§6.3 Analytic Capacity

6.3 Analytic Capacity

The theory of rational approximation is essentially built upon the concept of analytic capacity, which was introduced in 1940 by Finnish mathematician Lars Ahlfors. Our purpose here is to give a brief and elementary introduction. Despite its importance, still many trivially simple results remain conjecture.

The uses of analytic capacity are present in many other topics of complex analysis. Analytic capacity serves as a natural framework for general rational approximation theory. Our purpose here is to hint at how analytic capacity theory relates to the proof of Theorem 6.2.7 $Theorem 6.2.7 Theorem 6.2.7$ and pertinent problems in general.

Definition 6.3.1 (Analytic Capacity).

Let $𝐾 \subseteq ℂ$ be compact. The analytic capacity of $𝐾$ is defined as

𝛾 (𝐾) = sup {| 𝑓^{'} (\infty) | : [\begin{matrix} 𝑓 is holomorphic on \hat{ℂ} ∖ 𝐾 \\ 𝑓 (\hat{ℂ} ∖ 𝐾) \subseteq \overline{𝔻} \\ 𝑓 (\infty) = 0 \end{matrix}]}, (6.3 . 1)

where $𝑓^{'} (\infty)$ is defined as in (6.2.1) $(6.2.1) (6.2.1)$ . For an arbitrary set $𝑈 \subseteq ℂ$ , we define

sup {𝛾 (𝐾) : 𝐾 \subseteq 𝑈 \land 𝐾 is compact} .

Intuitively, $𝛾$ measures the extent to which bounded analytic functions outside $𝐾$ can deviate from constancy. Generally, the “larger” $𝐾$ is, the greater the capacity is.

Proposition 6.3.2.

𝐾 \subset ℂ

is a compact set of discrete points, then

𝜆 (𝐾) = 0

Proof.

For any

𝑓 : \hat{ℂ} ∖ 𝐾 \to ℂ

holomorphic with

𝑓 (\hat{ℂ} ∖ 𝐾) \subseteq \overline{𝔻}

, since

𝑓

is bounded, the Riemann’s Theorem for removable singularities (Theorem 3.2.9

Theorem 3.2.9 Theorem 3.2.9

) allows for an analytic continuation onto all of

\hat{ℂ}

. Then Liouville’s Theorem (Theorem 3.2.5

Theorem 3.2.5 Theorem 3.2.5

) implies that

𝑓

is constant and

𝑓^{'} (\infty) = 0

. Hence

𝛾 (𝐾) = 0

. ⁠

□

Theorem 6.3.3.

For

𝐾_{1} \subseteq 𝐾_{2}

both compact in

ℂ

𝛾 (𝐾_{1}) \leq 𝛾 (𝐾_{2})

Proof.

This follows directly from the definition and the fact that any function holomorphic on

\hat{ℂ} ∖ 𝐾_{1}

is also holomorphic on

\hat{ℂ} ∖ 𝐾_{2}

. ⁠

□

The preceding results above hint at the monotonous behavior of capacity. However, currently it is not known whether a general subadditivity property holds for analytic capacity, or that

𝛾 (𝐾_{1} \cup 𝐾_{2}) \overset{?}{\leq} 𝛾 (𝐾_{1}) + 𝛾 (𝐾_{2}) .

Recent results hint the affirmative, as many special cases of the relation have been proved; the question of subadditivity has been proved in the affirmative for disjoint compact continua, and recent findings by Xavier Tolsa show that capacity is (countably) semi-(sub)additive (the existence of an absolute constant $𝐶$ such that $𝛾 (𝐾_{1} \cup 𝐾_{2}) \leq 𝐶 [𝛾 (𝐾_{1}) + 𝛾 (𝐾_{2})]$ ).

We now give some quantifying examples of how analytic capacity measures a type of “largeness” of compact sets, (rather much like area, which satisfies the subadditivity relation). First we define a specific classification of compact sets.

