Gareth Speight
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View article: Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups
Directional Pliability, Whitney Extension, and Lusin Approximation for Curves in Carnot Groups Open
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…
View article: Universal Differentiability Sets in Laakso Space
Universal Differentiability Sets in Laakso Space Open
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 d…
View article: Higher order Whitney extension and Lusin approximation for Horizontal curves in the Heisenberg group
Higher order Whitney extension and Lusin approximation for Horizontal curves in the Heisenberg group Open
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. Combine…
View article: Maximal Directional Derivatives in Laakso Space
Maximal Directional Derivatives in Laakso Space Open
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 differentiabi…
View article: A $C^{m,ω}$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group
A $C^{m,ω}$ Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group Open
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^{…
View article: Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry
Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry Open
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 bo…
View article: A C^k Lusin Approximation Theorem For Real-Valued Functions on Carnot Groups
A C^k Lusin Approximation Theorem For Real-Valued Functions on Carnot Groups Open
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. …
View article: Function Spaces via Fractional Poisson Kernel on Carnot Groups and Applications
Function Spaces via Fractional Poisson Kernel on Carnot Groups and Applications Open
We provide a new characterization of homogeneous Besov and Sobolev spaces in Carnot groups using the fractional heat kernel and Poisson kernel. We apply our results to study commutators involving fractional powers of the sub-Laplacian.
View article: Universal differentiability sets in Carnot groups of arbitrarily high step
Universal differentiability sets in Carnot groups of arbitrarily high step Open
We show that any Carnot group G with sufficiently many deformable directions contains a measure zero set N such that every Lipschitz map f: G→ R is differentiable at some point of N. We also prove that model filiform groups satisfy this co…
View article: A $C^m$ Whitney extension theorem for horizontal curves in the Heisenberg group
A $C^m$ Whitney extension theorem for horizontal curves in the Heisenberg group Open
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 co…
View article: Universal differentiability sets and maximal directional derivatives in Carnot groups
Universal differentiability sets and maximal directional derivatives in Carnot groups Open
We show that every Carnot group G of step 2 admits a Hausdorff dimension one 'universal differentiability set' N such that every Lipschitz map f:G→R is Pansu differentiable at some point of N. This relies on the fact that existence of a ma…
View article: Domains in metric measure spaces with boundary of positive mean\n curvature, and the Dirichlet problem for functions of least gradient
Domains in metric measure spaces with boundary of positive mean\n curvature, and the Dirichlet problem for functions of least gradient Open
We study the geometry of domains in complete metric measure spaces equipped\nwith a doubling measure supporting a $1$-Poincar\\'e inequality. We propose a\nnotion of \\emph{domain with boundary of positive mean curvature} and prove\nthat, …
View article: Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient
Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient Open
We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincaré inequality. We propose a notion of \emph{domain with boundary of positive mean curvature} and prove that, for suc…
View article: Universal differentiability sets and maximal directional derivatives in Carnot groups
Universal differentiability sets and maximal directional derivatives in Carnot groups Open
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 existe…
View article: A Measure Zero UDS in the Heisenberg Group
A Measure Zero UDS in the Heisenberg Group Open
We show that the Heisenberg group contains a measure zero set N such that every real-valued Lipschitz function is Pansu differentiable at a point of N.
View article: Structure of porous sets in Carnot groups
Structure of porous sets in Carnot groups Open
We show that any Carnot group contains a closed nowhere dense set which has measure zero but is not $\\sigma $-porous with respect to the Carnot–Carathéodory (CC) distance. In the first Heisenberg group, we observe that there exist sets wh…
View article: Lusin approximation and horizontal curves in Carnot groups
Lusin approximation and horizontal curves in Carnot groups Open
We show that, given an absolutely continuous horizontal curve \gamma in the Heisenberg group, there is a C^{1} horizontal curve \Gamma such that \Gamma=\gamma and \Gamma'=\gamma' outside a set of small measure. Conversely, we construct an …
View article: THE p-WEAK GRADIENT DEPENDS ON p
THE p-WEAK GRADIENT DEPENDS ON p Open
Given α > 0, we construct a weighted Lebesgue measure on \mathbb{R}^n for which the family of nonconstant curves has p-modulus zero for p ≤ 1 + α but the weight is a Muckenhoupt A_p weight for p > 1 + α. In particular, the p-weak gradient …
View article: Structure of Porous Sets in Carnot Groups
Structure of Porous Sets in Carnot Groups Open
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 …
View article: Weighted Sobolev spaces on metric measure spaces
Weighted Sobolev spaces on metric measure spaces Open
We investigate weighted Sobolev spaces on metric measure spaces ( X , d , 𝔪 ) {(X,\mathrm{d},\mathfrak{m})} . Denoting by ρ the weight function, we compare the space W 1 , p ( X , d , ρ 𝔪 ) {W^{1,p}(X,\mathrm{d},\rho\ma…
View article: Lusin approximation for horizontal curves in step 2 Carnot groups
Lusin approximation for horizontal curves in step 2 Carnot groups Open
A Carnot group $\mathbb{G}$ admits Lusin approximation for horizontal curves if for any absolutely continuous horizontal curve $γ$ in $\mathbb{G}$ and $\varepsilon>0$, there is a $C^1$ horizontal curve $Γ$ such that $Γ=γ$ and $Γ'=γ'$ outsi…
View article: The $p$-weak gradient depends on $p$
The $p$-weak gradient depends on $p$ Open
Given a>0, we construct a weighted Lebesgue measure on R^n for which the family of non constant curves has p-modulus zero for p\leq 1+a but the weight is a Muckenhoupt A_p weight for p>1+a. In particular, the p-weak gradient is trivial for…