Sharp bounds for composition with quasiconformal mappings in Sobolev spaces

Let ϕ be a quasiconformal mapping, and let T_ϕ 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 T_ϕ on L^p and W^1,p for 1<p<∞. This cases are well understood but alternative proofs of some known results are provided. Using interpolation techniques it is seen that compactly supported Bessel potential functions in H^s,p are sent to H^s,q whenever 0<s<1 for appropriate values of q. The techniques used lead to sharp results and they can be applied to Besov spaces as well.