Whitney’s extension theorem for curves in the Heisenberg group

Image
Analysis and Operator Theory Seminar
February 2, 2023
11:30AM - 12:30PM
Location
MW 154

Date Range
Add to Calendar 2023-02-02 11:30:00 2023-02-02 12:30:00 Whitney’s extension theorem for curves in the Heisenberg group Title:  Whitney’s extension theorem for curves in the Heisenberg group Speaker:  Scott Zimmerman (The Ohio State University) Speaker's URL:  https://www.zimmermanmath.com/ Abstract:  Given a compact set $K$ in $\mathbb{R}^n$ and a continuous, real valued function $f$ on $K$, when is there a $C^{m,\omega}$ function $F$ defined on $\mathbb{R}^n$ such that $F|_K = f$? ($C^{m,\omega}$ is the space of $C^m$ functions whose $m$th order derivatives are uniformly continuous with modulus of continuity $\omega$.) Whitney famously answered this question in 1934 when extra data is provided on the derivatives of the extension on $K$, and less famously answered the original question for subsets of $\mathbb{R}$. The question was answered in full by Charles Fefferman in 2009. The Heisenberg group $H$ is $\mathbb{R}^3$ with a sub-Riemannian and metric structure generated by a class of admissible curves. We will consider Whitney's original question for curves in $H$: given a compact set $K$ in $\mathbb{R}$ and a continuous, map $f:K \to H$, when is there a $C^{m,\omega}$ admissible curve $F$ such that $F|_K = f$? I will present a project with Andrea Pinamonti and Gareth Speight in which we address this question. URL associated with Seminar:  https://u.osu.edu/aots/ MW 154 Department of Mathematics math@osu.edu America/New_York public
Description

Title:  Whitney’s extension theorem for curves in the Heisenberg group

Speaker:  Scott Zimmerman (The Ohio State University)

Speaker's URL:  https://www.zimmermanmath.com/

Abstract:  Given a compact set $K$ in $\mathbb{R}^n$ and a continuous, real valued function $f$ on $K$, when is there a $C^{m,\omega}$ function $F$ defined on $\mathbb{R}^n$ such that $F|_K = f$? ($C^{m,\omega}$ is the space of $C^m$ functions whose $m$th order derivatives are uniformly continuous with modulus of continuity $\omega$.) Whitney famously answered this question in 1934 when extra data is provided on the derivatives of the extension on $K$, and less famously answered the original question for subsets of $\mathbb{R}$. The question was answered in full by Charles Fefferman in 2009.

The Heisenberg group $H$ is $\mathbb{R}^3$ with a sub-Riemannian and metric structure generated by a class of admissible curves. We will consider Whitney's original question for curves in $H$: given a compact set $K$ in $\mathbb{R}$ and a continuous, map $f:K \to H$, when is there a $C^{m,\omega}$ admissible curve $F$ such that $F|_K = f$? I will present a project with Andrea Pinamonti and Gareth Speight in which we address this question.

URL associated with Seminar:  https://u.osu.edu/aots/

Events Filters: