Memorial José Luis Rubio de Francia

We consider the eigenfunctions of the Schrödinger operator with growing potential on the Euclidean space. To localize the eigenfunctions on the phase space, we construct a decomposition of the phase space into rectangles and show that each eigenfunction has some non-trivial mass concentrated near the corresponding energy shell. This is an ongoing joint project with Stefano Decio, Jaume De Dios Pont, and Svetlana Mayboroda.

The quest to find the sharp forms of functional inequalities has always been a beautiful and challenging theme in analysis. Among the most celebrated achievements of the last century one could certainly include the sharp forms of the Hausdorff-Young inequality for the Fourier transform, Young’s inequality for convolutions, the Hardy-Littlewood-Sobolev inequality and the classical Sobolev embeddings.

In this talk we will discuss a few sharp inequalities related to the Fourier transform, within two themes: Fourier restriction theory and Fourier uncertainty principles. In the first theme, we will be mainly concerned with the classical restriction setup for quadratic surfaces (spheres, paraboloids, cones and hyperboloids) arising from the seminal work of Strichartz of 1977. Related questions involving mixed norms and stability issues will also be addressed. In the second theme, we will address the problem of prescribing the sign of a function and its Fourier transform at infinity, and doing this in an optimal way (in an appropriate sense). This phenomenon was introduced by Bourgain, Kahane and Clozel in 2010 under the name of "sign Fourier uncertainty", and brings interesting connections to the sphere packing problem.

In the 1920’s Besicovitch studied linearly measurable sets in the plane, that is sets with locally finite ”length”. The basic question he addressed was whether the infinitesimal properties of the ”length” of a set E in the plane yield geometric information about E itself. This simple question marks the beginning of the study of the geometry of measures and the associated field, Geometric Measure Theory.

In this talk I will present several examples of measures that arise naturally in different contexts, for example PDEs and free boundary regularity problems. I will discuss how the infinitesimal properties of a measure yield a great deal of information about the measure and its support. In turn this sheds light on the original problem which gave rise to the measure in question.

A Kakeya (or Besicovitch) set is a set of Lebesgue measure zero in a Euclidean space containing a unit line segment in all directions. The Kakeya conjecture asks whether these must have full Hausdorff dimension. This is known in the plane and open in higher dimensions. It is related to many questions of modern Fourier analysis. In the talk I shall discuss its relations to questions how Hausdorff dimension is transformed under projections.

By the 1950’s the theory of absolutely convergent Fourier series had been well-developed and people started to ask what happens to theorems/theories about absolutely convergent Fourier series when one relaxes the condition to uniform convergence. There are two interesting instances where such a ques- tion was posed and then investigated; one concerning Sidon sets and another about homomorphisms of the Fourier algebra.

Interestingly, these two problems have near to exact counterparts to problems in Calderon-Zygumd theory and the problems were developed inde- pendently for a period of time. In this talk, we will describe this historical deve- lopment and discuss some recent work.

A question on incompressible flows leads to interesting problems on multilinear singular integral forms of Christ-Journé type. I'll discuss the background and present some recent results. This is joint work with Charles Smart and Brian Street.

The classical interpolation methods are fundamental tools in harmonic analysis, with interpolation of Lp-spaces being a prime example.

It turns out that in special circumstances one can introduce interpolation on Lp-spaces with complex exponents p. With such tools available we then e.g. obtain optimal integral estimates for nonlinear differential expressions, or describe the multifractal spectra of quasiconformal mappings.

The presentation is based on joint works with T. Iwaniec, I. Prause and E. Saksman.