Research Problems in Function Theory — Problem 7.55

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.

Let $n\ge2$, $B^n=\{x\in\mathbb R^n:|x|<1\}$ and $S^{n-1}=\partial B^n$. Let $D\subset\mathbb R^n$ be a Jordan domain and let $f:B^n\to D$ be a quasiconformal homeomorphism with a homeomorphic boundary extension $\bar f:\overline{B^n}\to\overline D$. Here quasiconformal means $f\in W^{1,n}_{\rm loc}$ and $\|Df\|^n\le K|\det Df|$ almost everywhere for some finite $K$. Assume $\mathcal H^{n-1}(\partial D)<\infty$, where $\mathcal H^{n-1}$ is Hausdorff surface measure. Boundary absolute continuity is the following preservation of surface-measure-zero Borel sets; the inverse is tested in the opposite direction.

Statement recorded in the supplied source.

Must both of the following hold? \[\mathcal H^{n-1}(E)=0,\ E\subset S^{n-1}\quad\Longrightarrow\quad\mathcal H^{n-1}(\bar f(E))=0, \] \[\mathcal H^{n-1}(F)=0,\ F\subset\partial D\quad\Longrightarrow\quad\mathcal H^{n-1}(\bar f^{-1}(F))=0? \] Retain also the inverse question in dimension $3$ under the additional assumption that $f$ extends quasiconformally to $\mathbb R^3$. The source records affirmative planar conformal results, planar quasiconformal counterexamples, and Gehring's affirmative forward result in this three-dimensional extension case. [S2]

Short English statement

Determine whether a quasiconformal boundary map, and its inverse, preserve surface-null sets when the target boundary has finite surface measure, including the inverse question for extendable three-dimensional maps.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2307055 (AMR-022-7055). 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 7.55, its complete update and relevant preceding definitions. https://arxiv.org/abs/1809.07200.