Global smoothness of quasiconformal mappings in the Triebel-Lizorkin scale

We study quasiconformal mappings in planar domains $\mathrm{\Omega }$ and their regularity properties described in terms of Sobolev, Bessel potential or Triebel-Lizorkin scales. This leads to optimal conditions, in terms of the geometry of the boundary $\mathrm{\partial }\mathrm{\Omega }$ and of the smoothness of the Beltrami coefficient, that guarantee the global regularity of the mappings in these classes. In the Triebel-Lizorkin class with smoothness below $1$, the same conditions give global regularity in $\mathrm{\Omega }$ for the principal solutions with Beltrami coefficient supported in $\mathrm{\Omega }$.