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

K. S. Senthil Raani

Published 2015-06-14Version 1

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 $d\sigma$ be the surface measure on the sphere $S^{n-1}\subset\mathbb{R}^n$. Then $$|\widehat{fd\sigma}(\xi)|\leq\ C\ (1+|\xi|)^{-\frac{n-1}{2}}.$$ It follows that $\widehat{fd\sigma}\in L^p(\mathbb{R}^n)$ 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 $0<\alpha<n$ for $p\leq 2n/\alpha$. In 2004, Agranovsky and Narayanan proved that if $\mu$ is a measure supported in a $C^1$-manifold of dimension $d<n$, then $\widehat{fd\mu}\notin L^p(\mathbb{R}^n)$ for $1\leq p\leq \frac{2n}{d}$. We prove that the Fourier transform of a measure $\mu_E$ supported in a set $E$ of fractal dimension $\alpha$ does not belong to $L^p(\mathbb{R}^n)$ for $p\leq 2n/\alpha$. We also study $L^p$-asymptotics of the Fourier transform of fractal measures $\mu_E$ under appropriate conditions on $E$ and give quantitative versions of the above statement by obtaining lower and upper bounds for the following: $$\underset{L\rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|\xi|\leq L}|\widehat{fd\mu_E}(\xi)|^pd\xi.$$