Ryan Alvarado - A characterization of the Sobolev embedding theorem in metric-measure spaces

Ryan Alvarado (Amherst College)

Historically, the Sobolev embedding theorem on domains has played a
key role in establishing many fundamental results in the area of
analysis and it is well known that the geometry of the underlying
domain is intimately linked to the availability of these embeddings.
In fact, certain geometrical characterizations of domains which
support Sobolev embeddings have been obtained in the Euclidean
setting. In this talk, we will revisit these embedding theorems in the
more general context of metric-measure spaces and discuss some recent
work which identifies a measure theoretic condition that is both
necessary and sufficient to ensure their veracity. A measure
characterization of Sobolev extension domains in the metric setting as
well as applications of our methods to spaces supporting p-Poincaré
inequalities will also be discussed. This talk is based on joint work
with Przemysław Górka (Warsaw University of Technology), Piotr Hajłasz
(University of Pittsburgh).