**Winter Seminars on Analysis and Differential Equations.**

Research Area A: Geometry of differential operators: from local to global properties

Seminars are every Friday at 14:15 in Raum 0.011, Endenicher Allee 60.

Kalender

**OCTOBER**

**12 - Organizing day**

**19 - Immanuel Zachhuber (Universität Bonn)**

*The continuous Anderson Hamiltonian and related stochastic evolution equations.**Abstract*: We analyse nonlinear Schrödinger and wave equations whose linear part is given by the renormalised Anderson Hamiltonian in two and three dimensional periodic domains.

**26 - João Ramos (Universität Bonn)**

*Maximal functions and two-dimensional restriction, combined.**Abstract*: The restriction conjecture for the Fourier transform has been a very active topic in Fourier analysis for over the past 40 years, with new machinery being developed continuously to deal with its subtleties. However, its most basic manifestation, the two-dimensional case, has been solved for over 40 years, as the paper by Carleson and Sjölin sheds light into why the phenomenon occurs. More specifically, if 1 <= p <4/3, they show that it make sense to talk about the restriction to the unit circle of the Fourier transform of a function in L^p.

Nevertheless, in 2016, Müller, Ricci and Wright have considered a different, yet related, stronger property: what can be actually said about the pointwise definition of the restriction operator? Is there a suitable notion of Lebesgue points of the Fourier transform over a curve? As their result would show, the answer is affirmative: in the restricted range 1 <= p < 8/7, besides having a restriction operator, they show that almost every point of the unit circle is a Lebesgue point for the Fourier transform, with respect to the (affine) arc length measure.

The aim of this talk is to extend the Müller-Ricci-Wright result to the whole range of exponents 1 <= p <4/3. We do so by finding a clever way to introduce a bilinearization of our operator, along with the Carleson-Sjölin machinery, which allows us to bypass several technical details contained in their paper. Time allowing, we shall also discuss some interesting open problems and high-dimensional generalizations.

**NOVEMBER**

**2 - Danylo Radchenko (Max-Planck-Institut für Mathematik)**

*Exotic Fourier summation formulas**Abstract*: I will talk about a family of summation formulas for functions on the real line. As special cases, this family contains the classical Poisson summation formula and also a curious summation formula originally discovered by Guinand. These summation formulas arise naturally in a certain interpolation problem for Schwartz functions, and can be explicitly described using modular forms for the Hecke theta group, and the Gaussian hypergeometric function.

**9 - Pekka Pankka (University of Helsinki)**

*Constructions of quasiregular mappings in higher dimensions**Abstract*: In his 1985 paper on the sharpness of the Picard theorem for quasiregular mappings, Rickman introduced new piece-wise linear methods to construct quasiregular mappings in the 3-space. One of the methods is deformation of 2-dimensional Alexander maps, that is, a branched cover extension method for piece-wise linear branched covers from a (planar) surface to the 2-sphere having three critical values. In this talk I will discuss a version of Rickman's deformation theory in higher dimensions and its role in extending various constructions of Rickman, and Heinonen and Rickman beyond dimension 3. This is joint work with Jang-Mei Wu.

**16 - Polona Durcik (Caltech)**

*Singular Brascamp-Lieb inequalities**Abstract*: Brascamp-Lieb inequalities are L^p estimates for certain multilinear forms on functions on Euclidean spaces. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions is replaced by a Calderon-Zygmund kernel. We focus on a family of multilinear forms on R^n with a certain cubical structure and discuss some L^p estimates for them. Joint work with C. Thiele.

**23 - Marco Vitturi (Université de Nantes)**

*Maximally truncated bilinear square functions**Abstract*: We bring together two topics from time-frequency analysis: on the one hand, maximal truncations of bilinear singular integrals as originally studied by Lacey in relation to the bilinear Hardy-Littlewood maximal function; on the other hand, bilinear square functions as studied by Bernicot - whose associated multipliers are characteristic functions of identical diagonal strips in the frequency plane that are uniformly spaced. We combine the two objects into maximally truncated bilinear square functions and prove boundedness in the local L^2 range (sharp, up to the endpoints), with constant logarithmic in the number of strips considered. This serves two purposes: one is to provide some tools that should help in studying the general problem of bilinear square functions (for arbitrary strips), which is still wide open; the other is to make a first step towards establishing a bilinear version of an important lemma by Bourgain for maximal Fourier projections. Joint work with Cristina Benea.

**30 - Juan José Marín (ICMAT)**

*Boundary value problems for elliptic systems on SKT domains**Abstract*: We use the theory of layer potentials to study boundary value problems on bounded SKT domains, a family of domains introduced by S. Semmes, C. Kenig and T. Toro. We consider the case of elliptic systems with constant coefficients in which the boundary data belongs to L^p(Bd Ω,w), where w is in A_p(Bd Ω) is a Muckenhoupt weight. This extends previous results by S. Hofmann, M. Mitrea and M. Taylor. This approach relies on the invertibility of layer potentials defined on the boundary of the domain and therefore several tools are developed for this purpose, such as a quantitative interpolation theorem for compact operators. Moreover, these techniques can also be used for the cases of boundary data in other spaces, such as variable Lebesgue spaces or rearrangement-invariant Banach function spaces. Finally, we will discuss how to adapt this theory to the case in which Bd Ω is unbounded. This is joint work with J.M. Martell, D. Mitrea and M. Mitrea.