An alternative perspective of this “largeness” pertains to a certain removability of sets. A compact set $𝐾 \subset ℂ$ is considered to be removable if every bounded holomorphic function on the complement can be extended to $ℂ$ . For instance, the analytic capacity $𝛾 ({𝑎})$ of any singleton ${𝑎}$ (any singular point) or set of discrete points is 0, as evidenced by Proposition 6.3.2 $Proposition 6.3.2 Proposition 6.3.2$ ; and moreover, any singleton or compact set of discrete points is a removable set. In a heuristic sense, analytic capacity measures the irremovability of a set, and larger sets tend to be “less removable.”

A compact set $𝐾 ⋐ ℂ$ is a continuum if it is connected, $ℂ ∖ 𝐾$ is connected, and if it is not a singleton ( $𝐾$ contains at least 2 distinct points).

Proposition 6.3.4.

Let

𝐾 ⋐ ℂ

be a continuum. Then

𝛾 (𝐾) = | 𝑓^{'} (\infty) |

where

𝑓 : \hat{ℂ} ∖ 𝐾 \to 𝔻

is a biholomorphism satisfying

𝑓 (\infty) = 0

(i. e. the maximal

| 𝑓^{'} (\infty) |

in the supremum of the definition of analytic capacity is attained when

𝑓

is biholomorphic).

Proof.

Let $𝑓$ be the biholomorphism, $𝑔 : \hat{ℂ} ∖ 𝐾 \to 𝔻$ be holomorphic (not necessarily surjective) mapping $\infty$ to $0$ . Since $ℎ \equiv 𝑔 \circ 𝑓^{- 1} : 𝔻 \to 𝔻$ and maps $0$ to $0$ , the Schwarz Lemma (Lemma 3.5.1 $Lemma 3.5.1 Lemma 3.5.1$ ) implies that

| ℎ (𝑧) | \leq | 𝑧 |

for all $𝑧 \in 𝔻$ . Thus, $| 𝑔 (𝑧) | \leq | 𝑓 (𝑧) |$ , and

| 𝑔^{'} (\infty) | = lim_{𝑧 \to \infty} | 𝑧 𝑔 (𝑧) | \leq lim_{𝑧 \to \infty} | 𝑧 𝑓 (𝑧) | = | 𝑓^{'} (\infty) | . □

Proposition 6.3.5.

The analytic capacity of any closed disk is the radius.

Proof.

Since $\overline{𝐷 (𝑎, 𝑟)}$ is a continuum, a biholomorphism $𝑓 : \hat{ℂ} ∖ \overline{𝐷 (𝑎, 𝑟)} \to 𝔻$ such that $𝑓 (\infty) = 0$ . One such biholomorphism is given by

𝑓 (𝑧) = \frac{𝑟}{𝑧 - 𝑎}, 𝑓^{'} (\infty) = lim_{𝑧 \to 0} \frac{d}{d 𝑧} (\frac{𝑟}{\frac{1}{𝑧} - 𝑎}) = 𝑟 .

Hence, Proposition 6.3.4

Proposition 6.3.4 Proposition 6.3.4

, gives that

𝛾 (\overline{𝐷 (𝑎, 𝑟)}) = 𝑟

. ⁠

□

Proposition 6.3.6.

If $𝐾 \subset ℂ$ is a continuum, then

\frac{diam 𝐾}{4} \leq 𝛾 (𝐾) \leq diam 𝐾 .

Proof.

Assume $𝑓 : \hat{ℂ} ∖ 𝐾 \to 𝔻$ is a biholomorphism mapping $\infty$ to $0$ . The lower bound follows directly from Proposition 6.2.2 $Proposition 6.2.2 Proposition 6.2.2$ . Let $𝑝 \in 𝐾$ be arbitrary, then for any $𝑞 \in 𝐾$ , we obtain $| 𝑝 - 𝑞 | \leq diam 𝐾$ , implying that $𝐾 \subseteq \overline{𝐷 (𝑝, diam 𝐾)}$ . By Proposition 6.3.5 $Proposition 6.3.5 Proposition 6.3.5$ , we have $𝛾 (\overline{𝐷 (𝑝, diam 𝐾)}) = diam 𝐾$ , and Theorem 6.3.3 $Theorem 6.3.3 Theorem 6.3.3$ consequently gives the upper bound of

𝛾 (𝐾) \leq diam 𝐾 . □

We outline the precise connections to rational approximation:

Theorem 6.3.7.

