# Trimester Seminar

Venue: HIM lecture hall, Poppelsdorfer Allee 45
Organizers: Herbert Koch, Daniel Tataru, Christoph Thiele

## Monday, May 19

11:00 - 12:00 Guozhen Lu: An alternative proof for the Lp estimates for the multi-parameter Coifman-Meyer Fourier multipliers with limited smoothness
Abstract: In this talk, we will report some recent work on a Hormander type theorem of Lp estimates of the multi-parameter Coifman-Meyer Fourier multiplier operators with limited smoothness on the symbols. We will present an alternative proof of the remarkable Lp estimates established by C. Muscalu, J. Pipher, T. Tao and C. Thiele for the multi-parameter Fourier multiplier (Acta Math. 2004 and Revista Matematica Iberoamericaan 2006). Weighted L estimates for the symbols with limited smoothness are also established. This is joint work with J. Chen.

15:00 - 16:00 Alexandru Ionescu: Water waves - local and global regularity and formation of singularities (Part II)
Abstract: I will discuss some recent work on the water waves equation in 2 dimensions. We will be interested in several aspects: formulation of the evolution problem, the local regularity theory, existence of global solutions, and the dynamical formation of singularities.

## Tuesday, May 20

15:00 - 16:00 Alexandru Ionescu: Water waves - local and global regularity and formation of singularities (Part III)
Abstract: I will discuss some recent work on the water waves equation in 2 dimensions. We will be interested in several aspects: formulation of the evolution problem, the local regularity theory, existence of global solutions, and the dynamical formation of singularities.

## Wednesday, May 21

14:00 - 15:00 Chris Sogge: Focal points and sup-norms of eigenfunctions

15:00 - 16:00 Bartosz Trojan: Littlewood-Paley theory for $\tilde{A}_2$ buildings

## Thursday, May 22

15:00 - 16:00 Andrew Morris: Huygens' Principle for Hyperbolic Equations with $L^\infty$ coefficients via First-Order Systems
Abstract: We prove that strongly continuous groups generated by first-order systems on Riemannian manifolds have finite propagation speed. Our procedure provides a new direct proof for self-adjoint systems that has an extension to operators on metric measure spaces. As an application, we present a new approach to the weak Huygens' principle for second-order hyperbolic equations with $L^\infty$ coefficients. This is joint work with Alan McIntosh.

## Friday, May 23

14:00 - 15:00 David Cruz-Uribe: Regularity of solutions of degenerate p-Laplacian equations

15:00 - 16:00 Mariusz Mirek: Discrete maximal functions in higher dimensions
Abstract: We present a higher dimensional counterpart of Bourgain's pointwise ergodic theorem along an arbitrary integer valued polynomial mapping. We achieve this by proving variational estimates Vr on Lp spaces for all p > 1 and r > max{p,p/(p-1)}. Moreover, we obtain the estimates which are uniform in the coefficients of a polynomial mapping of fixed degree.

## Monday, May 26

15:00 - 16:00 Tuomas Hytönen: Recent advances on weighted estimates in harmonic analysis

## Tuesday, May 27

15:00 - 16:00 Detlef Müller: On Lp-multipliers for subelliptic operators

## Wednesday, May 28

15:00 - 16:00 Jiqiang Zheng: Linear adjoint restriction estimates for cone and paraboloid
Abstract: In this talk, we will present some recent work on the restriction theorem for cone and paraboloid in harmonic analysis. Based on the spherical harmonics expansion and the asymptotic behavior of the Bessel function, we show that a modified linear adjoint restriction estimate holds for all Schwartz functions compactly supported on the cone or paraboloid.

## Thursday, May 29

15:00 - 16:00 Stefan Buschenhenke: A Fourier restriction estimate for a surface of finite type

## Friday, May 30

15:00 - 16:00 Tanja Eisner: A generalisation of the Wiener-Wintner theorem

## Monday, June 2

14:00 - 15:00 Paata Ivanishvili: Bellmann functions and the Young inequality

15:00 - 16:00 Jose Maria Martell: The Dirichlet problem for elliptic systems
with non-smooth data
Abstract:
Take an arbitrary second-order, homogeneous, elliptic system, with constant complex coecients (e.g., the Laplacian or the Lame system of elasticity). Consider the associated Dirichlet problem in the upperhalf space with non-smooth boundary data in some class of functions (e.g., Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions). We present a general method, written in the framework of Kothe function spaces, establishing that the well-posedness of the corresponding boundary value problems is equivalent to the boundedness of the Hardy-Littlewood maximal function. Joint work with D. Mitrea, I. Mitrea, and M. Mitrea.

