Homeomorphic Sobolev extensions and integrability of hyperbolic metric

Very recently, it was proved that if the hyperbolic metric of a planar Jordan domain is $L^q$-integrable for some $q\in (1,\infty)$, then every homeomorphic parametrization of the boundary Jordan curve via the unit circle can be extended to a Sobolev homeomorphism of the entire disk. This naturally raises the question of whether the extension holds under more general integrability conditions on the hyperbolic metric. In this work, we examine the case where the hyperbolic metric is $\phi$-integrable. Under appropriate conditions on the function $\phi$, we establish the existence of a Sobolev homeomorphic extension for every homeomorphic parametrization of the Jordan curve. Moreover, we demonstrate the sharpness of our result by providing an explicit counterexample.