This blogpost is loosely based on a series of talk in IISERB during August-December 2016..
We try to learn the relation between geometric measure theory and Fourier analysis on. We will be concentrating on those parts of Fourier analysis on
where Hausdorff dimension plays role. However, we discuss a few topics in geometric measure theory in detail. Let us see how restriction conjecture is related to Kakeya type problems.
What is measure theory? The study of measures, generalizing the intuitive notions of length, area, volume.. We are aware of the Lebesgue measures. In general it is the study of measures on any general space, not just Euclidean spaces.
Lebesgue Measure: : Given a set
, we define the measure (outer measure)
where infimum is taken over , the countable collection of boxes
whose union covers
. In other words instead of collection of boxes we can consider
the countable collection of balls
of radius
whose union covers
:
We recall a very beautiful measure:
Hausdorff Measure: : Given a set
, we define the measure
with ,
where the infimum is taken over the countable collection of balls
of radius
whose union covers
.
Recall the following:
- Measure on a set $X$, we mean usually by outer measure $\mu$: a non-negative, monotone, countably subadditive (that is,
) function on
that gives value zero for the empty set.
- Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement.
- A measure
is called Borel measure if the Borel sets are measurable.
- A Borel measure
is Borel regular if for any
, there exists a Borel set
such that
and
.
- A Borel measure is locally finite if compact sets have finite measure.
- Locally finite Borel measures are often called Radon measures.
Example: Lebesgue measure is Radon measure. Counting measure on any metric space is Borel regular but it is Radon only if the space is discrete. Hausdorff measure is not Radon.
Vitali type covering theorem for Lebesgue measure: Let and
be a family of closed balls such that every point of
is contained in an arbitrarily small ball in
, that is,
for
. Then there are disjoint balls
such that
.
Consider the example: Let be a Radon measure on
:
, that is,
denotes the length measure on
axis. The family
covers
in the sense of the above theorem. But fails the conclusion, since for any countable collection
we have
.
For Radon measures, the theorem holds if we assume to be a family of closed balls such that each point of A is the centre of arbitrarily small balls of
.
Geometric measure theory is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth.
When we mean smoothness, on a careful observation, what we see is the smoothness of the boundary. For a given curve in , can we find a surface of minimal area with that curve as the boundary? This is nothing but the Plateau’s problem, posed in 1760 and completely solved in 1930. While studying this, geometric measure theory was developed.
How about sets which has ‘roughness’ at every point? Can we still study geometry of these sets? We will start with some examples of non-smooth sets:
Generalized Cantor set: Let . Fix a set
of
finite numbers in
with
for all
. Fix
small such that
for all
and
; also such that
. We construct a sequence of sets
:
is the union of
intervals of length
with starting points in the set
. In other words we have removed
in the interval
. We call the removed intervals as black intervals and the considered intervals as white intervals. Now on each interval
in
we have
points:
. By the assumption on
we can see that
. Hence we continue to construct
as the set of
intervals of length
. Proceeding like this, at
step we have
intervals of length
. The generalized Cantor set is $Elatex =\cap_nE_n$.
It is an easy ‘Basic real analysis’-exercise to check that this set is nowhere dense perfect non-empty uncountable set.
Salem sets: Similar to generalized Cantor sets, we have Salem sets (constructed by Salem 1950). The purpose of this set will be described later. Let . Fix a set
of
finite numbers in
with
for all
. Fix
small such that
for all
and
; also such that
. Choose a sequence
such that
. We construct a sequence of sets
:
is the union of
intervals of length
with starting points in the set
. In other words we have removed
intervals in the interval
. We call the removed intervals as black intervals and the considered intervals as white intervals. Now on each interval
in $E_1$ we have $N$ points:
. By the assumption on
we can see that
. Hence we continue to construct
as the set of
intervals of length
. Proceeding like this, at
step we have
intervals of length
. The Salem set is
.
Brownian Motion: Consider the space of all continuous functions
with
such that the increments
and $\omega(t_4)-\omega(t_3)$ are independent of
and
.
For Example: For any fixed , consider the random variable
given by
with the probability density function
, that is with
). We call the random variables
defined by
, Brownian motion. Zeroes of the Brownian motion has Hausdorff dimension
when we consider
. (What are zero sets of the Brownian motion? For each
,
“Hitting back at the starting point”). We then have the probability measure
on
.
Does these sets have any geometry to it? We can right away observe that there is an in-built measure to these sets. Are these measures absolutely continuous with respect to the measures like Hausdorff measures? Is there an analogue of ‘smoothness’ to be spoken on these sets?
The main references are : Fourier analysis and Hausdorff dimension and Geometry of sets and measures in Euclidean Spaces by Pertti Mattila; Decay of the Fourier transform by Alex Iosevich and Elijah Liflyand.
(to be continued….)