In the mathematical discipline ofcomplex analysis, theanalytic capacity of acompact subsetK of thecomplex plane is a number that denotes "how big" aboundedanalytic function onC \ K can become. Roughly speaking,γ(K) measures the size of theunit ball of the space of bounded analytic functions outsideK.

It was first introduced byLars Ahlfors in the 1940s while studying the removability ofsingularities of bounded analytic functions.

LetK ⊂C becompact. Then its analytic capacity is defined to be

${\displaystyle \gamma (K)=sup\{|f^{\prime }(\mathrm{\infty })|;f\in {\mathcal{H}}^{\mathrm{\infty }}(\mathbf{C}\setminus K),\Vert f{\Vert }_{\mathrm{\infty }}\le 1,f(\mathrm{\infty })=0\}}$

Here,${\displaystyle {\mathcal{H}}^{\mathrm{\infty }}(U)}$ denotes the set ofbounded analyticfunctionsU →C, wheneverU is anopen subset of thecomplex plane. Further,

${\displaystyle f^{\prime }(\mathrm{\infty }):=\underset{z\to \mathrm{\infty }}{lim}z\left(f(z)-f(\mathrm{\infty })\right)}$

${\displaystyle f(\mathrm{\infty }):=\underset{z\to \mathrm{\infty }}{lim}f(z)}$

Note that${\displaystyle f^{\prime }(\mathrm{\infty })=g^{\prime }(0)}$, where${\displaystyle g(z)=f(1/z)}$. However, usually${\displaystyle f^{\prime }(\mathrm{\infty })\ne \underset{z\to \mathrm{\infty }}{lim}{f}^{\prime }(z)}$.

Equivalently, the analytic capacity may be defined as^[1]

${\displaystyle \gamma (K)=sup\left|\frac{1}{2\pi }{\int }_{C}f(z)dz\right|}$

whereC is a contour enclosingK and the supremum is taken overf satisfying the same conditions as above:f is bounded analytic outsideK, the bound is one, and${\displaystyle f(\mathrm{\infty })=0.}$

IfA ⊂C is an arbitrary set, then we define

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

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

It is easy to see thatK is removable if and only ifγ(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.

For each compactK ⊂C, there exists a unique extremal function, i.e.${\displaystyle f\in {\mathcal{H}}^{\mathrm{\infty }}(\mathbf{C}\setminus K)}$ such that${\displaystyle \Vert f\Vert \le 1}$,f(∞) = 0 andf′(∞) =γ(K). This function is called theAhlfors function ofK. Its existence can be proved by using a normal family argument involvingMontel's theorem.

Let dim_H denoteHausdorff dimension andH¹ denote 1-dimensionalHausdorff measure. ThenH¹(K) = 0 impliesγ(K) = 0 while dim_H(K) > 1 guaranteesγ(K) > 0. However, the case when dim_H(K) = 1 andH¹(K) ∈ (0, ∞] is more difficult.

Given the partial correspondence between the 1-dimensional Hausdorff measure of a compact subset ofC and its analytic capacity, it might be conjectured thatγ(K) = 0 impliesH¹(K) = 0. However, this conjecture is false. A counterexample was first given byA. G. Vitushkin, and a much simpler one byJohn B. Garnett in his 1970 paper. This latter example is thelinear four corners Cantor set, constructed as follows:

LetK₀ := [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 ofK₀. In general,K_n is the union of 4ⁿ squares (denoted by${\displaystyle Q_n^j}$) of side length 4⁻ⁿ, each${\displaystyle Q_n^j}$ being in the corner of some${\displaystyle Q_{n-1}^k}$. TakeK to be the intersection of allK_n then${\displaystyle H^1(K)=\sqrt{2}}$ butγ(K) = 0.

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

${\displaystyle \gamma (K)=0⟺{\int }_0^{\pi }{\mathcal{H}}^1({\mathrm{proj}}_{\theta }(K))d\theta =0}$

where${\displaystyle {\mathrm{proj}}_{\theta }(x,y):=x\cos \theta +y\sin \theta }$ 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 andH¹(K) < ∞. In 2002,Xavier Tolsa proved that analytic capacity is countably semiadditive. That is, there exists an absolute constantC > 0 such that ifK ⊂C is a compact set and${\displaystyle K=\bigcup _{i=1}^{\mathrm{\infty }}{K}_i}$, where eachK_i is a Borel set, then${\displaystyle \gamma (K)\le C\sum _{i=1}^{\mathrm{\infty }}\gamma (K_i)}$.

David's and Tolsa's theorems together imply that Vitushkin's conjecture is true whenK isH¹-sigma-finite.

In the nonH¹-sigma-finite case, Pertti Mattila proved in 1986^[2] that the conjecture is false, but his proof did not specify which implication of the conjecture fails. Subsequent work by Jones and Muray^[3] 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.^[4]

• Capacity of a set – In Euclidean space, a measure of that set's "size"
• Conformal radius

• 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.

• Analytic functions