Geodesic
Energy of measures on compact Riemannian manifolds
Kathryn E. Hare1 ; Maria Roginskaya2
1 Department of Pure Mathematics University of Waterloo Waterloo, Ontario, N2L 3G1, Canada
2 Department of Mathematics Chalmers TH & Göteborg University Eklandagatan 86 SE 412 96, Göteborg, Sweden
Studia Mathematica, Tome 159 (2003) no. 2, pp. 291-314 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/sm159-2-9
Keywords: investigate energy measures positive signed compact riemannian manifolds formula given relating energy integral positive measure projections measure eigenspaces laplacian formula analogous classical formula comparing energy measure euclidean space weighted norm its fourier transform boundedness modified energy integral signed measures gives bounds hausdorff dimension measure refined energy integrals hausdorff dimensions studied applied investigate singularity riesz product measures dimension

Kathryn E. Hare 1 ; Maria Roginskaya 2

1 Department of Pure Mathematics University of Waterloo Waterloo, Ontario, N2L 3G1, Canada
2 Department of Mathematics Chalmers TH & Göteborg University Eklandagatan 86 SE 412 96, Göteborg, Sweden 
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Kathryn E. Hare; Maria Roginskaya. Energy of measures 
 on compact Riemannian manifolds. Studia Mathematica, Tome 159 (2003) no. 2, pp. 291-314. doi: 10.4064/sm159-2-9

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