Clemson Analysis and PDE Seminar

Abstract: In this talk, we revisit classical fractional Sobolev embedding theorems and the algebra property of the fractional Sobolev space $H^s(\mathbb{R})$ from a dyadic point of view. Inspired by previous work due to Aimar et al. we use Haar functions and dyadic decompositions to obtain the dyadic analogues of these classic results. Especially, in low regularities we provide explicit examples for the failure of the algebra property without using the Fourier transform.

Abstract: There are different notions of dimension, each suited for different purposes. The Assouad dimension is such a metric notion that quantifies the homogeneity of a metric space, similarly to how a doubling measure does so in spaces of homogeneous type. While it was originally introduced to address the embeddability properties of metric spaces, it has recently gained more popularity within the analysis community due to its connection to many other areas. In this talk, we discuss applications of the Assouad dimension and the Assouad spectrum (a recently introduced dimension "function") in the context of quasiconformal classification and rigidity.

Abstract: TBA