We will describe the method of energy channels, developed with Duyckaerts and Merle, and some of its applications to linear and nonlinear wave equations, with emphasis on the soliton resolution conjecture for energy critical nonlinear equations.

We show local and global regularity properties of Besov functions defined on a space of homogeneous type. More precisely, by using a characterization of Besov spaces with wavelet coefficients we obtain an upper bound for the spectrum of singularities. We further show that this bound is almost always attained in the sense of prevalence. Lastly, we establish a trace theorem for Besov spaces and study the optimal regularity for certain cases.

Let T be a linear bounded operator on a Banach space. It is said to be that T is Cesàro bounded if its means \frac{1}{n+1}\sum_{j=0}^n T^j are uniformly bounded on \mathcal{B}(X). This concept extends to the power-boundedness and, in the last years, some authors have considered the analogous for the fractional order means.

In this talk, I will present ergodic results of the orbits and means of Cesàro bounded operators. Also, I will show examples, some specific techniques and tools needed to face the mentioned results, and the sketch of some proofs.

The topic of differentiation of integrals arises with the origins of calculus, yet in many respects remains mysterious with many open problems. This informal talk will give a broad overview of the subject of differentiation of integrals, highlighting results from the seventeenth through the twenty-first centuries. An emphasis will be given on open problems and why they are difficult. Recent applications to partial differential equations will also be discussed. This work is joint with Ioannis Parissis.

We will consider the Kakeya conjecture; the hope that $\delta$-tubes, which  point in directions which are separated by $\delta$, cannot be compressed very much by positioning them strategically. This can be formulated precisely as a lower bound on the measure of any set that contains the tubes. On the one hand, we will see that the conjectured bound holds when the containing set is a neighborhood of any real algebraic variety, confirming a conjecture of Guth. The proof employs tools from the theory of semialgebraic sets including Gromov's algebraic lemma and Tarski's projection theorem. On the other hand, we will use polynomial partitioning to prove that the conjectured bound holds when there is a total absence of algebraic structure. Balancing between the two cases yields improved bounds in certain intermediate dimensions. This is joint work with Jonathan Hickman and Nets Katz.

In this talk I will show the strong unique continuation property for solutions of higher order fractional Schrödinger operators, including the case of variable coefficients and  the presence of Hardy type gradient potentials. The proof relies on a generalised Caffarelli-Silvestre extension for the higher order fractional Laplacian and on Carleman estimates. I will also discuss applications of the unique continuation results in the context of nonlocal Calderón problems.

This is a joint work with Angkana Rüland.

In this talk I will introduce the backscattering inverse problem in which one tries to recover information of a potential  by looking how it scatters back free plane waves.

Since the recovery of the  whole potential is still an open question, many  works have studied the  problem of recovery of singularities, which basically consist of trying to get as much information as possible about the singular part of the potential. In the second half of the talk, we will see that under certain assumptions one can recover the singularities up to a one derivative gain in the Sobolev scale, and that in general this is the best possible result. 

We shall begin the seminar with a brief introduction to the theory of bilinear Fourier multipliers in euclidean spaces. Our aim is to discuss bilinear analogues of some problems from the classical Fourier multiplier theory concerning the Fourier multipliers of the form of the \(e^{i \phi(\xi)}\), where \(\phi\) is a real-valued function. In particular, we shall prove a negative result about the bilinear multiplier operators associated with unimodular functions \(e^{i \phi(\xi-\eta)}\) for non-linear real functions \(\phi\). This is based on a joint work with K. Jotsaroop. 

In one of its forms in the linear case, Rubio de Francia's extrapolation theorem states that if an operator \(T\) is bounded on on \(L^q(w)\) for a single \(1\leq q<\infty\) for all weights \(w\) in the Muckenhoupt class \(A_q\), then \(T\) is in fact bounded on \(L^p(w)\) for all \( 1 < p < \infty\) for all w in the Muckenhoupt class \(A_p\). In recent developments, motivated by operators such as the bilinear Hilbert transform, multilinear versions of this result have appeared. In this talk I will discuss the recent multilinear extrapolation result I obtained. The proof here differs from the proofs given in the works of Cruz-Uribe, Martell [2017], Li, Martell, Ombrosi [2018], and the recent work of Li, Martell, Martikainen, Ombrosi, Vuorinen [2019] in that it does not rely on any linear off-diagonal extrapolation techniques. Rather, a multilinear analogue of the Rubio de Francia algorithm was developed, leading to an extrapolation theorem that includes the endpoints as well as a sharp dependence result with respect to the involved multilinear Muckenhoupt constants.

In this talk we will describe the smoothing effect of bilinear fractional integral operators and similar multipliers, in the sense that they improve the regularity of functions. We will also consider bilinear singular multiplier operators which preserve regularity and use them to obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces. If time permit we will also discuss related results for bilinear maximal functions. The talk is based on joint work with Jarod Hart and Xinfeng Wu.

We establish a set of necessary conditions and a set of sufficient conditions for boundedness of a family of Brascamp-Lieb forms in Lorentz spaces and L^p-spaces with power weights. The conditions are close to optimal. This  is joint work with Carl Lee and Katharine Ott. 

