New Developments in Parabolic Uniform Rectifiability

Abstract: In the 1980’s the $L^2$ boundedness of the Cauchy integral was established (by Coifman, McIntosh and Meyer) and this $L^2$ boundedness was quickly generalized to `nice’ singular integral operators (Coifman, David and Meyer / David) on Lipschitz graphs. David and Semmes then asked the natural question: For what sets are all `nice’ singular integral operators $L^2$ bounded? They were remarkably successful in this endeavor, providing more than 15 equivalent notions and called these sets “uniformly rectifiable” or UR. These sets are still studied extensively today, most recently in their connection to elliptic partial differential equations.

Among the characterizations of UR sets provided by David and Semmes is a quadratic estimate on the so-called $\beta$-numbers, which measure the flatness of the set at a particular location and scale. In a paper of Hofmann, Lewis and Nyström a notion of “parabolic uniform rectictifiable sets” was introduced by taking the definition as a “quadratic estimate on the parabolic $\beta$-numbers”. There are no correct proofs of ANY of the analogues of the David Semmes theory; for instance, it is not known if $L^2$ boundedness of parabolic singular integral operators characterizes parabolic uniformly rectifiable sets.

In this talk I will discuss some recent progress in the direction of establishing the parabolic David-Semmes theory and some open problems that remain. On one hand, we have made significant progress and provided some useful characterizations of parabolic uniform rectifiability. On the other hand, we have also discovered that many of the `elliptic’ characterizations do not hold in this parabolic setting. This is joint work with J. Hoffman, S. Hofmann, J.L. Luna and K. Nyström.

analysis of PDEsclassical analysis and ODEsfunctional analysismetric geometry

Audience: researchers in the topic

HA-GMT-PDE Seminar

Series comments: Description: A senior graduate student/postdoc series in harmonic analysis, geometric measure theory, and partial differential equations.

Meetings will be weekly, and usually will occur on Monday, with some exceptions. For each talk, the Zoom link is made available in the website too. Contact Bruno Poggi at poggi008@umn.edu if you would like to subscribe to the seminar mailing list.