# Analysis Seminar

Analysis Seminar

The seminar will meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.

# 2017-2018 Analysis Seminar Schedule

 date speaker title host(s) institution September 8 in B239 (Colloquium) Tess Anderson UW Madison A Spherical Maximal Function along the Primes Tonghai September 19 Brian Street UW Madison Convenient Coordinates Betsy September 26 Hiroyoshi Mitake Hiroshima University Derivation of multi-layered interface system and its application Hung October 3 Joris Roos UW Madison A polynomial Roth theorem on the real line Betsy October 10 Michael Greenblatt UI Chicago Maximal averages and Radon transforms for two-dimensional hypersurfaces Andreas October 17 David Beltran Basque Center of Applied Mathematics Fefferman-Stein inequalities Andreas Wednesday, October 18, 4:00 p.m. in B131 Jonathan Hickman University of Chicago Factorising X^n Andreas October 24 Xiaochun Li UIUC Recent progress on the pointwise convergence problems of Schroedinger equations Betsy Thursday, October 26, 4:30 p.m. in B139 Fedor Nazarov Kent State University The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp Sergey, Andreas Friday, October 27, 4:00 p.m. in B239 Stefanie Petermichl University of Toulouse Higher order Journé commutators Betsy, Andreas Wednesday, November 1, 4:00 p.m. in B239 (Colloquium) Shaoming Guo Indiana University Parsell-Vinogradov systems in higher dimensions Andreas November 14 Naser Talebizadeh Sardari UW Madison Quadratic forms and the semiclassical eigenfunction hypothesis Betsy November 28 Xianghong Chen UW Milwaukee Some transfer operators on the circle with trigonometric weights Betsy Monday, December 4, 4:00, B139 Bartosz Langowski and Tomasz Szarek Institute of Mathematics, Polish Academy of Sciences Discrete Harmonic Analysis in the Non-Commutative Setting Betsy Wednesday, December 13, 4:00, B239 (Colloquium) Bobby Wilson MIT Projections in Banach Spaces and Harmonic Analysis Andreas Monday, February 5, 3:00-3:50, B341 (PDE-GA seminar) Andreas Seeger UW Singular integrals and a problem on mixing flows February 6 Dong Dong UIUC Hibert transforms in a 3 by 3 matrix and applications in number theory Betsy February 13 Sergey Denisov UW Madison Spectral Szegő theorem on the real line February 20 Ruixiang Zhang IAS (Princeton) The (Euclidean) Fractal Uncertainty Principle Betsy, Jordan, Andreas February 27 Detlef Müller University of Kiel On Fourier restriction for a non-quadratic hyperbolic surface Betsy, Andreas Wednesday, March 7, 4:00 p.m. Winfried Sickel Friedrich-Schiller-Universität Jena On the regularity of compositions of functions Andreas March 20 Betsy Stovall UW Two endpoint bounds via inverse problems April 10 Martina Neuman UC Berkeley Gowers-Host-Kra norms and Gowers structure on Euclidean spaces Betsy Friday, April 13, 4:00 p.m. (Colloquium, 911 VV) Jill Pipher Brown Mathematical ideas in cryptography WIMAW April 17 Title April 24 Lenka Slavíková University of Missouri $L^2 \times L^2 \to L^1$ boundedness criteria Betsy, Andreas May 1 Xianghong Gong UW Smooth equivalence of deformations of domains in complex euclidean spaces May 7 in B223 Ebru Toprak UIUC Dispersive estimates for massive Dirac equations Betsy May 15 Gennady Uraltsev Cornell TBA Andreas, Betsy May 16-18, Workshop in Fourier Analysis Betsy, Andreas

# Abstracts

### Brian Street

Title: Convenient Coordinates

Abstract: We discuss the method of picking a convenient coordinate system adapted to vector fields. Let X_1,...,X_q be either real or complex C^1 vector fields. We discuss the question of when there is a coordinate system in which the vector fields are smoother (e.g., C^m, or C^\infty, or real analytic). By answering this in a quantitative way, we obtain coordinate charts which can be used as generalized scaling maps. When the vector fields are real this is joint work with Stovall, and continues in the line of quantitative sub-Riemannian geometry initiated by Nagel, Stein, and Wainger. When the vector fields are complex one obtains a geometry with more structure which can be thought of as "sub-Hermitian".

