Absolutely continuous functions of n variables were recently introduced by J. Malý [5]. We introduce a more general definition, suggested by L. Zajíček. This new absolute continuity also implies continuity, weak differentiability with gradient in Lⁿ, differentiability almost everywhere and the area formula. It is shown that our definition does not depend on the shape of balls in the definition.
In [4], a class of absolutely continuous functions of d-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of [4] are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls.
Let Ω,Ω’ ⊂ ℝⁿ be domains and let f: Ω → Ω’ be a homeomorphism. We show that if the composition operator maps the Sobolev-Lorentz space to for some q ≠ n then f must be a locally bilipschitz mapping.
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