This mini-course aims to introduce and present recent developments on the Mizohata-Takeuchi conjecture for the Fourier extension operator. We informally refer to an inequality which controls some integral taken on Rn by integrals taken on lower dimensional subspaces (such as the X-ray transform) as a tomographic bound, and this conjecture is a classical example of such bound. This inequality can be motivated by its connections to well-posedness of some PDEs, as well as to the endpoint multilinear Fourier restriction conjecture. After covering the necessary background, we will motivate and discuss our recent work on a Sobolev variant of Mizohata-Takeuchi, in which a good understanding of the classical Wigner transform plays a crucial role.

1. Lecture: Harmonic Analysis background

We will cover the basic theory of the Fourier transform in Rn, the Wigner transform, X-ray and Radon transforms, fractional integration and the Kenig-Stein operator.

2. Lecture: The Fourier restriction problem

We will cover the Stein-Tomas theory and applications to PDE, discuss some of the tools used over the last 50 years to study the restriction conjecture and understant its connection to the Kakeya problems.

3. Lecture: The Stein and Mizohata-Takeuchi conjectures I

The plan for this lecture is to introduce Stein’s and Mizohata-Takeuchi’s conjectures, build our intuition through simple cases and cover the basics of spherical harmonics and Bessel functions that will be necessary for the next lecture.

4. Lecture: The Stein and Mizohata-Takeuchi conjectures II

The plan is to verify the Mizohata-Takeuchi conjecture for the sphere when whe underlying weight is radial. This will require the previously covered background on spherical harmonics and Bessel functions.

5. Lecture: The Sobolev-Mizohata-Takeuchi problem

The plan is to present recent progress in our joint work with Bennett, Gutierrez and Nakamura on a Sobolev version of the Mizohata-Takeuchi problem. This is a weaker version of the conjecture that can be studied in connection with certain nonlinear variants of the Wigner transform and of the Kenig-Stein operator.

References

1. J. A. Barcel´o, J. Bennett, A. Carbery, A  note  on  localised  weighted  estimates  for  the  extension operator, J. Aust. Math. Soc. 84 (2008), 289–299.
2. J.  A.  Barcel´o,   J.  Bennett,   A.  Carbery,   A.  Ruiz,   M.  Vilela,   Some  special  solutions  to  the Schr¨odinger  equation,
3. J. A. Barcel´o, J. Bennett, A. Ruiz, Spherical  perturbations  of  Schr¨odinger  equations, J. Fourier Analysis and Applications, 12, Issue 3 (2006), 269–290.