The study of uniform rectifiability started as a hunt for optimal geometric conditions for various aspects of Calderón-Zygmund type harmonic analysis. More recently, the emphasis of the field has been on exploring the connections between partial differential equations and boundary geometry. In this talk, we discuss the origins of and some recent trends in uniform rectifiability, related topics and Varopoulos type approximability properties of harmonic functions. These properties help us to overcome some problems related failure of certain types of Carleson measure estimates. The talk is partially based on my previous and on-going work with S. Hofmann and S. Bortz.

It is well known that the centered Hardy-Littlewood maximal operator is of weak type (1,1) for every reasonable measure on $\mathbb{d}^d$, with bounds that depend only on the dimension. We explore the analogous question in the setting of separable metric spaces.

Sobolev spaces of manifold-valued maps arise naturally in the study of some variational problems coming from materials science. Indeed, the behaviour of some materials (such as nematic liquid crystals) can be described by a map $u$ from a Euclidean domain $\Omega$ to a compact, smooth Riemannian manifold $N$, which parametrises the possible local configurations for the material. The map $u$ is required to belong to a suitable energy space --- for instance, the gradient of $u$ must be $p$-th power integrable. We are then led to define the Sobolev space $W^{1,p}(\Omega, N)$. We will focus on a functional-analytical question: are smooth maps dense in $W^{1,p}(\Omega, N)$? The answer turns out to depend on topological obstructions. In this talk, which is designed as an introduction to the topic, we will review a few classical density and non-density results.

One of the archetypical aggregation-diffusion models is the so-called classical parabolic-elliptic Patlak-Keller-Segel (PKS for short) model. This model was classically introduced as the simplest description for chemotatic bacteria movement in which linear diffusion tendency to spread fights the attraction due to the logarithmic kernel interaction in two dimensions. For this model there is a well-defined critical mass. In fact, here a clear dichotomy arises: if the total mass of the system is less than the critical mass, then the long time asymptotics are described by a self-similar solution, while for a mass larger than the critical one, there is finite time blow-up. In this talk we will show some recent results concerning the symmetry of the stationary states for a nonlinear variant of the PKS model, of the form

(1) ∂tρ = ∆ρm + ∇ · (ρ∇(W ∗ ρ)), being W ∈ C^1(R^d \ {0}), d ≥ 2, a suitable aggregation kernel, in the assumptions of dominated diffusion, i.e. when m > 2−2/d. In particular, if W represents the classical logarithmic kernel in the two-dimensional case, we will show that there exists a unique stationary state for the model (1) and it coincides, according to one of the main results in the work [1], with the global minimizer of the free energy functional associated to (1). In the case d = 2 we will also show how such steady state coincides with the asymptotic profile of (1). Finally, we will also discuss some recent results concerning the model (1) with a Riesz potential aggregation, namely when W(x) = cd,s|x|2s−d for s ∈ (0, d/2), again in the diffusion dominated regime, i.e. for m > 2 − (2s)/d. In particular, all stationary states of the model are shown to be radially symmetric decreasing and that global minimizers of the associated free energy are compactly supported, uniformly bounded, Hölder regular, and smooth inside their support. These results are objects of the joint works [2], [3].

References

[1] J. A. Carrillo, D. Castorina, B. Volzone, Ground States for Diffusion Dominated Free Energies with Logarithmic Interaction, SIAM J. Math. Anal. 47 (2015), no. 1, 1–25.

[2] J. A. Carrillo, S. Hittmeir, B. Volzone, Y. Yao, Nonlinear Aggregation-Diffusion Equations: Radial Symmetry and Long Time Asymptotics,, arXiv:1603.07767.

[3] J. A. Carrillo, F. Hoffmann, E. Mainini, B. Volzone, Ground States in the Diffusion-Dominated Regime, arXiv:1705.03519.

Consider a local energy solution to an inhomogeneous elliptic equation  in divergence form. Classical results in regularity theory tell that when the source term has regularity slightly better than what is required for solvability, the regularity of the solution itself is also better than what is assumed a priori. This is traditionally seen as a consequence of Gehring's lemma about open-ended property of reverse Hölder classes. In this talk, I discuss a functional analytic point of view on the topic with focus on extensions to parabolic PDEs.

The classical Liouville Theorem states that bounded harmonic functions are constant. In this talk we will revisit this result for a class of symmetric nonlocal operators (Lévy-Khinchine). This class of operators naturally includes the fractional Laplacian, Relativistic Schrodinger operators, convolution operators, as well as discretizations of both local and nonlocal symmetric diffusion operators.

First, we will treat the one dimensional case. Here we give a precise classification for which the Liouville Theorem holds. The condition will be related to irrational numbers (\cite{AldTEnJa18}).

In the multi dimensional case such a characterization is also proved. This time it will be given in terms of additives subgroups of $\mathbb{R}^N$ (\cite{AldTEnJa18}). 

