can assume that each of the measures μ_{k} has support entirely within one of the three sectors \{z : −π + \frac{Θ}{2} ≤ arg (z) ≤ −\frac{Θ}{2}\}, \{z : −\frac{Θ}{2} ≤ arg (z) ≤ \frac{Θ}{2}\}, \{z : \frac{Θ}{2} ≤ arg (z) ≤ π − \frac{Θ}{2}\}.

Let

γ_{k} = 2^{min(k,log_{2} \frac{1}{|λ|})}C(A_{k} \ Ω)^{\frac{1}{2}} .

For some conjugacy class modulo 3, the sum of γ_{k}^{2} for k in this conjugacy class is comparable to ∑_{k=1}^{∞} γ_{k}^{2}. We shall choose only those μ_{k} for k in this conjugacy class, and set the other μ_{k}’s to zero (the point of this is to ensure the supports are well separated).

Let

f_{k}(z) = ∫ \frac{dμ_{k}(w)}{z − w}

be the Cauchy transform of μ_{k}. Let us fix some λ between −Γ and 0, and let N be the closest integer to log_{2} \frac{1}{|λ|}. By the way we have chosen the measures, we have that |f_{k}(λ)|

10 JOHN E. M^{c}CARTHY AND LIMING YANG

is comparable to 2^{k}∥μ_{k}∥ for k < N and to 2^{N}∥μ_{k}∥ for k > N; For k = N, we can only say that it dominates 2^{N}∥μ_{k}∥. Thus we have, for all k,

|f_{k}(λ)| ≥ F γ_{k}.  (5)

We also have that ∫_{Ω} |f_{k}|^{2}dA ≤ 1, and the estimate

|f_{n}(z)| ≤ ∥μ_{n}∥ [ dist(z, A_{n})]^{−1}

gives, for |n − k| ≥ 2,

∫_{A_{k}} |f_{n}|^{2}dA ≤ F C(A_{n} \ Ω)2^{2 min(n,k)−2k}.  (6)

Let f = ∑ α_{n}f_{n}, where we shall choose the α’s later. We have

\begin{aligned}∫_{Ω} |f|^{2}dA & = ∑_{k=1}^{∞} ∫_{A_{k}∩Ω} | ∑_{|n−k|≥2} α_{n}f_{n} + α_{k}f_{k}|^{2}dA \\ & ≤ 2[∑_{k=1}^{∞} ∫_{A_{k}∩Ω} |α_{k}f_{k}|^{2}dA + ∑_{k=1}^{∞} ∫_{A_{k}∩Ω} | ∑_{|n−k|≥2} α_{n}f_{n}|^{2}dA]. \end{aligned}

Now, by the inequalities of Minkowski and Cauchy-Schwartz,

\begin{aligned}∫_{A_{k}∩Ω} | ∑_{|n−k|≥2} α_{n}f_{n}|^{2}dA & ≤ [ ∑_{|n−k|≥2} |α_{n}|(∫_{A_{k}∩Ω} |f_{n}|^{2}dA)^{\frac{1}{2}} ]^{2} \\ & ≤ ( ∑_{|n−k|≥2} |α_{n}|^{2})( ∑_{|n−k|≥2} ∫_{A_{k}∩Ω} |f_{n}|^{2}dA). \end{aligned}  (7)

Using this and inequality (6), we get that

∫_{Ω} |f|^{2}dA ≤ F (∑ |α_{n}|^{2})(1 + ∑_{n,k:|n−k|≥2} C(A_{n} \ Ω)2^{2 min(n,k)−2k}).

But this last factor is dominated by

∑_{n=1}^{∞} nC(A_{n}\Ω)

which is finite by hypothesis. So we finally get that

∫_{Ω} |f|^{2}dA ≤ F (∑ |α_{n}|^{2}).  (8)

Now choose α_{k} so that |α_{k}| = γ_{k}, and so that α_{k}f_{k}(λ) ≥ 0. By inequalities (5) and (8), we have that

\frac{|f(λ)|^{2}}{∫_{Ω} |f|^{2}dA} ≥ F \frac{(∑ |α_{k}|γ_{k})^{2}}{∑ |α_{k}|^{2}} = F ∑_{k=1}^{∞} γ_{k}^{2}.

