On Fourier asymptotics of a generalized Cantor measure
Volume 119 / 2010
Abstract
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~$p$.
