arXiv 1506.070052015-06-14math.CAmath.FA

分形测度的傅里叶变换的$L^p$ -渐近性

$L^p$-Asymptotics of Fourier transform of fractal measures

K. S. Senthil Raani

备注：: Thesis submitted in Indian Institute of Science, Bangalore
发表信息：: Electronic Theses and Dissertations of Indian Institute of Science 2016

摘要：

调和分析的基本问题之一是研究$\mathbb{R}^n$中薄集上测度或分布的傅里叶变换的衰减性质。当支持是一个足够光滑的流形，几乎完整的图片是可用的。这个方向的早期结果之一如下：设$f\in C_c^{\infty}(\mathbb{R}^n)$和$dσ$是球面$S^{n-1}\subset\mathbb{R}^n$上的曲面测度。然后是$$|\widehat{fdσ}(ξ)|\leq\ C\ (1+|ξ|)^{-\frac{n-1}{2}}.$$。因此，所有$p>\frac{2n}{n-1}$的$\widehat{fdσ}\in L^p(\mathbb{R}^n)$。在适当的曲率假设下，这个结果可以推广到PDMATHAMT维流形上的紧支撑测度。类似的结果是已知的措施支持在$\mathbb{R}^n$在适当的曲率条件下的低维流形。然而，分形测度的图景还远未完成。本文研究了$p\leq 2n/α$的分形维数$0<α<n$集上测度的傅里叶变换的渐近性质。在2004年，Agranovsky和Narayanan证明了如果$μ$是一个在维数为$d<n$的$C^1$ -流形中支持的测度，那么$\widehat{fdμ}\notin L^p(\mathbb{R}^n)$对于$1\leq p\leq \frac{2n}{d}$。证明了在分形维数为$α$的集合$E$中支持的测度$μ_E$的傅里叶变换不属于$p\leq 2n/α$的$L^p(\mathbb{R}^n)$.我们还研究了$E$上分形测度$μ_E$在适当条件下的傅里叶变换的$L^p$ -渐近性，并通过得到以下的上下界给出了上述陈述的定量形式：$$\underset{L\rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|ξ|\leq L}|\widehat{fdμ_E}(ξ)|^pdξ.$$

英文摘要：

One of the basic questions in harmonic analysis is to study the decay properties of the Fourier transform of measures or distributions supported on thin sets in $\mathbb{R}^n$. When the support is a smooth enough manifold, an almost complete picture is available. One of the early results in this direction is the following: Let $f\in C_c^{\infty}(\mathbb{R}^n)$ and $dσ$ be the surface measure on the sphere $S^{n-1}\subset\mathbb{R}^n$. Then $$|\widehat{fdσ}(ξ)|\leq\ C\ (1+|ξ|)^{-\frac{n-1}{2}}.$$ It follows that $\widehat{fdσ}\in L^p(\mathbb{R}^n)$ for all $p>\frac{2n}{n-1}$. This result can be extended to compactly supported measure on $(n-1)$-dimensional manifolds with appropriate assumptions on the curvature. Similar results are known for measures supported in lower dimensional manifolds in $\mathbb{R}^n$ under appropriate curvature conditions. However, the picture for fractal measures is far from complete. This thesis is a contribution to the study of asymptotic properties of the Fourier transform of measures supported in sets of fractal dimension $0<α<n$ for $p\leq 2n/α$. In 2004, Agranovsky and Narayanan proved that if $μ$ is a measure supported in a $C^1$-manifold of dimension $d<n$, then $\widehat{fdμ}\notin L^p(\mathbb{R}^n)$ for $1\leq p\leq \frac{2n}{d}$. We prove that the Fourier transform of a measure $μ_E$ supported in a set $E$ of fractal dimension $α$ does not belong to $L^p(\mathbb{R}^n)$ for $p\leq 2n/α$. We also study $L^p$-asymptotics of the Fourier transform of fractal measures $μ_E$ under appropriate conditions on $E$ and give quantitative versions of the above statement by obtaining lower and upper bounds for the following: $$\underset{L\rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|ξ|\leq L}|\widehat{fdμ_E}(ξ)|^pdξ.$$

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文