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

Let f: ω ⊂ r n → r n be a quasiconformal mapping whose jacobian is denoted by j f and let e x p (ω) be the space of exponentially integrable functions on ω. we give an explicit bound for the norm of the composition operator t f: u e x p (ω) → u � f-1 e x p (f (ω)) and,as a related question,we study the behaviour of the norm of log j f in the exponential class. the a ∞ property of j f 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. � 2016 fernando farroni and raffaella giova.