Harmonic Analysis and PDE’s Summer School

We will discuss various aspects of the incompressible Euler equation:
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We will focus primarily on the properties and construction of steady solutions.

In this lecture, we present a modern perspective on functional smoothness and regularity by employing sharp tools from harmonic analysis. We first move beyond classical pointwise analysis to explore regularity through the lens of local averages. This approach provides a more flexible and robust framework, particularly in settings where traditional derivatives either do not exist or cannot be defined. Subsequently, we apply these techniques to derive improved local Poincaré-Sobolev estimates, which are instrumental in proving the celebrated De Giorgi regularity theorem. This framework also yields a self-contained proof of the John–Nirenberg theorem for BMO functions. A central theme of the discussion is the “self-improving” property—a remarkable phenomenon where modest local control over oscillations leads to significantly stronger global integrability. As further applications, we present an extension of the Nash inequality, providing an alternative path to De Giorgi’s theorem, as well as a generalization of the Gagliardo–Nirenberg–Sobolev theorem within the setting of Campanato spaces. This lecture is based on joint works with Iker Gardeazabal and Emiel Lorist, and with Alejandro Claros and Linfei Zheng.

Harmonic measure is a widely studied topic in analysis that arises naturally in connection with the Dirichlet problem for the Laplacian on a domain. It can also be interpreted in terms of the first-passage probability of Brownian motion.
This measure is supported on the boundary of the domain, and its interplay with the surface measure contains deep geometrical and analytical information. These connections have been studied since the paper of F. and M. Riesz (1916), in which they proved that harmonic measure is mutually absolutely continuous with respect to arc length in the case of a simply connected planar domain with rectifiable boundary.
In 2016, Azzam, Hofmann, Martell, Mayboroda, Mourgoglou, Tolsa, and Volberg proved that the absolute continuity of harmonic measure implies rectifiability of the boundary. In this course I will discuss the techniques involved in the proof of this result, which rely heavily on harmonic analysis, and I will also discuss related problems such as generalizations to elliptic PDEs in divergence form under suitable regularity assumptions on the coefficients.

In 1980, B. Muckenhoupt conjectured that the weighted inequality connecting the Hilbert transform and the Hardy-Littlewood maximal operator holds if and only if the weight w satisfies the so-called  condition. Very recently, in joint work with F. Nazarov and S. Ombrosi, we gave a positive answer to this conjecture. In this minicourse, I plan to give an overview of the proof of this result and to explain several of the main ideas and technical ingredients behind it.

This mini-course introduces periodic coherent structures in the two- and three-dimensional incompressible Euler equations. We begin with the finite-dimensional viewpoint of autonomous dynamical systems, Hamiltonian flows, conserved quantities, periodic orbits, and point-vortex dynamics. We then develop the bifurcation tools needed in infinite-dimensional settings and show how they lead to classical families of rigidly rotating vortex patches. We also discuss the desingularization
of rigid point-vortex configurations via concentrated vortex patches, using the implicit function theorem in Banach spaces.

In the second part of the course, we turn to non-rigid motions, with particular emphasis on leapfrogging vortex dynamics. We explain how the planar leapfrogging motion can be desingularized at the level of vortex patches, and why this problem requires more delicate techniques such as KAM theory and Nash–Moser iteration. Finally, we describe how these ideas can be adapted to the three-dimensional axisymmetric Euler equations in order to construct leapfrogging vortex rings.