Quasiconformal mappings and exponentially integrable functions

We prove that a if-quasiconformal mapping f: R2 → R 2 which maps the unit disk D onto itself preserves the space EXP(D) of exponentially integrable functions over D, in the sense that u ∈ EXP(D) if and only if u o f-1 ∈ EXP(D). Moreover, if / is assumed to be conformal outside the unit disk and principal, we provide the estimate 1/1+K log K ≤ ||u o f-1||EXP(D)/||u||EXP(D)≤1+K log K for every u ε EXP(D). Similarly, we consider the distance from L∞ in EXP and we prove that if f: Ω → Ω' is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then 1/K ≤ distEXP(f(G))(u o f-1, L∞(f(G)))/distEXP(f(G))(u, L∞(G)) ≤ K for every u ∈ EXP(G). We also prove that the last estimate is sharp, in the sense that there exist a quasiconformal mapping f: D → D, a domain G ⊂ ⊂ D and a function u ∈ EXP(G) such that distEXP(f(G))(u o f-1, L∞(f(G))) = K distEXP(f(G))(u, L∞(G)).