3
$\begingroup$

It's stated in Väisälä's 'Lectures on n-dimensional quasiconformal mappings' (p. 20) that, in the geometric definition of a quasiconformal mapping, that the modulus of a family of curves associated to a ring is unaffected by considering admissible functions $\rho$ which are continuous and not just measurable. No further explanation was given there. I skimmed through some papers by Gehring, Väisälä, etc. from the time period and couldn't find any elaboration on this. Could someone suggest a reference, preferably with proof?

The question could also be interpreted in terms of the regularity of admissible functions for capacities.

A reminder of the definitions: for any disjoint connected continua $E, F \subset \mathbb{R}^n$, we consider the family $\Gamma$ of curves with initial point in $E$ and terminal point in $F$. A measurable function $\rho: \mathbb{R}^n \rightarrow [0, \infty]$ is admissible if $\int_\gamma \rho ds \geq 1$ for all rectifiable $\gamma \in \Gamma$. Then $\text{Mod}_p \Gamma = \inf\{ \int_{\mathbb{R}^n} \rho^p dm\}$, the infimum taken over all admissible $\rho$.

$\endgroup$
2
$\begingroup$

Gehring shows (for p=n=3) in the first equality of Theorem 1 in his paper

Extremal length definitions for the conformal capacity of rings in space. https://projecteuclid.org/euclid.mmj/1028998672

that the conformal modulus of a ring domain $R$ (defined via measurable functions) can equivalently be defined as the infimum of $\int |Df|^p$ where $f$ varies among $ACL$ functions having boundary values 0 and 1 on the two boundary components of $R$. Moreover he cites on p.138 another paper of himself showing that $ACL$ may be replaced by $C^1$.

It seems quite likely that a corresponding characterization also holds for general n and $p$ (not necessarily equal to $n$), maybe even the same proof goes through, see edit below.

Assuming that these results also hold for general $p$ and $n$, it follows immediately from the argument on p.142 of the same paper that the conformal modulus of $R$ can also be reached as the infimum over continuous admissible functions.

Edit: Actually the whole argument seems to be essentially spelled out in Rickman's book on quasiregular mappings, starting on p.53, section 10.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.