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$L^{p}$-improving properties of measures of positive energy dimension

100%

Hare K., Roginskaya M.

Colloquium Mathematicum

2005

tom 102

nr 1

73-86

A measure is called $L^{p}$-improving if it acts by convolution as a bounded operator from $L^{p}$ to $L^{q}$ for some q > p. Positive measures which are $L^{p}$-improving are known to have positive Hausdorff dimension. We extend this result to complex $L^{p}$-improving measures and show that even their energy dimension is positive. Measures of positive energy dimension are seen to be the Lipschitz measures and are characterized in terms of their improving behaviour on a subset of $L^{p}$-functions.

Energy of measures on compact Riemannian manifolds

100%

Hare K., Roginskaya M.

Studia Mathematica

2003

tom 159

nr 2

291-314

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

2 Hare K.

2 Roginskaya M.

1 2005

1 2003

Wyniki wyszukiwania

$L^{p}$-improving properties of measures of positive energy dimension

Energy of measures on compact Riemannian manifolds