### Hiroyoshi Mitake

Title: Derivation of multi-layered interface system and its application

Abstract: In this talk, I will propose a multi-layered interface system which can be formally derived by the singular limit of the weakly coupled system of the Allen-Cahn equation. By using the level set approach, this system can be written as a quasi-monotone degenerate parabolic system. We give results of the well-posedness of viscosity solutions, and study the singularity of each layers. This is a joint work with H. Ninomiya, K. Todoroki.

### Joris Roos

Title: A polynomial Roth theorem on the real line

Abstract: For a polynomial P of degree greater than one, we show the existence of patterns of the form (x,x+t,x+P(t)) with a gap estimate on t in positive density subsets of the reals. This is an extension of an earlier result of Bourgain. Our proof is a combination of Bourgain’s approach and more recent methods that were originally developed for the study of the bilinear Hilbert transform along curves. This talk is based on a joint work with Polona Durcik and Shaoming Guo.

### Michael Greenblatt

Title: Maximal averages and Radon transforms for two-dimensional hypersurfaces

Abstract: A general local result concerning L^p boundedness of maximal averages over 2D hypersurfaces is described, where p > 2. The surfaces are allowed to have either the traditional smooth density function or a singularity growing as |(x,y)|^{-t} for some 0 < t < 2. This result is a generalization of a theorem of Ikromov, Kempe, and Mueller. Similar methods can be used to show sharp L^p to L^p_a Sobolev estimates for associated Radon transform operators when p is in a certain interval containing 2.

### David Beltran

Title: Fefferman Stein Inequalities

Abstract: Given an operator T, we focus on obtaining two-weighted inequalities in which the weights are related via certain maximal function. These inequalites, which originated in work of Fefferman and Stein, have been established in an optimal way for different classical operators in Harmonic Analysis. In this talk, we survey some classical results and we present some recent Fefferman-Stein inequalities for pseudodifferential operators and for the solution operators to dispersive equations.

### Jonathan Hickman

Title: Factorising X^n.

Question: how many ways can the polynomial $X^n$ be factorised as a product of linear factors? Answer: it depends on the ring... In this talk I will describe joint work with Jim Wright investigating certain exponential sum estimates over rings of integers modulo N. This theory serves as a discrete analogue of the (euclidean) Fourier restriction problem, a central question in contemporary harmonic analysis. In particular, as part of this study, the question of counting the number of factorisations of polynomials over such rings naturally arises. I will describe how these number-theoretic considerations can themselves be approached via methods from harmonic analysis.

### Xiaochun Li

Title: Recent progress on the pointwise convergence problems of Schrodinger equations

Abstract: Recently, Guth, Du and I solved the pointwise convergence problem of Schrodinger equations in two-dimensional case. We proved that the solution to free Schrodinger equation in R^2 converges to its initial data, provided the initial data belongs to H^s for s larger than 1/3. This result is sharp, up to the end point, due to Bourgain's example. The proof relies on the polynomial partitioning method and the decoupling method. In addition, the pointwise convergence problem is closely related to Fourier restriction conjecture.

### Fedor Nazarov

Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden conjecture is sharp.

Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for the norm of the Hilbert transform on the line as an operator from $L^1(w)$ to $L^{1,\infty}(w)$ cannot be improved in general. This is a joint work with Andrei Lerner and Sheldy Ombrosi.

### Stefanie Petermichl

Title: Higher order Journé commutators

Abstract: We consider questions that stem from operator theory via Hankel and Toeplitz forms and target (weak) factorisation of Hardy spaces. In more basic terms, let us consider a function on the unit circle in its Fourier representation. Let P_+ denote the projection onto non-negative and P_- onto negative frequencies. Let b denote multiplication by the symbol function b. It is a classical theorem by Nehari that the composed operator P_+ b P_- is bounded on L^2 if and only if b is in an appropriate space of functions of bounded mean oscillation. The necessity makes use of a classical factorisation theorem of complex function theory on the disk. This type of question can be reformulated in terms of commutators [b,H]=bH-Hb with the Hilbert transform H=P_+ - P_- . Whenever factorisation is absent, such as in the real variable setting, in the multi-parameter setting or other, these classifications can be very difficult.

Such lines were begun by Coifman, Rochberg, Weiss (real variables) and by Cotlar, Ferguson, Sadosky (multi-parameter) of characterisation of spaces of bounded mean oscillation via L^p boundedness of commutators. We present here an endpoint to this theory, bringing all such characterisation results under one roof.

