Self-affine measures that are $L^{p}$-improving
Volume 139 / 2015
Colloquium Mathematicum 139 (2015), 229-243
MSC: Primary 28A80; Secondary 43A05, 42A38.
DOI: 10.4064/cm139-2-5
Abstract
A measure is called $L^{p}$-improving if it acts by convolution as a bounded operator from $L^{q}$ to $L^{2}$ for some $q<2$. Interesting examples include Riesz product measures, Cantor measures and certain measures on curves. We show that equicontractive, self-similar measures are $L^{p}$-improving if and only if they satisfy a suitable linear independence property. Certain self-affine measures are also seen to be $L^{p}$-improving.
