Regularized Distances and Geometry of Measures

Abstract: I will present joint work with Max Engelstein and Svitlana Mayboroda where we generalize the notion of the regularized distance function $$ D_{\mu,\alpha}(x)= \left(\int |x-y|^{d-\alpha}\, d\mu(y)\right)^{1/\alpha}, $$

to functions with more general integrands. We provide a large class of integrands for which the corresponding distance functions contain geometric information about $\mu$. In particular, we produce examples that are in some sense far from the original kernel $|x-y|^{d-\alpha}$ but still characterize the geometry of $\mu$ since they have nice symmetries with respect to flat sets. In co-dimension 1, these examples are explicit, but in higher co-dimensions, our proof of existence of such examples is non-constructive, and thus we have no additional information about their structure.

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.