On Fourier asymptotics of a generalized Cantor measure

Bérenger Akon Kpata¹ ; Ibrahim Fofana¹ ; Konin Koua¹

¹ UFR Mathématiques et Informatique Université de Cocody 22 BP 582 Abidjan 22, Côte d'Ivoire

Colloquium Mathematicum, Tome 119 (2010) no. 1, pp. 109-122

Résumé

Let $d$ be a positive integer and $\mu $ a generalized Cantor measure satisfying $\mu=\sum_{j=1}^{m}a_{j}\mu \circ S_{j}^{-1}$, where $0 a_{j} 1$, $\sum_{j=1}^{m}a_{j}=1$, $S_{j}=\rho R+b_{j}$ with $0 \rho 1$ and $R$ an orthogonal transformation of $\mathbb{R} ^{d}$. Then $$\cases{ 1 p\leq2\ \Rightarrow\cr\displaystyle\sup_{r>0}\, r^{d( {1}/{\alpha ^{\prime }}-{1}/{{p^{\prime}}}) } \bigg(\int_{J_{x}^{r}}\vert \widehat{\mu}( y) \vert ^{p^{\prime}}\,dy\bigg) ^{{1}/{p^{\prime}}}\leq D_{1} \rho ^{-{d}/{\alpha ^{\prime }}},\ x \in \mathbb{R}^{d}, \cr p=2\ \Rightarrow\ \displaystyle\inf_{r\geq 1}\, r^{d( {1}/{\alpha ^{\prime }}-{1}/{2}) }\bigg( \int_{J_{0}^{r}}\vert \widehat{\mu}( y) \vert ^{2}\,dy\bigg) ^{{1}/{2}}\geq D_{2} \rho ^{{d}/{\alpha ^{\prime }}}, } $$ where $J_{x}^{r}= \prod_{i=1}^{d}( x_{i}-{r}/{2},x_{i}+{r}/{2}) $, $\alpha ^{\prime }$ is defined by $\rho ^{{d}/{\alpha ^{\prime }}}=( \sum_{j=1}^{m} a_{j}^{p}) ^{{1}/{p}}$ and the constants $D_{1}$ and $D_{2}$ depend only on $d$ and~$p$.

Détail

DOI : 10.4064/cm119-1-6

Keywords: positive integer generalized cantor measure satisfying sum circ where sum rho rho orthogonal transformation mathbb cases leq rightarrow displaystyle sup alpha prime prime bigg int vert widehat vert prime bigg prime leq rho alpha prime mathbb rightarrow displaystyle inf geq alpha prime bigg int vert widehat vert bigg geq rho alpha prime where prod alpha prime defined rho alpha prime sum constants depend only