Research Problems in Function Theory — Problem 7.76

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For compact $E\subset\mathbb C$, continuous analytic capacity is \[\alpha(E)=\sup\{|f'(\infty)|:f\in C(\widehat{\mathbb C}),\ f\in\mathcal O(\widehat{\mathbb C}\setminus E),\ |f|\le1,\ f(\infty)=0\}. \] Here $f'(\infty)=\lim_{z\to\infty}zf(z)$. On arbitrary bounded sets use inner capacity $\alpha(A)=\sup\{\alpha(L):L\subset A,\ L\text{ compact}\}$. A $C^1$ arc is the image of an injective regular $C^1$ parametrization of a compact interval; $C^{1+\varepsilon}$ additionally means its derivative is Holder continuous with positive exponent $\varepsilon$.

Statement recorded in the supplied source.

If $K\subset\mathbb C$ is compact and $\alpha(K)=0$, must $\alpha(E\setminus K)=\alpha(E)$ for every compact $E\subset\mathbb C$? Retain the special case where $K$ is a $C^1$ arc. The source distinguishes this from analytic capacity and records the $C^{1+\varepsilon}$-arc case as settled. [S2]

Short English statement

Does deleting a compact set of zero continuous analytic capacity preserve this capacity, including deletion of a continuously differentiable arc?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307076 (AMR-022-7076). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Walter K. Hayman and Edward F. Lingham, Research Problems in Function Theory (2018), Problem 7.76, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.