A convolution property of some measures with self-similar fractal support
Volume 109 / 2007
Colloquium Mathematicum 109 (2007), 171-177
MSC: Primary 43A05; Secondary 43A15.
DOI: 10.4064/cm109-2-1
Abstract
We define a class of measures having the following properties:
$\bullet$ the measures are supported on self-similar fractal subsets of the unit cube $I^{M}=[0,1)^{M}$, with 0 and 1 identified as necessary;
$\bullet$ the measures are singular with respect to normalized Lebesgue measure $m$ on $I^{M}$;
$\bullet$ the measures have the convolution property that $ \mu * L^{p} \subseteq L^{p + \varepsilon} $ for some $ \varepsilon = \varepsilon (p) > 0 $ and all $ p \in (1, \infty ) $.
We will show that if $({1}/{p},{1}/{q})$ lies in the triangle with vertices $(0,0)$, $(1,1)$ and $({1}/{2},{1}/{3})$, then $\mu * L^{p} \subseteq L^{q}$ for any measure $\mu$ in our class.