Let $𝐾 \subseteq ℂ$ be compact such that $\exists 𝑐 > 0$ such that $\forall 𝛿 > 0$ , $\forall 𝑝 \in 𝜕 𝐾$ ,

𝛾 (𝐷 (𝑝, 𝛿) ∖ 𝐾) \geq 𝑐 𝛿 .

Then

\forall 𝑓 \in 𝐶 (𝐾)

holomorphic on

\overset{⚬}{𝐾}

can be uniformly and rationally approximated on

𝐾

with poles in

\hat{ℂ} ∖ 𝐾

Corollary 6.3.7.1.

Let

𝐾 \subseteq ℂ

be compact. If the connected components

𝑈_{𝑗}

\hat{ℂ} ∖ 𝐾

give the uniform existence of some

𝛿 > 0

such that

\forall 𝑗

diam 𝑈_{𝑗} \geq 𝛿

, then

\forall 𝑓 \in 𝐶 (𝐾)

holomorphic on

\overset{⚬}{𝐾}

can be uniformly and rationally approximated on

𝐾

with poles in

\hat{ℂ} ∖ 𝐾

Notice here that no restrictions are imposed on the finiteness of the number of connected components of the complement. The general conclusion given for Mergelyan’s Theorem is not true for more general compact sets.

The counterexample we now provide due to [1], we provide the construction of the compact set $𝐾$ .

Example 6.3.8.

There exists a compact set

𝐾 \subseteq ℂ

and

𝑓 \in 𝐶 (𝐾)

, such that

𝑓

is holomorphic on

\overset{⚬}{𝐾}

and cannot be rationally approximated on

𝐾

Proof.

Let $𝑆 = {𝑠_{𝑘}}_{𝑘 \in ℕ}$ be a countable dense set of points in $𝔻$ (use a bijection $ℕ \to ℚ$ and Cantor’s pairing function $ℕ \to ℕ^{2}$ to get a surjection $ℕ \to ℚ^{2} \cap 𝔻$ ).

Fix $0 < 𝜀^{'} < 1$ . Let $𝑧_{1} = 𝑠_{1}$ and $𝑟_{1} = \frac{𝜀}{2} < 𝜀^{'}$ , $𝑟_{1}^{2} < \frac{1}{2}$ . For each $𝑘 \in ℕ$ , let $𝑧_{𝑘}$ be the first to be the first point in the dense sequence $𝑆$ such that

𝑐_{𝑘} \notin ⋃_{𝑗 = 1}^{𝑘 - 1} \overline{𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})} .

Then choose $𝑟_{𝑘}$ such that $𝐷 (𝑧_{𝑘}, 𝑟_{𝑘})$ lies in $𝔻$ and does not intersect any previous $𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})$ for $𝑗 < 𝑘$ (possible by the fact that each $𝑧_{𝑗}$ does not lie on the boundary of the previous disks) and so that

0 < 𝑟_{𝑘} \leq \frac{𝜀}{2} - \frac{1}{2} \sum_{𝑗 = 1}^{𝑘 - 1} 𝑟_{𝑗} < 𝜀^{'} - \sum_{𝑗 = 1}^{𝑘 - 1} 𝑟_{𝑗} ⟹ \sum_{𝑗 = 1}^{𝑘} < 𝜀^{'}

under the inductive hypothesis that $\sum_{𝑗 = 1}^{𝑘 - 1} 𝑟_{𝑗} < 𝜀^{'}$ . Apply another bound, so that

