Mathematical Sciences Department Analysis/PDE Seminar: Ryan Alvarado, Amherst College

Title: Optimal conditions for embeddings and extensions of Sobolev, Besov, and Triebel-Lizorkin spaces in quasi-metric measure spaces

Abstract: Embedding and extension properties of Sobolev functions defined on a given subset of $\mathbb{R}^n$ have been instrumental in establishing fundamental results in analysis. Since the 1990s, significant efforts have been made to generalize the classical Euclidean theory of Sobolev spaces to broader contexts, such as (quasi-)metric measure spaces, where embedding and extension theorems remain central. In this talk, we will discuss recent work that identifies necessary and sufficient conditions under which embedding and extension results hold for certain classes of Sobolev, Besov, and Triebel--Lizorkin spaces in the general framework of quasi-metric measure spaces. An interesting facet of this work is how the geometrical characteristics of the quasi-metric space intervene in the embedding and extension properties of these function spaces.