Comparisons of relative BV-capacities and Sobolev capacity in metric spaces

We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak $(1,1)$-Poincar\'e inequality. We prove the equality of 1-modulus and 1-capacity, extending the known results for $1 < p < \infty$ to also cover the more geometric case $p = 1$. Then we give alternative definitions for variational BV-capacities and obtain equivalence results between them. Finally we study relations between total 1-capacity and versions of BV-capacity.Comment: 30 page

On Carrasco Piaggio's theorem characterizing quasisymmetric maps from compact doubling spaces to Ahlfors regular spaces

In this note we deconstruct and explore the components of a theorem of Carrasco Piaggio, which relates Ahlfors regular conformal gauge of a compact doubling metric space to weights on Gromov-hyperbolic fillings of the metric space. We consider a construction of hyperbolic filling that is simpler than the one considered by Carrasco Piaggio, and we determine the effect of each of the four properties postulated by Carrasco Piaggio on the induced metric on the compact metric space.Comment: 23 page

A Rigidity Property of Some Negatively Curved Solvable Lie Groups

We show that for some negatively curved solvable Lie groups, all self quasiisometries are almost isometries. We prove this by showing that all self quasisymmetric maps of the ideal boundary (of the solvable Lie groups) are bilipschitz with respect to the visual metric. We also define parabolic visual metrics on the ideal boundary of Gromov hyperbolic spaces and relate them to visual metrics

Homogeneous Newton-Sobolev spaces in metric measure spaces and their Banach space properties

In this note we prove the Banach space properties of the homogeneous Newton-Sobolev spaces $HN^{1,p}(X)$ of functions on an unbounded metric measure space $X$ equipped with a doubling measure supporting a $p$-Poincar\'e inequality, and show that when $1<p<\infty$, even with the lack of global $L^p$-integrability of functions in $HN^{1,p}(X)$, we have that every bounded sequence in $HN^{1,p}(X)$ has a strongly convergent convex-combination subsequence. The analogous properties for the inhomogeneous Newton-Sobolev classes $N^{1,p}(X)$ are proven elsewhere in existing literatureComment: 10 page