From lower measure bound to Sobolev embedding and back again

Thursday, May 16, 2019 - 15:00
Speaker Information
Ryan Alvarado
Prof. Dr.
Amherst College

Abstract or Additional Information

Historically, the Sobolev embedding theorem has played a key role in establishing many basic results in the area of analysis. Typically, sufficient conditions on the underlying measure have been imposed in order to guarantee the availability of the aforementioned theorem. In this talk, we will revisit this result in the context of metric measure spaces, and discuss some recent work which identifies a set of conditions on the measure that are both necessary and sufficient to ensure its veracity. A measure characterization of Sobolev extension domains as well as applications of our methods to  spaces support $p$-Poincar\'e inequalities will also be discussed. This talk is based on joint work with Przemys\l{}aw G\'orka (Warsaw University of Technology), Piotr Haj\l{}asz (University of Pittsburgh).

Research Area