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
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….)