**DECEMBER**

**7 - Pavel Zorin-Kranich (Universität Bonn)**

*Decoupling for moment manifolds**Abstract*: Using the parabola as an example I will indicate how multilinear estimates and transversality appear in the Bourgain-Demeter argument for decoupling inequalities. Understanding transversality in high dimension and codimension remains a challenge. I will report on the latest progress in this direction obtained in joint work with S. Guo.

**14 - Gael Diebou (Universität Bonn)**

*Dirichlet Problems for Weakly Harmonic Maps With Rough Data**Abstract*: Given a map u:M-->N between two Riemannian manifolds, one can associate a natural object called energy. Critical points of the latter (in the class of mappings with finite energies) are known as weakly harmonic maps (WHM). Equivalently, they are defined as solutions in the sense of distributions of the associated Euler-Lagrange equation. In this talk, we present new results that pertain to the solvability of the Dirichlet problems for WHM from the upper half-space to any arbitrary compact manifold N with rough data (measurable bounded and BMO). Unlike earlier existing contributions (where the smoothness of data is required) which are essentially based on the direct method of calculus of variations, our method deals with a non-variational approach. The techniques that we will present, in the framework of Elliptic PDEs are new and originate from the celebrated work of Koch and Tataru on the well-posedness for the Navier-Stokes equations. If time allows, we will show how our approach extends beyond the context of the harmonic maps by considering a much larger class of nonlinearities.

**21 - Anne Franzen (Technical University of Lisbon)**

*Flat Friedmann-Lemaitre-Robertson-Walker and Kasner Big Bang Singularities analysed on the level of scalar waves**Abstract*:We consider the wave equation, Box_g Psi=0, in fixed flat Friedmann-Lemaitre-Robertson-Walker and Kasner spacetimes with topology R_+ X T^3. We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface {t=0}. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate L^2-sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the L^2(T^3) norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

**JANUARY**

**11 - Joris Roos (University of Wisconsin-Madison)**

*A maximal operator associated with families of parabolas**Abstract*: Let $H^{(u)}$ be the Hilbert transform along the parabola $(t,ut^2)$ where $u$ is a real number. This talk will be about the maximal operator $\mathcal{H}^U f = \sup_{u\in U} |H^{(u)} f|$, where $U$ is any set of positive reals. We obtain essentially optimal lower and upper bounds for the $L^p$ operator norm of $\mathcal{H}^U$ when $p\in (2,\infty)$. Our methods also apply to curves $(t,ut^b)$ for any real number $b>1$. This is joint work with Shaoming Guo, Andreas Seeger and Po-Lam Yung.

**18 - Mateus Sousa (Universität München)**

*Recent progress in sharp Fourier restriction theory**Abstract*: In this talk we will consider some current research problems in sharp Fourier restriction theory. We will discuss the problem of finding sharp constants for restriction estimates, as well as the questions of existence and classification of extremizers of these estimates. The main focus will be on the recent developments related to restriction inequalities for hyperboloids and spheres.

**18 - Patrick Gérard (Université Paris-Sud)**

*A nonlinear Fourier transform for the Benjamin-Ono equation on the circle**Abstract*: The Benjamin-Ono equation is a a dispersive equation in one space dimension which admits a Lax pair involving non local operators on the Hardy space. I will report on some work in progress, in collaboration with Thomas Kappeler, devoted to the spectral theory of such operators on the circle, leading to the construction of a nonlinear Fourier transform for the Benjamin-Ono evolution.

**25 - Panu Lahti (Universität Augsburg)**

*BV functions and Federer's characterization of sets of finite perimeter in metric spaces**Abstract*: We consider the theory of functions of bounded variation (BV functions) in the general setting of a complete metric space equipped with a doubling measure and supporting a PoincarĂ© inequality. Such a theory was first developed by Ambrosio (2002) and Miranda (2003). I will give an overview of the basic theory and then discuss a metric space proof of Federer's characterization of sets of finite perimeter, i.e. sets whose characteristic functions are BV functions. This characterization states that a set is of finite perimeter if and only if the n-1-dimensional (in metric spaces, codimension one) Hausdorff measure of the set's measure-theoretic boundary is finite. The proof relies on fine potential theory in the case p=1, much of which seems to be new even in Euclidean spaces.

**FEBRUARY**

**1 - Olaf Post (Universität Trier)**

*Convergence of resonances of manifolds shrinking to a metric graph**Abstract*: In this talk, we give an overview on results concerning Laplacians on a family of non-compact manifolds with cylindrical ends shrinking towards a metric graph. We show convergence of spectra, resolvents and resonances. One problem arising here is that the operators act in different spaces. The resonances are defined via the so-called complex scaling method.

**1 - Giuseppe Negro (Basque Center for Applied Mathematics)**

*Sharp estimates for linear and nonlinear wave equations via the Penrose transform**Abstract*: The Penrose transform is a conformal compactification 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 R^(1+3).