Warped products, solid hyperbolic fillings, and the identity $D^{1,p}=N^{1,p}+\mathbb{R}$

Abstract

We construct a large class of metric measure spaces $Z$ which satisfy the identity $D^{1,p}(Z)=N^{1,p}(Z)+\mathbb{R}$, i.e.\ any measurable function $u:Z\to \mathbb{R}$ with an $L^p$-integrable upper gradient is a constant term away from being $L^p$-integrable. To do so, we construct a family of hyperbolic fillings ${\mathbb{H}}_{\alpha ,\beta }(Y)$, $\alpha ,\beta \in (0,\infty )$, of a metric measure space $Y$, via a warped product of $Y$ with an exponentially weighted positive real line. We then show that for certain classes of $Y$, the above identity is satisfied for $Z={\mathbb{H}}_{\alpha ,\beta }(Y)$ when $1\le p\le \beta /\alpha$. We also show that under mild assumptions on $Y$, the warped product ${\mathbb{H}}_{\alpha ,\beta }(Y)$ is Gromov hyperbolic as a metric space and the Gromov boundary of ${\mathbb{H}}_{\alpha ,\beta }(Y)$ is quasisymmetric to $Y$.

47 pages, 1 figure