Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class

Let : Ω ⊂ R → R be a quasiconformal mapping whose Jacobian is denoted by 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 : ∈ EXP(Ω) → ∘ −1 ∈ EXP((Ω)) and, as a related question, we study the behaviour of the norm of log in the exponential class. The ∞ property of 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.