Compact embeddings of Hajłasz-Sobolev spaces

Artur Słabuszewski

supervisor: Przemysław Górka

In the 90s of the last century Hajłasz has introduced the so called Hajłasz-Sobolev spaces as a generalisation of the Sobolev spaces to the setting of arbitrary metric-measure space. Since then, the theory is still being developed. One of the fundamental results in the theory of Sobolev spaces is the Rellich-Kondrachov theorem, which states that for bounded domains with sufficiently regular boundary, any family of the functions with bounded Sobolev norm is pre-compact in L^p. The question which naturally arises is if there is a counterpart of this result in the theory of Hajłasz-Sobolev spaces and what do we need to assume on the underlying measure and metric. There are few papers in which L^p compactness of bounded Hajłasz-Sobolev functions was obtained with various assumption involving metric and measure (e.g. geometrically doubling condition).

However, in my recent article with Przemysław Górka we proved very general version of Rellich-Kondrachov theorem. It turns out that it is sufficient to assume only total boundedness of the metric space. On the other hand, I have found the example of a metric-measure space which is not totally bounded but the compact embedding still holds. In other words, a total boundedness is not a necessary condition.