The tools used go deep into modern advances in dyadic harmonic analysis, while preserving the Ansatz from classical operator theory.

### Shaoming Guo

Title: Parsell-Vinogradov systems in higher dimensions

Abstract: I will present a few results on counting the numbers of integer solutions of Parsell-Vinogradov systems in higher dimensions. Applications to Waring’s problem and to the problem of counting rational linear subspaces lying on certain hyper-surface will be discussed. Joint works with Jean Bourgain, Ciprian Demeter and Ruixiang Zhang.

Title: Quadratic forms and the semiclassical eigenfunction hypothesis

Abstract: Let $Q(X)$ be any integral primitive positive definite quadratic form in $k$ variables, where $k\geq4$, and discriminant $D$. For any integer $n$, we give an upper bound on the number of integral solutions of $Q(X)=n$ in terms of $n$, $k$, and $D$. As a corollary, we prove a conjecture of Lester and Rudnick on the small scale equidistribution of almost all functions belonging to any orthonormal basis of a given eigenspace of the Laplacian on the flat torus $\mathbb{T}^d$ for $d\geq 5$. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.

### Xianghong Chen

Title: Some transfer operators on the circle with trigonometric weights

Abstract: A transfer operator is an averaging operator over the preimages of a given map. Certain dynamical properties of the map can be studied through its associated transfer operator. In this talk we will introduce a class of weighted transfer operators associated to the Bernoulli maps on the circle (i.e. multiplication by a given integer, mod 1). We will illustrate how the spectral properties of these operators may depend on the specific weight chosen and demonstrate multiple phase transitions. We also present some results on evaluating the spectral radii and corresponding eigenfunctions of these operators, as well as their connections to Fourier analysis. This is joint work with Hans Volkmer.

### Bobby Wilson

Title: Projections in Banach Spaces and Harmonic Analysis

Abstract: In this talk, we will discuss the measure theoretic principles of orthogonal projections that follow from the classical Besicovitch-Federer projection theorem. The Besicovitch-Federer projection theorem offers a characterization of rectifiability of one-dimensional sets in R^d by the size of their projections to lines. We will focus on the validity of analogues to the Besicovitch-Federer projection theorem with respect to such sets in general Banach spaces. In particular, we will show that the projection theorem is false when the Banach space is infinite-dimensional and discuss related applications to questions in Harmonic Analysis. This is joint work with Marianna Csornyei and David Bate.

### Andreas Seeger

Title: Singular integrals and a problem on mixing flows

Abstract: The talk will be about results related to Bressan's mixing problem. We present an inequality for the change of a Bianchini semi-norm of characteristic functions under the flow generated by a divergence free time dependent vector field. The approach leads to a bilinear singular integral operator for which one proves bounds on Hardy spaces. This is joint work with Mahir Hadžić, Charles Smart and Brian Street.

### Dong Dong

Title: Hibert transforms in a 3 by 3 matrix and applications in number theory

Abstract: This talk could interest both analysts and number theorists. I will first present 35 variants of Hilbert transforms, with a focus on their connections with ergodic theory, number theory, and combinatorics. Then I will show how to use Fourier analysis tools to reduce a number theory problem (Roth theorem) to an algebraic geometry problem: this joint work Li and Sawin fully answers a question of Bourgain and Chang about three-term polynomial progressions in subsets of finite fields. I guarantee that a second-year graduate student can understand at least 50% of the talk.

### Sergey Denisov

Title: Spectral Szegő theorem on the real line

Abstract: For even measures on the real line, we give the criterion for the logarithmic integral to converge in terms of the corresponding De-Branges system (or Krein's string). The applications to probability (linear prediction for stationary Gaussian processes) will be explained. This is the joint result with R. Bessonov.

### Ruixiang Zhang

Title: The (Euclidean) Fractal Uncertainty Principle

Abstract: On the real line, a version of the uncertainty principle says: If a nonzero function f has its Fourier support lying in B and |A||B| is much smaller than 1, then the L^2 norm of f on A cannot be close to the whole L^2 norm of f. Recently, Bourgain and Dyatlov proved a Fractal Uncertainty Principle (FUP) which has a similar statement. The difference is that in FUP the product of |A| and |B| can be much bigger, but A and B both have to be porous at many scales. We will introduce the theorem and then discuss some unusual features of its proof, most notably the application of the Beurling-Malliavin Theorem. In the original work the dependence on the dimensions of both fractals was ineffective. We will also discuss why we can overcome this ineffectivity (joint work with Long Jin).

