Nonlocal problems and fractional function spaces on metric spaces via hyperbolic fillings

The project deals with first-order analysis (including gradients, Sobolev type spaces and partial differential equations) on metric spaces, but will have applications also for Euclidean domains and fractals.An aim is to transform fractional function spaces and nonlocal minimization problems (such as the fractional p-Laplacian) on general metric spaces into local problems on so-called hyperbolic fillings.

These are infinite graphs constructed so that they have a given metric space as the boundary at infinity.

In particular, this makes it possible to see Besov type spaces on general metric spaces as traces of Sobolev type spaces, with consequences also for the related minimization problems and differential operators.

In this way we will take advantage of the already developed theories for Sobolev spaces and p-harmonic functions on well-behaved metric spaces to obtain new results also for fractional spaces and problems in less regular situations.

We will also study how such function spaces and minimization problems transform under the procedures of uniformization and sphericalization, which efficiently map unbounded spaces to bounded ones, so that results for one class of spaces can be transfered to other spaces and vice versa.