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 `$f\in C_c^{\infty}(d\sigma)$`

, where `$d\sigma$`

is the surface measure on the sphere `$S^{n-1}\subset\mathbb{R}^n$`

. Then

It follows that `$\widehat{fd\sigma}\in L^p(\mathbb{R}^n)$`

for all
`$p>2n/(n-1)$`

. 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:

\(\underset{L\Rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{|\xi|\leq L}|\widehat{fd\mu_E}(\xi)|^pd\xi,\) \(\underset{L\Rightarrow\infty}{\limsup} \frac{1}{L^k} \int_{L\leq |\xi|\leq 2L}|\widehat{fd\mu_E}(\xi)|^pd\xi.\)

- All seminars.
- Seminars for 2015

Last updated: 18 May 2024