A convolution property of some measures with self-similar fractal support

Denise Szecsei ¹

¹ Department of Mathematics Stetson University P.O. Box 7532 Daytona Beach, FL 32116, U.S.A.

Colloquium Mathematicum, Tome 109 (2007) no. 2, pp. 171-177

Résumé

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.

Détail

DOI : 10.4064/cm109-2-1

Keywords: define class measures having following properties bullet measures supported self similar fractal subsets unit cube identified necessary bullet measures singular respect normalized lebesgue measure bullet measures have convolution property * subseteq varepsilon varepsilon varepsilon infty lies triangle vertices * subseteq measure class