We explore the extent to which the Fourier transform of an \(L^p\) density supported on the sphere in \(\mathbb{R}^n\) can have large mass on affine subspaces, placing particular emphasis on lines and hyperplanes. This involves understanding quantities of the form \(X(|\widehat{gd\sigma}|^2)\) and \(\mathcal{R}(|\widehat{gd\sigma}|^2)\), where \(X\) and \(\mathcal{R}\) denote the classical X-ray and Radon transforms respectively. This is joint work with Shohei Nakamura (Tokyo Metropolitan University).

The extension operator in the sphere Wf is well defined for any distribution in the sphere f and it is a solution of the Helmholtz equation in all the euclidean space. A useful and well known situation is when f belongs to \(L^2\) of the sphere, then Wf is called a Herglotz wave function. There are several characterizations of solutions of the Herglotz wave functions, starting with the classical one proved by Hartman and Wilcox. In this talk I will speak about the characterization of solutions of the Helmholtz arising from Sobolev spaces in the sphere through the extension operator. The results required the characterization of the Sobolev spaces in the sphere via multidimensional square functions that I will explain. These are the results of collaborations with Juan Antonio Barceló, Magali Folch, Teresa Luque and Maricruz Vilela.

The problem of restriction of the Fourier transform to hypersurfaces (or more generally to submanifolds in \(\mathbb {R}^n)\) was posed by Stein in the seventies. This operator, in its adjoint form, gives the solution of dispersive equations in terms of the Fourier transform of the initial data. Also, the restriction operator can be thought as a model case for more complicated oscillatory integral operators. We will make a review of this problem, which is still open. We will present some new results for the case of surfaces with negative curvature. This is part is a joint work with Stefan Buschenhenke and Detlef Müller, from Kiel.

In this talk I will present some recent results ---obtained in collaboration with Andoni García--- on point-source scattering theory in the presence of critically-singular and delta-shell potentials. These potentials consist of a combination of compactly supported functions with local singularities in the critical Lebesgue space and measures supported on compact hypersurfaces. To address the forward and inverse problems, I will show some new spaces adapted to the resolvent operator. These spaces turn to be very convenient to treat such potentials, and we hope that they could also help to deal with more singular perturbations.

We give a survey of affine invariant harmonic analysis as it relates to the Fourier restriction problem. This survey will develop a certain perspective.

In this talk, we consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian white noise. We show how wave packets can be used to partially recover the observable from the measurements almost surely. Furthermore, we point out the limitation of wave packets to recover the remaining part of the observable, and show how the errors hide the signal coming from the observable. The recovery results are based on an ergodicity property of the errors produced by wave packets. (Joint work with Pedro Caro).

I shall present estimates from below of oscillatory integrals that are related to some Partial Differential Equations.

We will review some classic and new results about resolvent estimates for Laplace and Dirac operators, motivated by the study of the spectra of some non self-adjoint perturbations. As applications, we will show localization or absence of eigenvalues as consequences of the Birman-Schwinger Principle.

We recall the conjectures of Stein and of Mizohata and Takeuchi and observe a certain self-similarity structure.