This nonlocal result will allow us (\cite{AldTEnJa19}) to give a full characterization of the Liouville property for any linear operator (local + nonlocal, and not necessarily symmetric) with constant coefficients satisfying the maximum principle (see \cite{Cou64}).

[AldTEnJa18] N.~Alibaud, F.~del Teso, J.~Endal, and E.~R. Jakobsen. Characterization of nonlocal diffusion operators  satisfying the Liouville theorem. Irrational numbers and subgroups of $\mathbb{R}^d$. (Preprint: arXiv:1807.01843).

[AldTEnJa19] N.~Alibaud, F.~del Teso, J.~Endal, and E.~R. Jakobsen. The Liouville theorem and linear operators satisfying the maximum principle. A complete characterization in the constant coefficient case. (Work in progress).

[Cou64] P.~Courrège. Générateur infinitésimal d'un semi-groupe de convolution sur $R^{n}$, et formule de Lévy-Khinchine. Bull. Sci. Math. (2), 88:3--30, 1964.

Joint work with N. Alibaud (University of Besan\c con), J. Endal and E. R. Jakobsen (Norwegian University of Science and Technology).

Given the smoothness index α \in R, we give two characterizations of the Sobolev space H^α on the unit sphere S^{d-1}. The first one is obtained via the L^2 boundedness of a quadratic multiscale operator, that in the particular case 0<α<2, is a square function. This characterization is based on [1].

The second characterization is in terms of the entire solutions of the Helmholtz equation. The result is motivated by [2] and [3].

This is a joint work with J. A. Barceó and S. Pérez-Esteva and it is still in progress.

[1] R. Alabern, J. Mateu, J. Verdera, \emph{A new characterization of Sobolev spaces on R^n, Math. Ann. , 2012, 354 (2): 589-–626.

[2] P. Hartman and C. Wilcox, Reproducing kernel for the Herglotz functions in 

and solutions of the Helmholtz equation, Math. Zeitschr., 1961, 75, 228--255.

[3] S. Pérez-Esteva and S. Valenzuela-Díaz, Reproducing kernel for the Herglotz functions in R^n and solutions of the Helmholtz equation, J. Fourier Anal. Appl., 2017, 23 (4): 834--862.

In a joint work with Pedro Caro (BCAM), we consider the inverse Calder\'on problem consisting of determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually assumes the data to be given by such map. This situation corresponds to having access to infinite-precision measurements, which is totally unrealistic. In this work, we study the Calderón problem assuming the data to contain measurement errors and provide formulas to reconstruct the conductivity and its normal derivative on the surface. Additionally, we state the rate convergence of the method. Our approach is theoretical and has a stochastic flavour.

In this talk we will explore some recent results on boundedness of maximal functions of convolution type associated to smooth kernels on Hardy-Sobolev spaces, as well as discuss sharpness of such bounds.

We consider a nonlinear Schrödinger type equation with nonlocal nonlinearity, of a convolution type, called the generalized Hartree equation. In the focusing case we investigate global behavior of solutions and formation of stable singularities. In the inter-critical regime we first obtain a dichotomy for global vs finite time existing solutions exhibiting two methods of obtaining scattering: one via Kenig-Merle concentration - compactness and another one is using Dodson-Murphy approach via Morawetz on Tao’s scattering criteria. Next, we investigate stable blow-up solutions in a critical regime and describe the blow-up dynamics, which is similar to NLS. This work is a part of the PhD dissertation under the supervision of Svetlana Roudenko.

While the best constants in the Hausdorff-Young inequality on \mathbb R^n have long been known, the corresponding problem on general noncommutative Lie groups is still open. In joint work with Michael Cowling, Detlef Müller and Javier Parcet (arXiv:1807.04670), we establish a sharp local central version of the inequality for compact Lie groups, and extend known results for the Heisenberg group. In addition, we prove a universal lower bound to the best constants for general Lie groups.

We consider the weighted Hardy spaces H_w^p, w a Muckenhoupt weight and T a singular integral.

The goal of this talk is to give bounds for the norms ||T||_{H_w^p\rightarrow H_w^p} and ||T||_{H_w^p\rightarrow L_w^p} in terms of the characteristic of the weight w. First, we obtain results for weights in A_1 and singular integrals operator of homogeneous type. On the other hand, for certain convolution type operators we obtain bounds for weights in A_{\infty}. This is a joint work with Estefania Dalmasso from IMAL.

In this talk we will present some models for for free boundary incompressible flow. In particular, we will consider the motion of an incompressible irrotational fluid moving according to the Euler equation and the motion of an incompressible fluid in a porous medium moving following either Darcy's Law or Forchheimer's Law. We will show some numerical simulations and also present some mathematical results. These results are in collaboration with Arthur Cheng (NCU), Stefano Scrobogna (BCAM), Steve Shkoller (UC Davis) and Jon Wilkening (UC Berkeley).

The Penrose transform is a simple conformal transformation of the  Minkowski space-time. We show a connection with the best constant  problem for the Strichartz inequalities associated with the wave equation. This also has applications to nonlinear problems; we present a sharp estimate for the cubic wave equation on \(\mathbb R^{1+3}\).