arXiv subjects
Publications and source records attributed to Soheil Malekzadeh.
We prove that if $Φ:X\to Y$ a mapping of weak bounded length distortion from a quasiconvex and complete metric space $X$ to any metric space $Y$, then for any Lipschitz mapping $f:\mathbb{R}^k\supset E\to X$ we have that ${\mathcal H}^k(f(E))=0$ in $X$ if and only if ${\mathcal H}^k(Φ(f(E)))=0$ in $Y$. This generalizes an earlier result of Hajłasz and Malekzadeh where the target space $Y$ was a Euclidean space $Y=\mathbb{R}^N$.
In this paper, we present a new characterization of the mappings of bounded length distortion (BLD for short). In the original geometric definition it is assumed that a BLD mapping is open, discrete and sense preserving. We prove that the first two of the three conditions are redundant and the sense-preserving condition can be replaced by a weaker assumption that the Jacobian is non-negative.
We find necessary and sufficient conditions for a Lipschitz map $f:\mathbb{R}E\to X$, into a metric space to have the image with the $k$-dimensional Hausdorff measure equal zero, $H^k(f(E))=0$. An interesting feature of our approach is that despite the fact that we are dealing with arbitrary metric spaces, we employ a variant of the classical implicit function theorem. Applications include pure unrectifiability of the Heisenberg groups and that of more general Carnot-Carathéodory spaces.