Artur Słabuszewski
supervisor: Przemysław Górka
Let (X,d) be a metric space equipped with a Borel regular measure μ. It has been shown by R. Alvarado, P. Górka and P.Hajłasz that Sobolev embeddings for Hajłasz-Sobolev space M^{1,p} (X,d,μ) are equivalent with lower Ahlfors regularity of the measure. Similar results (obtained by Y. Zhou) are also known for Sobolev spaces of fractional order (also known as Słobodeckij space) defined on open subsets of Euclidean space. During the talk I will present generalisation of Zhou results obtained with Przemysław Górka. It turns out that if we consider fractional Sobolev space defined on the metric measure space, then with some additional assumptions on (X,d,μ) Sobolev embeddings holds if and only if μ is lower Ahlfors regular.