Difference between revisions of "Analysis Seminar"

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'''Analysis Seminar
+
'''Fall 2019 and Spring 2020 Analysis Seminar Series
 
'''
 
'''
  
 
The seminar will  meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
 
The seminar will  meet Tuesdays, 4:00 p.m. in VV B139, unless otherwise indicated.
  
If you wish to invite a speaker please  contact  Betsy at stovall(at)math
+
If you wish to invite a speaker please  contact  Brian at street(at)math
  
 
===[[Previous Analysis seminars]]===
 
===[[Previous Analysis seminars]]===
  
= 2017-2018 Analysis Seminar Schedule =
+
= Analysis Seminar Schedule =
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
Line 16: Line 16:
 
!align="left" | host(s)
 
!align="left" | host(s)
 
|-
 
|-
|September 8 in B239 (Colloquium)
+
|Sept 10
| Tess Anderson
+
| José Madrid
| UW Madison
+
| UCLA
|[[#linktoabstract A Spherical Maximal Function along the Primes]]
+
|[[#José Madrid On the regularity of maximal operators on Sobolev Spaces ]]
|Tonghai
+
| Andreas, David
|-
 
|September 19
 
| Brian Street
 
| UW Madison
 
|[[#Brian Street  |  Convenient Coordinates ]]
 
| Betsy
 
 
|-
 
|-
|September 26
+
|Sept 13 (Friday, B139)
| Hiroyoshi Mitake
+
| Yakun Xi
| Hiroshima University
+
| University of  Rochester
|[[#Hiroyoshi Mitake Derivation of multi-layered interface system and its application ]]
+
|[[#Yakun Xi Distance sets on Riemannian surfaces and microlocal decoupling inequalities ]]
| Hung
+
| Shaoming
 
|-
 
|-
|October 3
+
|Sept 17
 
| Joris Roos
 
| Joris Roos
 
| UW Madison
 
| UW Madison
|[[#Joris Roos  |  A polynomial Roth theorem on the real line ]]
+
|[[#Joris Roos  |  L^p improving estimates for maximal spherical averages ]]
| Betsy
+
| Brian
 
|-
 
|-
|October 10
+
|Sept 20 (2:25 PM Friday, Room B139 VV)
| Michael Greenblatt
+
| Xiaojun Huang
| UI Chicago
+
| Rutgers University–New Brunswick
|[[#Michael Greenblatt Maximal averages and Radon transforms for two-dimensional hypersurfaces ]]
+
|[[#linktoabstract A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries ]]
| Andreas
+
| Xianghong
 
|-
 
|-
|October 17
+
|Oct 1
| David Beltran
+
| Xiaocheng Li
| Basque Center of Applied Mathematics
+
| UW Madison
|[[#David Beltran |   Fefferman-Stein inequalities ]]
+
|[[#Xiaocheng Li | An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ ]]
| Andreas
+
| Simon
|-
 
|Wednesday, October 18, 4:00 p.m.  in B131
 
|Jonathan Hickman
 
|University of Chicago
 
|[[#Jonathan Hickman  |  Factorising X^n  ]]
 
|Andreas
 
 
|-
 
|-
|October 24
+
|Oct 8
| Xiaochun Li
+
| Jeff Galkowski
| UIUC
+
| Northeastern University
|[[#Xiaochun Li Recent progress on the pointwise convergence problems of Schroedinger equations ]]
+
|[[#Jeff Galkowski Concentration and Growth of Laplace Eigenfunctions ]]
 
| Betsy
 
| Betsy
 
|-
 
|-
|Thursday, October 26, 4:30 p.m. in B139
+
|Oct 15
| Fedor Nazarov
+
| David Beltran
| Kent State University
+
| UW Madison
|[[#Fedor Nazarov | The Lerner-Ombrosi-Perez bound in the Muckenhoupt Wheeden conjecture is sharp  ]]
+
|[[#David Beltran |   Regularity of the centered fractional maximal function ]]
| Sergey, Andreas
+
| Brian
 
|-
 
|-
|Friday, October 27, 4:00 p.m.  in B239
+
|Oct 22
| Stefanie Petermichl
+
| Laurent Stolovitch
| University of Toulouse
+
| University of Côte d'Azur
|[[#Stefanie Petermichl | Higher order Journé commutators  ]]
+
|[[#Laurent Stolovitch | Linearization of neighborhoods of embeddings of complex compact manifolds ]]
| Betsy, Andreas
+
| Xianghong
 