## Thursday, June 5

14:00 - 15:00 Pierre Portal: Functional calculus of Dirac operators and tent spaces

Abstract: In [2], Axelsson, Keith, and McIntosh have shown that results about Riesz transforms, such as the boundedness of the Cauchy integral on Lipschitz curves, and Kato's square root estimates, can be seen as instances of a perturbation result, in $L^2$, for the holomorphic functional calculus of certain first order differential operators. This perspective has since been proven to be particularly useful for the development of harmonic analysis on manifolds, and in the study of rough boundary value problems. It has been extended from $L^2$ to $L^p$ in two different ways: using an appropriate extrapolation method in [1], or a set of martingale techniques that provide $L^p$ analogues of the key techniques of [2] in [3].

In this talk, we present an alternative set of $L^p$ techniques based on Hardy spaces rather than martingale methods. This turns out to be simpler and give stronger results, and also points out an interesting phenomenon: the heart of the harmonic analysis in [2] actually extends from $L^2$ to $L^p$ for all $p \in (1,\infty)$, while the (necessary) restrictions in $p$ only come from an estimate that is trivial in $L^2$. Our approach is fundamentally based on Coifman-Meyer-Stein's theory of tent spaces, and on the current development of an operator-valued Calderón-Zygmund theory in these spaces.

This is joint work with D. Frey and A. McIntosh.

[1] P. Auscher, On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\mathbb R^n$ and related estimates. (2007

[2] A. Axelsson, S. Keith, A. McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators. (2006) 455--497.

[3] T. Hytönen, A. McIntosh, P. Portal, Kato's square root problem in Banach spaces. (2008) 675--726.

15:00 - 16:00 Dorothee Frey:  Heat kernel regularity on metric measure spaces

## Friday, June 6

15:00 - 16:00 Frederic Bernicot: Dispersive estimates through the heat semigroup and wave kernels

## Monday, June 16

15:00 - 16:00 Steve Hofmann: Uniform rectifiability and elliptic equations

## Tuesday, June 17

15:00 - 16:00 Francesco DiPlinio: An improved multi-frequency Calderon-Zygmund decomposition and applications to modulation invariant singular integrals

Abstract: We describe a strengthening of the multi-frequency Calderon-Zygmund decomposition of Nazarov, Oberlin and Thiele where, loosely speaking, the interaction of the "bad" part, i.e. having mean-zero with respect to N frequencies x1,...,xN,  with functions  localized in frequency near one of the xj is exponentially small in terms of the "good", i.e. L2, part. This decomposition is used to prove that the bilinear Hilbert transform maps into weak L2/3 up to a doubly logarithmic factor. Joint work with C. Thiele

## Thursday, June 19

15:00 - 16:00 Vjekoslav Kovac: Multilinear singular integrals with entangled structure and a few applications

## Friday, June 20

15:00- 16:00 Igor Verbitsky: Sublinear equations and weighted norm inequalities, and new potentials of Wolff type

Abstract: We will present sharp estimates of solutions and regularity results for certain sublinear elliptic equations, as well as their analogues involving the p-Laplacian and fractional Laplacian operators. Related weighted norm inequalities, sublinear analogues of Schur's lemma, and a new class of Wolff-type potentials which control solutions to such equations will be discussed.
This work is joint with Dat Tien Cao.

## Monday, June 23

15:00 - 16:00 Andreas Seeger: Multilinear singular integral forms related to a problem on mixing flows

## Tuesday, June 24

15:00 - 16:00 Michael Lacey: Two weight inequality for the Hilbert transform

## Wednesday, June 25

15:00 - 16:00 Frank Merle: Asymptotics for critical nonlinear dispersive equations and Universality properties

Abstract: We consider various examples of critical nonlinear partial differential equations which have the following common features: they are Hamiltonian, of dispersive nature, have a conservation law invariant by scaling, and have solutions of nonlinear type (their asymptotic behavior in time differs from the behavior of solutions of linear equations). The main questions concern the possible behaviors one can expect asymptotically in time. Are there many possibilities, or on the contrary very few universal behaviors depending on the type of initial data?
We shall see that the asymptotic behavior of solutions starting with general or constrained initial data is related to very few special solutions of the equation. This will be illustrated through different examples related to classical problems.

