Energy of measures on compact Riemannian manifolds

Kathryn E. Hare, and Maria Roginskaya. "Energy of measures on compact Riemannian manifolds." Studia Mathematica 159.2 (2003): 291-314. <http://eudml.org/doc/285238>.

@article{KathrynE2003,
abstract = {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 L² 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.},
author = {Kathryn E. Hare, Maria Roginskaya},
journal = {Studia Mathematica},
keywords = {energy; Hausdorff dimension; signed measure; Riesz product},
language = {eng},
number = {2},
pages = {291-314},
title = {Energy of measures on compact Riemannian manifolds},
url = {http://eudml.org/doc/285238},
volume = {159},
year = {2003},
}

TY - JOUR
AU - Kathryn E. Hare
AU - Maria Roginskaya
TI - Energy of measures on compact Riemannian manifolds
JO - Studia Mathematica
PY - 2003
VL - 159
IS - 2
SP - 291
EP - 314
AB - 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 L² 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.
LA - eng
KW - energy; Hausdorff dimension; signed measure; Riesz product
UR - http://eudml.org/doc/285238
ER -