Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class - Fernando Farroni; Raffaella Giova (Journal of Function Spaces)
Let f:Ω⊂Rn→Rn be a quasiconformal mapping whose Jacobian is denoted by Jf and let EXP(Ω) be the space of exponentially integrable functions on Ω. We give an explicit bound for the norm of the composition operator Tf: u∈EXP(Ω)↦u∘f-1∈EXP(f(Ω)) and, as a related question, we study the behaviour of the norm of log⁡Jf in the exponential class. The A∞ property of Jf is the counterpart in higher dimensions of the area distortion formula due to Astala in the plane and it is the key tool to prove the sharpness of our results.