Lower derivatives of functions of finite variation and generalized BCH sets (Q2367602)

In this paper one characterizes the sets of points of \(\mathbb{R}^ n\) where the generalized lower derivative \[ D_{\omega,\alpha} u(x)= \liminf_{t\to 0} {| u(x+ y, t)|\over \omega(| B(0,t)|)} \] is positive in case \(u\) is of finite \(\gamma\)-variation. The author denotes by \(| B(0,t)|\) the Lebesgue measure of the ball \(B(0,t)= \{y\in \mathbb{R}^ n;\;d(0,y)< t\}\), where \(d\) is a continuous, translation invariant pseudo-distance on \(\mathbb{R}^ n\) and by \(\omega\) a modulus of continuity. Defining the Beurling-Carleson-Hayman (BCH) set (or \((\omega,\alpha)\)- set) as a closed set \(E\subset \mathbb{R}^ n\) of Lebesgue measure zero such that for each bounded set \(B\) containing \(E\), \[ \int_ B {\omega(d^{\alpha r}(x, E))\over d^{\alpha r}(x, E)} dx \] the author shows that for \(\alpha\), \(\gamma\geq 1\), \(\{x: D_{\omega,\alpha} u(x)\geq 0\}\) is a countable union of \((\omega, \alpha)\)-sets. In the last part of the paper the author establishes sufficient and necessary conditions for the BCH sets to have a null Hausdorff measure.