So (∑_{k=1}^{∞} γ_{k}^{2})^{\frac{1}{2}} is a lower bound for k_{λ}, as desired. □

Proof of Corollary 2. (Sufficiency) If

lim sup_{k→∞} 2^{2k(1−α)}C(A_{k} \ Ω) < ∞.

then the hypotheses of Theorem 1 are satisfied. Using the estimate on ∥k_{λ}∥ from Theorem 1, together with the hypothesis that C(A_{k} \Ω) ≤ F 2^{−2k(1−α)}, one gets that |λ|^{α}∥k_{λ}∥ stays bounded.

(Necessity) Let f_{k} be as in the proof of Theorem 1. Then

\frac{|f_{k}(λ)|}{∥f_{k}∥} ≥ F 2^{min(k,log_{2} \frac{1}{|λ|})}C(A_{k} \ Ω)^{\frac{1}{2}} .

Letting λ = −2^{−k}, we get

|λ|^{α}∥k_{λ}∥ ≥ F 2^{k(1−α)}C(A_{k} \ Ω)^{\frac{1}{2}} ,

and the right-hand side must remain bounded, as desired. □

Proof of Theorem 3. This proceeds like the proof of Theorem 1, with appropriate modifications to Lemmata 5, 6 and 7 (in the proof of Lemma 5 the hypothesis that

lim_{k→∞} 2^{k(2−q)}C_{q}(A_{k} \ Ω) = 0

12 JOHN E. M^{c}CARTHY AND LIMING YANG

is used, and this is a sufficient condition for the upper bound to hold).

For the lower bound, there is one extra difficulty in getting the appropriate analogue of (7) - if one follows routinely, one gets ∑ |α|^{q} instead of ∑ |α|^{p}. To get around this, note that (6) becomes

∫_{A_{k}} |f_{n}|^{p}dA ≤ F Cq^{\frac{p}{q}} (A_{n} \ Ω)2^{p min(n,k)−2k}.  (6′)

Then (7) becomes

∫_{A_{k}∩Ω} | ∑_{|n−k|≥2} α_{n}f_{n}|^{p}dA ≤ ( ∑_{|n−k|≥2} |α_{n}|^{p})( ∑_{|n−k|≥2}[∫_{A_{k}∩Ω} |f_{n}|^{p}dA]^{\frac{q}{p}})^{\frac{p}{q}} .

Using (6′), we get that

∑_{k=1}^{∞} ∫_{A_{k}∩Ω} | ∑_{|n−k|≥2} α_{n}f_{n}|^{p}dA ≤ ∑_{k=1}^{∞}( ∑_{|n−k|≥2} |α_{n}|^{p})( ∑_{|n−k|≥2} C_{q}(A_{n} \ Ω)2^{q min(n,k)− \frac{2q}{p} k})^{\frac{p}{q}}

≤ (∑_{n=1}^{∞} |α_{n}|^{p}) ∑_{k=1}^{∞}( ∑_{|n−k|≥2} C_{q}(A_{n} \ Ω)2^{q min(n,k)− \frac{2q}{p} k}).

Now interchanging the order of summation and using the hypothesis of the theorem, (and the estimate ∫_{A_{n}∩Ω} |α_{n}f_{n}|^{p}dA ≤ |α_{n}|^{p} ) we finally get that

∫_{Ω} |f|^{p}dA ≤ F (∑ |α_{n}|^{p}).

Choosing |α_{k}| = γ_{k}^{q−1}, where γ_{k} = 2^{min(k,log_{2} \frac{1}{|λ|})}Cq^{\frac{1}{q}} (A_{k} \ Ω), yields the desired lower bound. □

GROWTH OF THE BERGMAN KERNEL ON PLANAR REGIONS 13

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[He] L. Hedberg, Non-linear Potentials and Approximation in the Mean by Analytic Func-

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