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

Abstract EN

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.