Sharp bounds for composition with quasiconformal mappings in Sobolev spaces

Abstract

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<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 Hs,p are sent to Hs,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.