Research Problems in Function Theory — Problem 7.75

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 $E\subset\mathbb C$ define its analytic capacity \[\gamma(E)=\sup\{|f'(\infty)|: f\in\mathcal O(\widehat{\mathbb C}\setminus E),\ |f|\le1,\ f(\infty)=0\}, \qquad f'(\infty)=\lim_{z\to\infty}zf(z). \] For an arbitrary bounded set $A$, its inner analytic capacity is $\gamma(A)=\sup\{\gamma(L):L\subset A,\ L\text{ compact}\}$.Identify $\mathbb C$ with $\mathbb R^2$. A $C^1$ diffeomorphism has a $C^1$ inverse; $\operatorname{GL}(2,\mathbb R)$ consists of invertible real linear maps. The four-corner quarter-square Cantor set is $C\times C$, where $C=\{\sum_{j\ge1}3\varepsilon_j4^{-j}:\varepsilon_j\in\{0,1\}\}$.

Statement recorded in the supplied source.

Prove or disprove each assertion: (a) $\gamma(E)=0$ if and only if $\gamma(\phi(E))=0$ for every compact $E$ and every $C^1$ diffeomorphism $\phi:\mathbb C\to\mathbb C$; (b) the same equivalence for every $\phi\in\operatorname{GL}(2,\mathbb R)$; (c) a universal $K>0$ satisfies $\gamma(E\cup F)\le K(\gamma(E)+\gamma(F))$ for all compact $E,F$, and in particular can $K=1$ be taken? (d) If compact $K$ has $\gamma(K)=0$, must $\gamma(E\setminus K)=\gamma(E)$ for every compact $E$, with the left side interpreted as inner capacity? Include the four-corner Cantor case of (d). The original update records Tolsa's proof of (c) for some universal constant, without resolving the proposed value $1$. [S2]

Short English statement

Study preservation of zero analytic capacity by smooth or linear changes of coordinates, universal subadditivity and its sharp constant, and removal of zero-capacity compact sets.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307075 (AMR-022-7075). 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 7.75, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.