
## 4. Regularity

1. ### Almost everywhere partial differentiability of mappings of integrable lower metric distortion

The lower metric dilatation of a homeomorphism $f:X\to Y$ of metric spaces is defined as $$h_f(x) = \liminf_{r\to 0} \frac{\sup \{ d(f(x),f(y)) : d(x,y) \le r \}}{\inf \{ d(f(x),f(z)) : d(x,z) \ge r \}}.$$

#### Problem 4.1.

[Pekka Koskela] Let $f$ be a homeomorphism of planar domains with $h_f \in L^1_{\scriptstyle{loc}}$ and $h_f$ finite off a set of $\sigma$-finite length. Does $f$ have partial derivatives almost everywhere?
By work of Sari Kallunki, the answer to the problem is yes if $L^1_{\scriptstyle{loc}}$ is replaced by $L^2_{\scriptstyle{loc}}$. The problem is also open if $L^1_{\scriptstyle{loc}}$ is replaced by $L^p_{\scriptstyle{loc}}$ for any $p<2$.
• ### Mapping properties of planar maps of exponentially integrable distortion

#### Problem 4.2.

[Pekka Koskela] Does there exist a homeomorphism $f$ of the plane with exponentially integrable distortion such that $f$ fixes the real axis and sends a set of positive length onto the $\tfrac13$ Cantor set?
• ### Approximation of Sobolev homeomorphisms by diffeomorphisms

#### Problem 4.3.

[Leonid Kovalev] Let $f$ be a homeomorphism of the plane such that $f$ and $f^{-1}$ lie in some Sobolev class, e.g., $W^{1,2}$. Can one approximate $f$ by diffeomorphisms $f_j$ so that $f_j$ and $f_j^{-1}$ converge in the Sobolev norm?
1. Remark. The one-sided approximation, namely $f_j \to f$ without the convergence of inverses, is possible in this case (Iwaniec–Kovalev–Onninen).
• Remark. Even the one-sided case remains open in the non-reflexive space $W^{1,1}$, or for any Sobolev space in dimension three.

Cite this as: AimPL: Mapping theory in metric spaces, available at http://aimpl.org/mappingmetric.