The Role of Gromov Hyperbolicity and Besov Spaces in Quasiconformal Analysis

This research project concerns the development of mathematical tools for analysis on metric measure spaces (spaces with well-defined notions of distance and volume), inspired by basic tools used in ordinary Euclidean space. The work has implications for many fields of mathematics, including dynamical systems, harmonic analysis, and partial differential equations.

One useful measurement in geometric analysis is the so-called Besov energy. This project will explore links between the nonlocal Besov-type energy in a compact metric measure space and local energy encoded in the space, as measured through a well-connected super-space that sees the compact space as its boundary. The work aims to use these links to explore how non-local energies are transformed by well-regulated changes in the metric, or distance.

Parts of this project will be done in collaboration with graduate students and other early career mathematicians, thus also providing training of the future STEM workforce.

In this project, the principal investigator will develop the potential theoretic tools related to Besov energies by using Gromov hyperbolic fillings of the compact metric measure space so that the compact space is realized as the boundary of the Gromov hyperbolic space. Such a Gromov hyperbolic space is of controlled geometry, and by exploiting this control, the principal investigator will seek to gain control of the nonlocal energies and to obtain information about how these energies are transformed by quasisymmetric maps.

The project will also augment the potential theoretic properties of Besov energy by developing the perspective of fractional p-Laplacian (nonlinear versions of the fractional Laplacian).

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.