Warped products, solid hyperbolic fillings, and the identity $D^{1,p} = N^{1,p} + \mathbb{R}$
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| Format: | Preprint |
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2025
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| _version_ | 1866909719455596544 |
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| author | Kangasniemi, Ilmari Kline, Josh Shanmugalingam, Nageswari Speight, Gareth |
| author_facet | Kangasniemi, Ilmari Kline, Josh Shanmugalingam, Nageswari Speight, Gareth |
| contents | 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 $u \colon Z \to \mathbb{R}$ 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 $\mathbb{H}_{α, β}(Y)$, $α, β\in (0, \infty)$, 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 $Z=\mathbb{H}_{α,β}(Y)$ when $1\le p\le β/α$. We also show that under mild assumptions on $Y$, the warped product $\mathbb{H}_{α, β}(Y)$ is Gromov hyperbolic as a metric space and the Gromov boundary of $\mathbb{H}_{α, β}(Y)$ is quasisymmetric to $Y$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_01857 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Warped products, solid hyperbolic fillings, and the identity $D^{1,p} = N^{1,p} + \mathbb{R}$ Kangasniemi, Ilmari Kline, Josh Shanmugalingam, Nageswari Speight, Gareth Metric Geometry Analysis of PDEs 46E36, 30L15, 53C23 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 $u \colon Z \to \mathbb{R}$ 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 $\mathbb{H}_{α, β}(Y)$, $α, β\in (0, \infty)$, 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 $Z=\mathbb{H}_{α,β}(Y)$ when $1\le p\le β/α$. We also show that under mild assumptions on $Y$, the warped product $\mathbb{H}_{α, β}(Y)$ is Gromov hyperbolic as a metric space and the Gromov boundary of $\mathbb{H}_{α, β}(Y)$ is quasisymmetric to $Y$. |
| title | Warped products, solid hyperbolic fillings, and the identity $D^{1,p} = N^{1,p} + \mathbb{R}$ |
| topic | Metric Geometry Analysis of PDEs 46E36, 30L15, 53C23 |
| url | https://arxiv.org/abs/2508.01857 |