Sharp bounds for composition with quasiconformal mappings in Sobolev spaces

Let ϕ be a quasiconformal mapping, and let Tϕ be the composition operator which maps f to f∘ϕ. Since ϕ may not be bi-Lipschitz, the composition operator need not map Sobolev spaces to themselves. The study begins with the behavior of Tϕ on Lp and W1,p for 1