Characterization of Asymptotically Smooth Chord-arc Curves

Abstract

We construct an explicit example of an asymptotically conformal chord-arc curve that fails to be asymptotically smooth. This implies that a function belonging to both the little Bloch space and $\mathrm{BMOA}$ does not necessarily lie in $\mathrm{VMOA}$, and that a strongly quasisymmetric homeomorphism which is symmetric is not necessarily strongly symmetric. We also provide a complete characterization of asymptotically smooth curves in terms of asymptotic conformality and uniform approximability.

arXiv commentWe added some remarks and explanations. This version is accepted by Revista Matemática Complutense