Journal ArticleBi-Sobolev ExtensionsKoski, Aleksis; Onninen, JaniAbstract We give a full characterization of circle homeomorphisms which admit a homeomorphic extension to the unit disk with finite bi-Sobolev norm. As a special case, a bi-conformal variant of the famous Beurling–Ahlfors extension theorem is obtained. Furthermore we show that the existing extension techniques such as applying either the harmonic or the Beurling–Ahlfors operator work poorly in the degenerated setting. This also gives an affirmative answer to a question of Karafyllia and Ntalampekos.2023-09-0110437961The Journal of Geometric Analysis3391050-6926https://doi.org/10.1007/s12220-023-01363-12154943National Science Foundation