Energy of measures on compact Riemannian manifolds

Volume 159 / 2003

Kathryn E. Hare, Maria Roginskaya Studia Mathematica 159 (2003), 291-314 MSC: Primary 58C35, 28A12; Secondary 42A55, 28A78. DOI: 10.4064/sm159-2-9


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.


  • Kathryn E. HareDepartment of Pure Mathematics
    University of Waterloo
    Waterloo, Ontario, N2L 3G1, Canada
  • Maria RoginskayaDepartment of Mathematics
    Chalmers TH & Göteborg University
    Eklandagatan 86
    SE 412 96, Göteborg, Sweden

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