Document Type
Presentation Abstract
Presentation Date
10-23-2017
Abstract
Singular integrals comprise a rich area of analysis, the most well known example being the Hilbert Transform. In this talk, we will discuss a singular integral that also intersects geometric measure theory. For functions f:\mathbb{R}^n\rightarrow\mathbb{R} that are C^{1,1} (i.e. the first derivative is Lipschitz continuous), for which 0 is a regular value (i.e. the gradient \nabla f does not vanish on the 0-level set), and whose 0-level set is bounded, there is a not too hard proof that our singular integral computes \mathcal{H}^{n-1}(\{f^{-1}(0)\}), the (n-1)-dimensional Hausdorff measure of the 0-level set of f. We will also briefly mention the simple analysis problem that inspired this research.
Download the attached PDF to see the abstract with proper math formatting.
Recommended Citation
Paxton, Laramie, "Measuring Level Sets of C^{1,1} Functions" (2017). Colloquia of the Department of Mathematical Sciences. 531.
https://scholarworks.umt.edu/mathcolloquia/531
Additional Details
Monday, October 23, 2017 at 3:00 p.m. in Math 103
Refreshments at 4:00 p.m. in Math Lounge 109