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Gareth Speight

Publications and source records attributed to Gareth Speight.

At least 19 recordsLinked to original sources
arXiv2026

Warped product spaces: Gromov hyperbolicity and identification of the visual boundary

In this paper, we consider warped product spaces $X\times_φY$, where $X$ is a complete geodesic Gromov hyperbolic space, $Y$ is a compact geodesic metric space, and the warping function $φ$ satisfies suitable exponential growth conditions. We prove that the warped product is Gromov hyperbolic and derive an explicit estimate for its hyperbolicity constant. We further establish a homeomorphism between its Gromov boundary and $\partial_GX\times Y$, with an explicit comparison formula for the visual metric on the Gromov boundary of $X\times_φY$ in terms of the visual metric $d_{\varepsilon, X}$ on $\partial_GX$ and $d_Y$ for suitable $\varepsilon>0$.

math.MG↗
arXiv2026

A Lipschitz curve in a Carnot group that is purely unrectifiable by smooth horizontal curves

We construct a Lipschitz curve in the free Carnot group of step 3 with 2 generators that meets every $C^{1}$ horizontal curve in a set of measure zero. This shows that the $C^{1}_{H}$-Lusin property fails in a strong sense in this group, and we deduce that such a curve must be purely $C^1_H$ 1-unrectifiable. Hence 1-rectifiability in Carnot groups is wildly different to its counterpart in Euclidean spaces, wherein the Whitney Extension Theorem guarantees that Lipschitz rectifiability and $C^1$ rectifiability are equivalent.

math.MG↗
arXiv2025

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

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 $Y$.

math.MG↗
arXiv2025

Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups

We show that, in arbitrary Carnot groups, pliability in a subset of directions is sufficient to guarantee the existence of a Whitney-type extension and a Lusin approximation for curves with tangent vectors in the same set of directions. We apply this to show that every horizontal curve in the Engel group must intersect a $C^{1}$ horizontal curve in a set of positive measure.

math.DG↗
arXiv2025

Universal Differentiability Sets in Laakso Space

We show that there exists a family of mutually singular doubling measures on Laakso space with respect to which real-valued Lipschitz functions are almost everywhere differentiable. This implies that there exists a measure zero universal differentiability set in Laakso space. Additionally, we show that each of the measures constructed supports a Poincaré inequality.

math.FA↗
arXiv2023

Higher order Whitney extension and Lusin approximation for Horizontal curves in the Heisenberg group

In the setting of horizontal curves in the Heisenberg group, we prove a $C^{m,ω}$ finiteness principle, a $C^{m,ω}$ Lusin approximation result, a $C^{\infty}$ Whitney extension result, and a $C^{\infty}$ Lusin approximation result. Combined with previous work, this completes the study of Whitney extension and Lusin approximation for horizontal curves of class $C^{m}$, $C^{m,ω}$, and $C^{\infty}$ in the Heisenberg group.

math.MG↗
arXiv2023

Weighted Sobolev Spaces on Metric Measure Spaces

We investigate weighted Sobolev spaces on metric measure spaces $(X,d,m)$. Denoting by $ρ$ the weight function, we compare the space $W^{1,p}(X,d,ρm)$ (which always concides with the closure $H^{1,p}(X,d,ρm)$ of Lipschitz functions) with the weighted Sobolev spaces $W^{1,p}_ρ(X,d,m)$ and $H^{1,p}_ρ(X,d,m)$ defined as in the Euclidean theory of weighted Sobolev spaces. Under mild assumptions on the metric measure structure and on the weight we show that $W^{1,p}(X,d,ρm)=H^{1,p}_ρ(X,d, m)$. We also adapt results by Muckenhoupt and recent work by Zhikov to the metric measure setting, considering appropriate conditions on $ρ$ that ensure the equality $W^{1,p}_ρ(X,d,m)=H^{1,p}_ρ(X,d,m)$.

math.AP↗
arXiv2022

Maximal Directional Derivatives in Laakso Space

We investigate the connection between maximal directional derivatives and differentiability for Lipschitz functions defined on Laakso space. We show that maximality of a directional derivative for a Lipschitz function implies differentiability only for a $σ$-porous set of points. On the other hand, the distance to a fixed point is differentiable everywhere except for a $σ$-porous set of points. This behavior is completely different to the previously studied settings of Euclidean spaces and Carnot groups.

math.FA↗
arXiv2022

A C^k Lusin Approximation Theorem For Real-Valued Functions on Carnot Groups

We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by $C^{k}_{\mathbb{G}}$ maps. We also prove that existence of an approximate (k-1)-Taylor polynomial almost everywhere is equivalent to admitting a Lusin approximation by maps in a suitable Lipschitz function space.

math.FA↗
arXiv2022

A $C^{m,ω}$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group

We characterize which mappings from a compact subset of $\mathbb{R}$ into the Heisenberg group can be extended to a $C^{m,ω}$ horizontal curve for a given modulus of continuity $ω$. We motivate our characterization by showing that the $C^{m,ω}$ extension property fails if we instead use a more direct analogue of the conditions from the $C^{m}$ case.

math.MG↗
arXiv2021

A $C^m$ Lusin Approximation Theorem for Horizontal Curves in the Heisenberg Group

We prove a $C^m$ Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is $m-1$ times $L^1$ differentiable almost everywhere coincides with a $C^m$ horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if $L^1$ differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.

math.MG↗
arXiv2020

Regularity of Solutions to the Fractional Cheeger-Laplacian on Domains in Metric Spaces of Bounded Geometry

We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,μ_X)$ satisfying a $2$-Poincaré inequality. Given a bounded domain $Ω\subset X$ with $μ_X(X\setminusΩ)>0$, and a function $f$ in the Besov class $B^θ_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^θ_{2,2}(X)$ such that $u=f$ in $X\setminusΩ$ and $\mathcal{E}_θ(u,u)\le \mathcal{E}_θ(h,h)$ whenever $h\in B^θ_{2,2}(X)$ with $h=f$ in $X\setminusΩ$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Hölder continuous on $Ω$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.

math.AP↗
arXiv2020

Universal differentiability sets and maximal directional derivatives in Carnot groups

We show that every Carnot group G of step 2 admits a Hausdorff dimension one `universal differentiability set' N such that every real-valued Lipschitz map on G is Pansu differentiable at some point of N. This relies on the fact that existence of a maximal directional derivative of f at a point x implies Pansu differentiability at the same point x. We show that such an implication holds in Carnot groups of step 2 but fails in the Engel group which has step 3.

math.FA↗
arXiv2019

Universal Differentiability Sets in Carnot Groups of Arbitrarily High Step

We show that every model filiform group $\mathbb{E}_{n}$ contains a measure zero set $N$ such that every Lipschitz map $f\colon \mathbb{E}_{n}\to \mathbb{R}$ is differentiable at some point of $N$. Model filiform groups are a class of Carnot groups which can have arbitrarily high step. Essential to our work is the question of whether existence of an (almost) maximal directional derivative $Ef(x)$ in a Carnot group implies differentiability of a Lipschitz map $f$ at $x$. We show that such an implication is valid in model Filiform groups except for a one-dimensional subspace of horizontal directions. Conversely, we show that this implication fails for every horizontal direction in the free Carnot group of step three and rank two.

math.FA↗
arXiv2018

Porosity and Differentiability of Lipschitz Maps from Stratified Groups to Banach Homogeneous Groups

Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $σ$-porous set. At density points of $A$ we establish a pointwise characterization of differentiability in terms of directional derivatives. We use these new results to obtain an alternate proof of almost everywhere differentiability of Lipschitz maps from subsets of stratified groups to Banach homogeneous groups satisfying a suitably weakened Radon-Nikodym property. As a consequence we also get an alternative proof of Pansu's Theorem.

math.FA↗
arXiv2018

A $C^m$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group

We characterize those mappings from a compact subset of $\mathbb{R}$ into the Heisenberg group $\mathbb{H}^{n}$ which can be extended to a $C^{m}$ horizontal curve in $\mathbb{H}^{n}$. The characterization combines the classical Whitney conditions with an estimate comparing changes in the vertical coordinate with those predicted by the Taylor series of the horizontal coordinates.

math.MG↗
arXiv2017

Structure of Porous Sets in Carnot Groups

We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $σ$-porous with respect to the Carnot-Carathéodory (CC) distance. In the first Heisenberg group we observe that there exist sets which are porous with respect to the CC distance but not the Euclidean distance and vice-versa. In Carnot groups we then construct a Lipschitz function which is Pansu differentiable at no point of a given $σ$-porous set and show preimages of open sets under the horizontal gradient are far from being porous.

math.MG↗