A Fourier Coefficients Approach to Hausdorff Dimension in the Heisenberg Group

Abstract

This paper establishes connections between the group-Fourier transform and the geometry of measures in the Heisenberg group. Firstly, it is shown that if the Fourier transform of a compactly supported, finite, Radon measure is square integrable, then the measure must have a square integrable density. If it's Fourier transform is integrable, the the measure must have a continuous density. In addition, an alternative formulation of the Fourier transform on the Heisenberg group is used to show that energies of measures can be computed via integrals on an appropriate frequency space. This in turns opens the possibility of using Fourier methods in the computation of Hausdorff dimension of sets.