## Thursday, June 26

15:00 - 16:00 Paul Mueller: Interpolatory Estimates, Riesz Transforms, Haar and Wavelet Projections

## Friday, June 27

10:00 - 11:00 Piero D'Ancona: Global existence of small equivariant wave maps on rotationally symmetric manifolds

Abstract: Joint work with Qidi Zhang (Shanghai). We introduce a class of rotationally invariant manifolds, which we call admissible, on which the wave flow satisfies smoothing and Strichartz estimates. We deduce the global existence of equivariant wave maps from admissible manifolds to general targets, for small initial data of critical regularity $H^{\frac n2}$. The class of admissible manifolds includes in particular asymptotically flat manifolds and perturbations of real hyperbolic spaces $\mathbb{H}^{n} for n\ge3$.

11:00 - 12:00 Jonas Lührmann: On the random data problem for nonlinear wave equations on Euclidean space

Abstract: Local well-posedness for wave equations with a defocusing energy-subcritical power-type nonlinearity on Euclidean space is well understood for initial data of subcritical or critical regularity. Despite significant efforts, global well-posedness has not yet been established down to critically regular initial data. In recent years, probabilistic methods have been used to investigate the behaviour of solutions in regimes where the deterministic techniques fail. In this talk, I will present an almost sure global existence result for such nonlinear wave equations. I will begin by introducing a randomization procedure for initial data in Sobolev spaces of low regularity which uses a unit-scale decomposition in frequency space. Based on improved almost sure space-time integrability properties of the free evolution of the randomized initial data as well as a probabilistic version of the high-low method by Colliander and Oh, we obtain almost sure global existence results. In particular, we will see that there exist global solutions for a large family of initial data with supercritical regularity. This is joint work with Dana Mendelson.

## Monday, June 30

15:00 - 16:00 Paco Villarroya Alvarez: T(1) Theory for compactness of Calderón-Zygmund operators

Abstract: In this talk, I will introduce new results about the characterization of non-convolution singular integral operators that extend compactly on $L^p$ spaces for $1<p<\infty$. I will also present the results in the corresponding endpoint cases and some applications.

## Wednesday, July 2

15:00 - 16:00 Xavier Tolsa: Rectifiability, densities, doubling conditions, and square functions

Abstract: Let E be a set with finite n-dimensional Hausdorff measure $H^n$ in $R^d$. A well known theorem of Preiss asserts that E is n-rectifiable if and only if the density $\lim_{r \to 0} H^n(B(x,r)) / r^n$exists and is different from 0 for $H^1$-almost every $x\in E$. In this talk I will explain some results which can be considered as square function versions of Preiss’ theorem. In particular, in the case n=1 we get that E is 1-rectifiable if and only if $\int_0^1 | \frac{H^1(E\cap B(x,2r))}{H^1(E\cap B(x,r))} - 2 |^2 \frac{dr}r < \infty$ for$H^1$-almost every $x\in E$.

Some of the results that will explain are joint works with Chousionis, Garnett, Le and Toro.

## Monday, July 7

15:00 - 16:00 Dmitriy Stolyarov: Bellman function for integral functionals on classes of functions with small mean oscillation (joint with Paata Ivanisvili, Nikolay Osipov, Vasilii Vayunin, and Pavel Zatitskiy)

Abstract: In 2003, V. I. Vasyunin found sharp constants in reverse Holder inequality for Muckenhoupt weights on an interval. The proof was based on solving a specific two-dimensional extremal boundary value problem that was dual to the initial infinite-dimensional problem, i.e. finding the exact Bellman function. Since then, his method has been transfered to many other inequalities (for example, various forms of the John--Nirenberg inequality) by Vasyunin, Slavin, Volberg, etc.. However, each time, it was a transfer, i.e. there was no general framework for such-type problems. Each time the researcher had to do lots of computations and magic guesses to solve the corresponding boundary value problem. We will give a general theory that includes the preceeding development. It occurs that notions from elementary differential and convex geometry (such as the torsion, the curvature, the Caratheodory theorem, etc.) are natural in dealing with these sort of Bellman functions. We also give a proof of the duality theorem between the two extremal problems (the initial extremal problem on a class of functions and a finite-dimensional Bellman function problem) that lies in the heart of the theory. Introduction of an appropriate class of martingales and a certain Bellman function on them is the key to the proof.

