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Absolutely continuous functions of several variables. (English) Zbl 0924.26008

The aim of the paper is to introduce a function space which seems to be the right extension to more than one independent variable of the classical space of absolutely continuous functions introduced by Vitali. A mapping \(\varphi:\Omega\subset{\mathbb R}^n\to {\mathbb R}^d\) is said to be \(n\)-absolutely continuous (in symbols, \(\varphi\in AC_n(\Omega)\)) if for any \(\varepsilon>0\) there exists \(\delta>0\) such that \[ \sum_i {\mathcal L}^n(B_i)<\delta \Longrightarrow \sum_i\bigl(\text{osc}_{B_i} \varphi\bigr)^n<\varepsilon \] whenever \(B_i\subset\Omega\) are pairwise disjoint. The author proves that any mapping in this class belongs to the Sobolev space \(W^{1,n}\) and that any mapping in \(W^{1,p}\) for some \(p>n\) is \(n\)-absolutely continuous. Moreover, many results previously known for \(W^{1,p}\) functions for some \(p>n\), or for functions whose weak gradient is in the Lorentz space \(L^{n,1}\) (differentiability a.e., Lusin property, degree, area and coarea formulas), can be extended to \(AC_n\) functions and proved in a unified way. It is also proved that spherically pseudomonotone functions are \(n\)-absolutely continuous.
Reviewer: L.Ambrosio (Pisa)

MSC:

26B30 Absolutely continuous real functions of several variables, functions of bounded variation
26A46 Absolutely continuous real functions in one variable
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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