Scott Zimmerman

activity

20142024

most citedAn implicit function theorem for Lipschitz mappings into metric spaces

4 citations · 5 across the 11 of their papers we have counts for

collaborators

16 papers

math.MG2024

On the equivalence of derivatives for maps between Carnot groups

Scott Zimmerman

This paper gives an alternate, elementary proof of a result of Magnani: maps between Carnot groups that preserve horizontal curves and are continuously differential in horizontal d…

math.MG2023

Bi-Lipschitz arcs in metric spaces with controlled geometry

Jacob Honeycutt, Vyron Vellis, Scott Zimmerman

We generalize a bi-Lipschitz extension result of David and Semmes from Euclidean spaces to complete metric measure spaces with controlled geometry (Ahlfors regularity and supportin…

math.MG2022

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

Gareth Speight, Scott Zimmerman

We characterize which mappings from a compact subset of $R$ into the Heisenberg group can be extended to a $C^{m, ω}$ horizontal curve for a given modulus of continuity $ω$ …

math.MG2021

Identifying 1-rectifiable measures in Carnot groups

Matthew Badger, Sean Li, Scott Zimmerman

We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, ex…

math.MG2021

Whitney's Extension Theorem and the finiteness principle for curves in the Heisenberg group

Scott Zimmerman

Consider the sub-Riemannian Heisenberg group $H$ . In this paper, we answer the following question: given a compact set $K \subseteq R$ and a continuous map $f:K \…

math.CA2020

Singular integrals on $C_{w^{*}}^{1, α}$ regular curves in Banach duals

Scott Zimmerman

The modern study of singular integral operators on curves in the plane began in the 1970's. Since then, there has been a vast array of work done on the boundedness of singular inte…

math.CA2020

Singular integrals on $C^{1, α}$ regular curves in Carnot groups

Vasileios Chousionis, Sean Li, Scott Zimmerman

Let $G$ be any Carnot group. We prove that if a convolution type singular integral associated with a $1$ -dimensional Calderón-Zygmund kernel is $L^{2}$ -bounded on horizontal…

math.MG2018

Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces

Vasileios Chousionis, Sean Li, Vyron Vellis +1

The Heisenberg group $H$ equipped with a sub-Riemannian metric is one of the most well known examples of a doubling metric space which does not admit a bi-Lipschitz embedd…

math.GT2018★ 4 cited

An implicit function theorem for Lipschitz mappings into metric spaces

Piotr Hajłasz, Scott Zimmerman

We prove a version of the implicit function theorem for Lipschitz mappings $f : R^{n + m} \supset A \to X$ into arbitrary metric spaces. As long as the pull-back of the Hausdor…

math.MG2018

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

Andrea Pinamonti, Gareth Speight, Scott Zimmerman

We characterize those mappings from a compact subset of $R$ into the Heisenberg group $H^{n}$ which can be extended to a $C^{m}$ horizontal curve in $\mathbb{H}^{…

math.MG2018

The Traveling Salesman Theorem in Carnot Groups

Vasileios Chousionis, Sean Li, Scott Zimmerman

Let $G$ be any Carnot group. We prove that, if a subset of $G$ is contained in a rectifiable curve, then it satisfies Peter Jones' geometric lemma with some natur…

math.MG2017

Weak BLD mappings and Hausdorff measure

Piotr Hajłasz, Soheil Malekzadeh, Scott Zimmerman

We prove that if $Φ : X \to Y$ a mapping of weak bounded length distortion from a quasiconvex and complete metric space $X$ to any metric space $Y$ , then for any Lipschitz mapping $f:…

math.MG2016

Sobolev extensions of Lipschitz mappings into metric spaces

Scott Zimmerman

Wenger and Young proved that the pair $(R^{m}, H^{n})$ has the Lipschitz extension property for $m \leq n$ where $H^{n}$ is the sub-Riemannian Heisenberg group.…

math.MG2015

The Whitney Extension Theorem for $C^{1}$ , horizontal curves in the Heisenberg group

Scott Zimmerman

For a real valued function defined on a compact set $K \subset R^{m}$ , the classical Whitney Extension Theorem from 1934 gives necessary and sufficient conditions for the ex…

math.CA2015★ 1 cited

The Dubovitski\uı-Sard Theorem in Sobolev Spaces

Piotr Hajłasz, Scott Zimmerman

The Sard theorem from 1942 requires that a mapping $f : R^{n} \to R^{m}$ is of class $C^{k}$ , $k > max (n - m, 0)$ . In 1957 Duvovitski\uı generalized Sard's theorem to the…

math.MG2014

Geodesics in the Heisenberg group

Piotr Hajłasz, Scott Zimmerman

We provide a new and elementary proof for the structure of geodesics in the Heisenberg group $H^{n}$ . The proof is based on a new isoperimetric inequality for closed curves…

On the equivalence of derivatives for maps between Carnot groups

Bi-Lipschitz arcs in metric spaces with controlled geometry

A Cm,ω Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group

Identifying 1-rectifiable measures in Carnot groups

Whitney's Extension Theorem and the finiteness principle for curves in the Heisenberg group

Singular integrals on Cw∗1,α​ regular curves in Banach duals

Singular integrals on C1,α regular curves in Carnot groups

Bi-Lipschitz embeddings of Heisenberg submanifolds into Euclidean spaces

An implicit function theorem for Lipschitz mappings into metric spaces

A Cm Whitney Extension Theorem for Horizontal Curves in the Heisenberg Group

The Traveling Salesman Theorem in Carnot Groups

Weak BLD mappings and Hausdorff measure

Sobolev extensions of Lipschitz mappings into metric spaces

The Whitney Extension Theorem for C1, horizontal curves in the Heisenberg group

The Dubovitski\uı-Sard Theorem in Sobolev Spaces

Geodesics in the Heisenberg group

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

Singular integrals on $C_{w^{*}}^{1, α}$ regular curves in Banach duals

Singular integrals on $C^{1, α}$ regular curves in Carnot groups

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

The Whitney Extension Theorem for $C^{1}$ , horizontal curves in the Heisenberg group