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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 \}}.

      Add remark

      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


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          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


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              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.