$L^p$-improving properties of measures of positive energy dimension
Volume 102 / 2005
Colloquium Mathematicum 102 (2005), 73-86
MSC: Primary 43A05; Secondary 42A38, 28A12.
DOI: 10.4064/cm102-1-7
Abstract
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.
