Energy of measures on compact Riemannian manifolds
Tom 159 / 2003
Studia Mathematica 159 (2003), 291-314
MSC: Primary 58C35, 28A12; Secondary 42A55, 28A78.
DOI: 10.4064/sm159-2-9
Streszczenie
We investigate the energy of measures (both positive and signed) on compact Riemannian manifolds. A formula is given relating the energy integral of a positive measure with the projections of the measure onto the eigenspaces of the Laplacian. This formula is analogous to the classical formula comparing the energy of a measure in Euclidean space with a weighted $L^{2}$ norm of its Fourier transform. We show that the boundedness of a modified energy integral for signed measures gives bounds on the Hausdorff dimension of the measure. Refined energy integrals and Hausdorff dimensions are also studied and applied to investigate the singularity of Riesz product measures of dimension one.
