Notes on Absolutely Continuous Functions of Several Variables

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Let $\Omega\subset\mathbb{r}^n$ be a domain. A result of J. Kauhanen, P. Koskela and J. Malý in 1999 states that a function $f:\Omega\to\mathbb{R}$ with a derivative in the Lorentz space $L^{n,1}(\Omega,\mathbb{R}^n)$ is $n$-absolutely continuous in the sense of J. Malý. We give an example of an absolutely continuous function of two variables, whose derivative is not in $L^{2,1}$. The boundary behavior of $n$-absolutely continuous functions is also studied.