0 < 𝑟_{𝑘}^{2} \leq \frac{1}{4} - \frac{1}{2} \sum_{𝑗 = 1}^{𝑘 - 1} 𝑟_{𝑗}^{2} < \frac{1}{2} - \sum_{𝑗 = 1}^{𝑘 - 1} 𝑟_{𝑗}^{2} ⟹ \sum_{𝑗 = 1}^{𝑘} 𝑟_{𝑗}^{2} < \frac{1}{2}

under the additional assumption that $\sum_{𝑗 = 1}^{𝑘 - 1} 𝑟_{𝑗}^{2} < \frac{1}{2}$ . Repeat this process inductively for all $𝑘 \in ℕ$ . Define

𝐾 = \overline{𝔻} ∖ ⋃_{𝑗 = 1}^{\infty} 𝐷 (𝑧_{𝑗}, 𝑟_{𝑗}),

which is compact. For any point $𝑧 \in 𝐾$ , no disk centered at $𝑧$ exists such that $𝐷 (𝑧, 𝛿)$ is contained in $𝐾$ , since a subsequence of $𝑆$ accumulating to $𝑧$ in $\overline{𝔻}$ is removed from $𝐾$ . Hence, $\overset{⚬}{𝐾} = ⌀$ . Hence, any $𝑓 \in 𝐶 (𝐾)$ is holomorphic on the interior.

(This general construction of $𝐾$ is known as the Swiss cheese set)

We now show that $𝑧 \mapsto \overline{𝑧}$ cannot be uniformly rationally approximated on $𝐾$ . By explicit calculation or Green’s Theorem (Theorem 3.1.2 $Theorem 3.1.2 Theorem 3.1.2$ ), we have

\begin{aligned} \oint_{𝜕 𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})} \overline{𝑧} d 𝑧 & = \iint_{𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})} d (\overline{𝑧} d 𝑧) = 2 i \iint_{𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})} d 𝑥 \land d 𝑦 \\ = 2 π i 𝑟_{𝑗}^{2} ⟹ | \sum_{𝑗 = 1}^{\infty} \oint_{𝜕 𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})} \overline{𝑧} d 𝑧 | \leq π, \end{aligned}

and by similar reasoning

| \oint_{𝜕 𝔻} \overline{𝑧} d 𝑧 | = 2 π ⟹ | (\oint_{𝜕 𝔻} - \sum_{𝑗 = 1}^{\infty} \oint_{𝜕 𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})}) \overline{𝑧} d 𝑧 | \geq π .

For any rational $𝜓$ with poles off $𝐾$ ,

(\oint_{𝜕 𝔻} - \sum_{𝑗 = 1}^{\infty} \oint_{𝜕 𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})}) 𝜓 (𝑧) d 𝑧 = 0

by Theorem 3.1.5 $Theorem 3.1.5 Theorem 3.1.5$ . The summation’s convergence follows from $| \sum \oint_{𝜕 𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})} 𝜓 (𝑧) d 𝑧 |$ being termwise absolutely bounded by $\sum 2 π 𝑀 𝑟_{𝑗}$ , which converges by construction. Now, if $| \overline{𝑧} - 𝜓 (𝑧) | < \frac{1}{4}$ on $𝐾$ , then

\begin{aligned} π & \leq | (\oint_{𝜕 𝔻} - \sum_{𝑗 = 1}^{\infty} \oint_{𝜕 𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})}) \overline{𝑧} d 𝑧 | \leq | \oint_{𝜕 𝔻} - \sum_{𝑗 = 1}^{\infty} \oint_{𝜕 𝐷 (𝑧_{𝑗}, 𝑟_{𝑗})} | | 𝜓 (𝑧) - \overline{𝑧} | | d 𝑧 | \\ \leq 2 π (1 - \sum_{𝑗 = 1}^{\infty} 𝑟_{𝑗}) \frac{1}{4} \leq \frac{π}{2}, \end{aligned}

which is impossible. ⁠ $□$