Analytic capacity

In the mathematical discipline of complex analysis, the analytic capacity of a compact subset K of the complex plane is a number that denotes "how big" a bounded analytic function on C \ K can become. Roughly speaking, γ(K) measures the size of the unit ball of the space of bounded analytic functions outside K.

It was first introduced by Lars Ahlfors in the 1940s while studying the removability of singularities of bounded analytic functions.

Definition

Let K ⊂ C be compact. Then its analytic capacity is defined to be

\gamma (K)=\sup\{|f'(\infty )|;\ f\in {\mathcal {H}}^{\infty }(\mathbf {C} \setminus K),\ \|f\|_{\infty }\leq 1,\ f(\infty )=0\}

Here, ${\mathcal {H}}^{\infty }(U)$ denotes the set of bounded analytic functions U → C, whenever U is an open subset of the complex plane. Further,

f'(\infty ):=\lim _{z\to \infty }z\left(f(z)-f(\infty )\right)

f(\infty ):=\lim _{z\to \infty }f(z)

Note that $f'(\infty )=g'(0)$ , where $g(z)=f(1/z)$ . However, usually $f'(\infty )\neq \lim _{z\to \infty }f'(z)$ .

If A ⊂ C is an arbitrary set, then we define

\gamma (A)=\sup\{\gamma (K):K\subset A,\,K{\text{ compact}}\}.

Removable sets and Painlevé's problem

The compact set K is called removable if, whenever Ω is an open set containing K, every function which is bounded and holomorphic on the set Ω \ K has an analytic extension to all of Ω. By Riemann's theorem for removable singularities, every singleton is removable. This motivated Painlevé to pose a more general question in 1880: "Which subsets of C are removable?"

It is easy to see that K is removable if and only if γ(K) = 0. However, analytic capacity is a purely complex-analytic concept, and much more work needs to be done in order to obtain a more geometric characterization.

Ahlfors function

For each compact K ⊂ C, there exists a unique extremal function, i.e. $f\in {\mathcal {H}}^{\infty }(\mathbf {C} \setminus K)$ such that $\|f\|\leq 1$ , f(∞) = 0 and f′(∞) = γ(K). This function is called the Ahlfors function of K. Its existence can be proved by using a normal family argument involving Montel's theorem.

Analytic capacity in terms of Hausdorff dimension

Let dim_H denote Hausdorff dimension and H¹ denote 1-dimensional Hausdorff measure. Then H¹(K) = 0 implies γ(K) = 0 while dim_H(K) > 1 guarantees γ(K) > 0. However, the case when dim_H(K) = 1 and H¹(K) ∈ (0, ∞] is more difficult.

Positive length but zero analytic capacity

Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset of C and its analytic capacity, it might be conjectured that γ(K) = 0 implies H¹(K) = 0. However, this conjecture is false. A counterexample was first given by A. G. Vitushkin, and a much simpler one by John B. Garnett in his 1970 paper. This latter example is the linear four corners Cantor set, constructed as follows:

Let K₀ := [0, 1] × [0, 1] be the unit square. Then, K₁ is the union of 4 squares of side length 1/4 and these squares are located in the corners of K₀. In general, K_n is the union of 4ⁿ squares (denoted by $Q_{n}^{j}$ ) of side length 4⁻ⁿ, each $Q_{n}^{j}$ being in the corner of some $Q_{n-1}^{k}$ . Take K to be the intersection of all K_n then $H^{1}(K)={\sqrt {2}}$ but γ(K) = 0.

Vitushkin's conjecture

Let K ⊂ C be a compact set. Vitushkin's conjecture states that

\gamma (K)=0\ \iff \ \int _{0}^{\pi }{\mathcal {H}}^{1}(\operatorname {proj} _{\theta }(K))\,d\theta =0

where $\operatorname {proj} _{\theta }(x,y):=x\cos \theta +y\sin \theta$ denotes the orthogonal projection in direction θ. By the results described above, Vitushkin's conjecture is true when dim_HK ≠ 1.

Guy David published a proof in 1998 of Vitushkin's conjecture for the case dim_HK = 1 and H¹(K) < ∞. In 2002, Xavier Tolsa proved that analytic capacity is countably semiadditive. That is, there exists an absolute constant C > 0 such that if K ⊂ C is a compact set and $K=\bigcup _{i=1}^{\infty }K_{i}$ , where each K_i is a Borel set, then $\gamma (K)\leq C\sum _{i=1}^{\infty }\gamma (K_{i})$ .

David's and Tolsa's theorems together imply that Vitushkin's conjecture is true when K is H¹-sigma-finite.

In the non H¹-sigma-finite case, Pertti Mattila proved in 1986[1] that the conjecture is false, but his proof did not specify which implication of the conjecture fails. Subsequent work by Jones and Muray[2] produced an example of a set with zero Favard length and positive analytic capacity, explicitly disproving one of the directions of the conjecture. As of 2023 it is not known whether the other implication holds but some progress has been made towards a positive answer by Chang and Tolsa.[3]

References

Mattila, Pertti (1986). "Smooth Maps, Null-Sets for Integralgeometric Measure and Analytic Capacity". Annals of Mathematics. 123 (2): 303–309. doi:10.2307/1971273. ISSN 0003-486X.
Jones, Peter W.; Murai, Takafumi. "Positive analytic capacity but zero Buffon needle probability" (PDF). Pacific Journal of Mathematics. 133 (1): 99–114.
Chang, Alan; Tolsa, Xavier (2020-10-05). "Analytic capacity and projections". Journal of the European Mathematical Society. 22 (12): 4121–4159. doi:10.4171/JEMS/1004. ISSN 1435-9855.

Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces. Cambridge University Press. ISBN 0-521-65595-1.
Pajot, Hervé (2002). Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral. Lecture Notes in Mathematics. Springer-Verlag.
J. Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 21 (1970), 696–699
G. David, Unrectifiable 1-sets have vanishing analytic capacity, Rev. Math. Iberoam. 14 (1998) 269–479
Dudziak, James J. (2010). Vitushkin's Conjecture for Removable Sets. Universitext. Springer-Verlag. ISBN 978-14419-6708-4.
Tolsa, Xavier (2014). Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory. Progress in Mathematics. Birkhäuser Basel. ISBN 978-3-319-00595-9.

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[1] Mattila, Pertti (1986). "Smooth Maps, Null-Sets for Integralgeometric Measure and Analytic Capacity". Annals of Mathematics. 123 (2): 303–309. doi:10.2307/1971273. ISSN 0003-486X.

[2] Jones, Peter W.; Murai, Takafumi. "Positive analytic capacity but zero Buffon needle probability" (PDF). Pacific Journal of Mathematics. 133 (1): 99–114.

[3] Chang, Alan; Tolsa, Xavier (2020-10-05). "Analytic capacity and projections". Journal of the European Mathematical Society. 22 (12): 4121–4159. doi:10.4171/JEMS/1004. ISSN 1435-9855.