A Kakeya Set is a set that contains a unit line segment for every direction. For example, a ball of radius one half is a Kakeya set.
The Kakeya Set Conjecture asserts that every compact Kakeya set has Hausdorff dimension .
The X-ray transform is the operator
where is a line, and the Lebesgue measure on the line. The manifold can be represented as
here is the space of lines in that pass through the origin, and represents a line that passes through with direction . We endow with a measure invariant under rigid motions. It is an open problem to determine the exponents for which the following inequality holds:
For brevity we will write .
We can write the X-ray transform of a function as
where . If is a line , and the projection to the plane , then is the pullback , where is the Dirac delta centered at .
The operator dual to , i.e. , acts on functions in as
here —this is a distribution in , do not confuse with the usual Dirac delta centered at . The distribution is the restriction of the measure in to the set of lines passing through .
The first bound we can get for the X-ray transform is
since is constant. Alternatively, we can use the fact that the integral of along all the lines with the same direction equals , which also implies that . Any other bound of the X-ray transform can be interpolated with this bound to get further inequalities.
Besides -spaces, we need to consider spaces of regular functions.
Definition
(Sobolev space)
Let be a smooth function with Fourier transform supported in , and let be a smooth function with Fourier transform supported in . Suppose that
where and , for . Let be the projection .
The Sobolev-Slobodeckij spaces are, for and , defined as
and
See (Triebel 1978, Sec. 2.3.1).
The relation between that X-ray transform and the Kakeya set conjecture is the content of the following Theorem
Theorem
(Bourgain 1991, Lemma 2.15)
Let . If for every function such that , for compact, it holds that
(1)
then the Hausdorff dimension of a Kakeya set is at least .
Proof
To compute the Hausdorff dimension we can use either balls or dyadic cubes , for integer. In fact, we can cover a cube with a ball of radius , and conversely we can cover a ball of radius with a few cubes , for .
Let be a Kakeya set. Give any covering of at scale , i.e. a covering such that for every cube its side-length is , the goal is to show that if then
where is the side-length of the cube.
Let be a covering of at scale . We denote by the collection of all the cubes in with side-length , and we denote by the union of all the cubes in ; therefore,
Since our hypotheses involve the spaces , we should mollify . We take a suitable smooth function with compact support, define the dilations , and replace by .
Since contains a unit line segment in every direction , then for every it holds that , and then that
By (1) we have that
In general, the -norm of is
see Lemma below. Hence,
where denotes the number of cubes in .
By hypothesis , then for every we can apply Hölder inequality to get
where . The statement of the Theorem follows.
Lemma
If is a smooth function and , for , then
(2)
where depends on .
Proof
If is an integer, then (2) follows from and Young Inequality for convolutions.
If integer, then we have to estimate the norm of ; recall that , where . If then by Young Inequality for convolutions
If then fix a number . We estimate the norm of as
As before we get . For the other term, since then we get
we conclude so that . Therefore, the -norm of is
which concludes the proof.
For a fixed direction , the X-ray transform maps to a function in . The function turns out to be more regular than .
Theorem
If has support , for compact, then
(3)
The Japanese bracket is defined as , and then .
Proof
We set , and for a fixed direction we compute the Fourier transform of the function . By rotational symmetry we can assume that , so
The Fourier transform is . If is the Dirac delta of the plane normal to , then we may write
Hence,
For high frequencies we have that
For low frequencies we have that
The statement of the Theorem follows.
Sobolev embedding Theorem , for , and the inequality (3) allow us to conclude that for every such that , for compact, it holds that
for . Hence, by Theorem the Hausdorff dimension of a Kakeya set is at least 2. This result is the best possible in (Córdoba 1977), but far away from the expected dimension for .
Definition
(Lorentz Spaces)
Let be a measure space. The quasinorm of the space , for and , is
and, for and , is
Theorem
(Drury 1983)
If is a measurable set, and is the characteristic function of , then
(4)
This inequality corresponds to the restricted weak version of the point .
Proof
Let and define the set , then
(5)
This establishes a lower bound for , and we will get now an upper bound.
Choose a point such that — is the center of the Bourgain’s bush. By translation symmetry we can assume that . If is the Dirac delta centered at the origin, then
(6)
Here we can think of as the distribution we defined earlier supported over the lines passing through the point . We can also justify the notation by a limiting process since .
On the other hand, we can also estimate the left side of (6) as
The function can be computed explicitly
where and is the line that passes through and . Then, we can estimate the norm of as
Then we get the bound
(7)
By (6) and (7) we get the desired upper bound for :
This and (5) allow us to conclude that
which implies the restricted weak inequality (4).
Drury’s Theorem and the bound imply that
for and . By Sobolev embedding Theorem , for , we get further
for , and . Hence, by Theorem the Hausdorff dimension of a Kakeya set is at least .
References
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Bourgain, J.?, Besicovitch type maximal function operators and applications to Fourier Analysis, Geom. Funct. Anal., 1, 1991
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Córdoba, A.?, The Kakeya Maximal Function and the Spherical Summation Multipliers, Amer. J. Math., 99, 1977
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Christ, M.?, Estimates for the -plane transform, Indiana Univ. Math. J., 33, 1984
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Drury, S. W.?, estimates for the X-ray transform, Illinois J. Math., 27, 1983
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Triebel, H?, Interpolation Theory, Function Spaces, Differential Operators, North-Holland publishing company, 1978