## Thursday, July 10

15:00 - 16-00 Maria Carmen Reguera: Lower bounds for fractional Riesz transforms on general Cantor sets

## Friday, July 11

15:00 - 16:00 Blazej Wrobel: Dimension free Lp estimates for Riesz transforms via an H infinity joint functional calculus
Abstract: In 1983 E. M. Stein proved dimension free $L^p$ bounds for the vector of classical Riesz transforms on $R^d$. Since then many authors studied the phenomenon of dimension free estimates for (vectors of) Riesz transforms defined in various contexts. In this talk we present a fairly general scheme for deducing the dimension free $L^p$ boundedness of single d-dimensional Riesz transforms from the $L^p$ boundedness of one-dimensional Riesz transforms. The crucial tool we use is an $H^{\infty}$ joint functional calculus for a pair of strongly commuting operators. The scheme is applicable to all Riesz transforms acting on 'product' spaces, e.g.: Riesz transforms connected with (classical) multi-dimensional orthogonal expansions and discrete Riesz transforms on products of groups having polynomial growth.

## Tuesday, July 22

15:00 - 16:00 Klaus Widmayer: First results on the global behavior of the anharmonic oscillator

## Wednesday, July 23

15:00 - 16:00 Jonathan Hickman: Uniform estimates for dilated averages over curves

## Thursday, July 24

15:00 - 16:00 Ioannis Angelopoulos: Spherically symmetric nonlinear waves on extremal Reissner-Nordström spacetimes

## Friday, July 25

15:00 - 16:00 Benjamin Krause: An approach to pointwise ergodic theorems

## Monday, July 28

15:00-16:00 Nguyen Cong Phuc: The Navier-Stokes equations in nonendpoint borderline Lorentz spaces

Abstract: It is shown both locally and globally that $L_t^{\infty}(L_x^{3,q})$ solutions to the three-dimensional Navier-Stokes equations are regular provided $q\not=\infty$. Here $L_x^{3,q}$, $0, is an increasing scale of Lorentz spaces containing $L^3_x$. Thus the result provides an improvement of a result by Escauriaza, Seregin and Sverák, which treats the case $q=3$. A new local energy bound and a new $\epsilon$-regularity criterion are combined with the backward uniqueness theory for parabolic equations to obtain the result.

## Tuesday, July 29

15:00-16:00 Eyvindur Palsson: Finite point configurations and multilinear Radon transforms

Abstract: As big data sets have become more common, there has been significant interest in finding and understanding patterns in them. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will present recent Falconer type theorems, established by myself and my collaborators, for a wide class of finite point configurations in any dimension. The techniques we used come from analysis and geometric measure theory, and the key step was to obtain bounds on multilinear analogues of generalized Radon transforms. Further study of these multilinear operators lead to a variety of interesting applications, such as multilinear analogs of Stein's spherical maximal theorem.

## Wednesday, July 30

15:00-16:00 Haakan Hedenmalm: tba

## Thursday, July 31

15:00-16:00 Oliver Dragicevic: On spectral multipliers and a heat flow method

## Friday, August 1

15:00-16:00 Brian Street: Multiparameter Singular Integrals

## Friday, August 15

11:00- 12:00 Michael Struwe: Scattering for a critical nonlinear wave equation in 2 space dimensions

## Monday, August 18

13:00 - 14:30 Lena Gal: On multiple Zeta values

15:00 - 16:00 Arpad Benyi: Modulation Spaces and Their Applications

Abstract: We provide a brief introduction to the time-frequency analysis surrounding the so-called modulation spaces and indicate its applications to topics spanning pseudo-differential operators, PDEs and the regularity of Brownian motion.

## Tuesday, August 19

15:00 - 16:00 Xi Chen: Spectral measure on asymptotically hyperbolic space

## Thursday, August 21

15:00 - 16:00 Felipe Gonçalves: Band-Limited Approximations and Interpolation Formulas

Abstract: In this talk we construct functions of exponential type that best approximates a given function by above (or by below). Also we explain the connection of this problem with the problem of reconstruct a function of exponential type from its values and the values of its derivative over a given sequence of real points