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

TL;DR

This paper investigates the decay and integrability properties of Fourier transforms of measures supported on fractal sets, extending classical results from smooth manifolds to fractal measures and providing quantitative asymptotic bounds.

Contribution

It establishes new non-integrability results for Fourier transforms of fractal measures and derives quantitative bounds on their $L^p$-asymptotics under specific conditions.

Findings

Fourier transform of fractal measures is not in $L^p$ for $p \,\leq\, 2n/\alpha$.

Provides bounds for the limsup of scaled $L^p$ norms of Fourier transforms.

Extends classical decay results from smooth manifolds to fractal sets.

Abstract

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: $$\underset{L\rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|\xi|\leq L}|\widehat{fd\mu_E}(\xi)|^pd\xi.$$