Geometric analysis on Gromov hyperbolic spaces, their boundaries and other metric measure spaces, with applications to p-harmonic functions

This project focuses on several distinct but related problems in first-order analysis on nonsmooth metric measure spaces which need not have a differentiable structure. The main objectives are:I.

Transformation of measures between a Gromov hyperbolic space X and its visual boundaryIncluding measures in the study of these objects is nontrivial and a major novelty of our approach. Another aim is to get sharp trace results for function spaces on X and its boundary. Given a compact metric measure space Z, one can construct a Gromov hyperbolic space X with Z as its boundary.

We will optimize this construction and use it to study:II.

Nonlocal minimization problems and properties of functions in Besov spaces on metric spacesNonlocal problems are known to be hard, but seeing Besov spaces as traces of Sobolev spaces provides new tools. We expect results such as existence, uniqueness and regularity of the minimizers in general situations.III.

Preservation of Poincaré inequalities and p-harmonic functions under sphericalizationThis will bring new tools (e.g. Poincaré inequalities) and results to unbounded spaces, by sphericalizing them into bounded ones. Such results for p-harmonic functions can be useful e.g. for:IV.

Classifying spaces using p-harmonic functions and quasiminimizersThe project deals with analysis on metric spaces but is relevant also for R^n, manifolds, Heisenberg groups etc. Many Euclidean domains and fractal sets are included as special cases.