Explicit Bounds and Sharp Results for the Composition Operators Preserving the Exponential Class. (3rd August 2016)

• Record Type: Journal Article
• Authors: Farroni, Fernando; Giova, Raffaella
• Other Names: Hencl Stanislav Academic Editor.
• Is Part Of: Journal of function spaces. Volume 2016(2016)
• Publication Date: 2016-08-03
• Subjects: Function spaces -- Periodicals; 515.7305
• DOI: 10.1155/2016/3769813
• Languages: English
• ISSNs: 2314-8896

Abstract

Abstract : 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.