Geodesic
$L^p$-improving properties of measures of positive energy dimension
Kathryn E. Hare 1   ; Maria Roginskaya 2
1 Department of Pure Mathematics University of Waterloo Waterloo, ON, N2L 3G1 Canada
2 Department of Mathematics Chalmers TH and Göteborg University Eklandagatan 86 Göteborg, SE 412 96 Sweden
Colloquium Mathematicum, Tome 102 (2005) no. 1, pp. 73-86 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/cm102-1-7
Keywords: measure called improving acts convolution bounded operator positive measures which improving known have positive hausdorff dimension extend result complex improving measures even their energy dimension positive measures positive energy dimension seen lipschitz measures characterized terms their improving behaviour subset functions

Kathryn E. Hare  1   ; Maria Roginskaya  2

1 Department of Pure Mathematics University of Waterloo Waterloo, ON, N2L 3G1 Canada
2 Department of Mathematics Chalmers TH and Göteborg University Eklandagatan 86 Göteborg, SE 412 96 Sweden 
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Kathryn E. Hare; Maria Roginskaya. $L^p$-improving properties of measures of
 positive energy dimension. Colloquium Mathematicum, Tome 102 (2005) no. 1, pp. 73-86. doi: 10.4064/cm102-1-7

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