### Detlef Müller

Title: On Fourier restriction for a non-quadratic hyperbolic surface

Abstract: In contrast to what is known about Fourier restriction for elliptic surfaces, rather little is known about hyperbolic surfaces. Hitherto, basically only the quadric $z=xy$ had been studied successfully. In my talk, after giving some background on Fourier restriction, I shall report on recent joint work with S. Buschenhenke and A. Vargas on a cubic perturbation of this quadric. Our analysis reveals that the geometry of the problem changes drastically in the presence of a perturbation term, and that new techniques, compared to the elliptic case, are required to handle more general hyperbolic surfaces.

### Winfried Sickel

Title: On the regularity of compositions of functions

Abstract: Let $E$ denote a Banach space of locally integrable functions on $\mathbb{R}$. To each continuous function $f:\mathbb{R} \to \mathbb{R}$ we associate the composition operator $T_f(g):= f\circ g$, $g\in E$. The properties of $T_f$ strongly depend on the chosen function space $E$. In my talk I will concentrate on Sobolev spaces $W^m_p$ and Slobodeckij spaces $W^s_p$. The main aim will consist in giving a survey on necessary and sufficient conditions on $f$ such that the composition operator maps such a space $E$ into itself.

### Martina Neuman

Title: Gowers-Host-Kra norms and Gowers structure on Euclidean spaces

Abstract: The investigation on Brascamp-Lieb data - their structure, their extremizability, their stability and regularity of their constants - has been an active one in Harmonic Analysis. In this talk, I'll present an example of a Brascamp-Lieb structure: a so-called Gowers structure on Euclidean spaces, together with the related Gowers-Host-Kra norms - these were originally tools in additive combinatorics context. I'll dissertate on what happens when a function nearly achieves its Gowers-Host-Kra norm in a Euclidean context - this can be seen as continuation of the work of Eisner-Tao - and a related stability result of the Gowers structure on Euclidean spaces.

### Jill Pipher

Title: Mathematical ideas in cryptography

Abstract: This talk does not assume prior knowledge of public key crypto (PKC). I'll talk about the history of the subject and some current areas of research, including homomorphic encryption.

### Lenka Slavíková

Title: $L^2 \times L^2 \to L^1$ boundedness criteria

Abstract: It is a consequence of Plancherel's identity that a linear multiplier operator associated with a function $m$ is bounded from $L^2$ to itself if and only if $m$ belongs to the space $L^\infty$. In this talk we will investigate the $L^2 \times L^2 \to L^1$ boundedness of bilinear multiplier operators which is as central in the bilinear theory as the $L^2$ boundedness is in the linear multiplier theory. We will present a sharp $L^2 \times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this function; precisely we will show that boundedness holds if and only if $q<4$. We will then discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. This is a joint work with L. Grafakos and D. He.

### Xianghong Gong

Title: Smooth equivalence of deformations of domains in complex euclidean spaces

Abstract: We prove that two smooth families of 2-connected domains in the complex plane are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct two smooth families of smoothly bounded domains in C^n for n>=1 that are equivalent under discontinuous families of biholomorphisms but not under any continuous family of biholomorphisms. Finally, we give sufficient conditions for the smooth equivalence of two smooth families of domains. This is joint work with Hervé Gaussier.

### Ebru Toprak

Title: Dispersive estimates for massive Dirac equations

Abstract: In this talk, I will cover some existing L^1 \rightarrow L^\infty dispersive estimates for the linear Schr\"odinger equation with potential and present a related study on the two and three dimensional massive Dirac equation. In two dimension, we show that the t^{-1} decay rate holds if the threshold energies are regular or if there are s-wave resonances at the threshold. We further show that, if the threshold energies are regular then a faster decay rate of t^{-1}(\log t)^{-2} is attained for large t, at the cost of logarithmic spatial weights, which is not the case for the free Dirac equation. In three dimension, we show that the solution operator is composed of a finite rank operator that decays at the rate t^{-1/2} plus a term that decays at the rate t^{-3/2}. This is a joint work with M.Burak Erdo\u{g}an and William Green.