On homeomorphisms with a fixed point onto domains with a Poincaré inequality

The article is devoted to the study of homeomorphisms with one normalization condition, which act onto some Ahlfors regular domain with a Poincaré inequality and distort the modulus of families of paths according to the Poletsky inequality. We consider the case when the majorant from the above inequality has finite averages over infinitesimal balls. We have obtained the Hölder boundary continuity of such homeomorphisms provided that the image of at least one inner point is located at a fixed distance from the boundary of the mapped domain. In the manuscript, we have considered the cases of good boundaries and domains with prime ends.