|-
 
|-
|Wednesday, November 1, 4:00 p.m. in B239 (Colloquium)
+
|<b>Wednesday Oct 23 in B129</b>
| Shaoming Guo
+
|Dominique Kemp
| Indiana University
+
|Indiana University
|[[#Shaoming Guo  |   Parsell-Vinogradov systems in higher dimensions ]]
+
|[[#Dominique Kemp | Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature]]
| Andreas
+
|Betsy
 
|-
 
|-
|November 14
+
|Oct 29
| Naser Talebizadeh Sardari
+
| Bingyang Hu
 
| UW Madison
 
| UW Madison
|[[#Naser Talebizadeh Sardari Quadratic forms and the semiclassical eigenfunction hypothesis ]]
+
|[[#Bingyang Hu  |    Sparse bounds of singular Radon transforms]]
 +
| Street
 +
|-
 +
|Nov 5
 +
| Kevin O'Neill
 +
| UC Davis
 +
|[[#Kevin O'Neill A Quantitative Stability Theorem for Convolution on the Heisenberg Group ]]
 
| Betsy
 
| Betsy
 
|-
 
|-
|November 28
+
|Nov 12
| Xianghong Chen
+
| Francesco di Plinio
| UW Milwaukee
+
| Washington University in St. Louis
|[[#Xianghong Chen Some transfer operators on the circle with trigonometric weights ]]
+
|[[#Francesco di Plinio Maximal directional integrals along algebraic and lacunary sets]]
| Betsy
+
| Shaoming
 
|-
 
|-
|Monday, December 4, 4:00, B139
+
|Nov 13 (Wednesday)
| Bartosz Langowski and Tomasz Szarek
+
| Xiaochun Li
| Institute of Mathematics, Polish Academy of Sciences
+
| UIUC
|[[#Bartosz Langowski and Tomasz Szarek Discrete Harmonic Analysis in the Non-Commutative Setting ]]
+
|[[#Xiaochun Li Roth's type theorems on progressions]]
| Betsy
+
| Brian, Shaoming
 
|-
 
|-
|Wednesday, December 13, 4:00, B239 (Colloquium)
+
|Nov 19
|Bobby Wilson
+
| Joao Ramos
|MIT
+
| University of Bonn
|[[#Bobby Wilson | Projections in Banach Spaces and Harmonic Analysis ]]
+
|[[#Joao Ramos  |   Fourier uncertainty principles, interpolation and uniqueness sets ]]
| Andreas
+
| Joris, Shaoming
 
|-
 
|-
| Monday, February 5, 3:00-3:50, B341  (PDE-GA seminar)
+
|Jan 21
| Andreas Seeger
+
| No Seminar
| UW
+
|  
|[[#Andreas Seeger |  Singular integrals and a problem on mixing flows]]
 
 
|
 
|
 +
|
 +
|-
 +
|Friday, Jan 31, 4 pm, B239, Colloquium
 +
| Lillian Pierce
 +
| Duke University
 +
|[[#Lillian Pierce  |  On Bourgain’s counterexample for the Schrödinger maximal function ]]
 +
| Andreas, Simon
 
|-
 
|-
|February 6
+
|Feb 4
| Dong Dong
+
| Ruixiang Zhang
| UIUC
+
| UW Madison
| [[#Dong Dong | Hibert transforms in a 3 by 3 matrix and applications in number theory]]
+
|[[#Ruixiang Zhang  |   Local smoothing for the wave equation in 2+1 dimensions ]]
|Betsy
+
| Andreas
 +
|-
 +
|Feb 11
 +
| Zane Li
 +
| Indiana University
 +
|[[#Zane Li  |  A bilinear proof of decoupling for the moment curve ]]
 +
| Betsy
 
|-
 
|-
|February 13
+
|Feb 18
| Sergey Denisov
+
| Sergey Denisov
 
| UW Madison
 
| UW Madison
| [[#Sergey Denisov | Spectral Szego theorem on the real line]]
+
|[[#linktoabstract  |   De Branges canonical systems with finite logarithmic integral ]]
|  
+
| Street
 
|-
 
|-
|February 20
+
|Feb 25
| Ruixiang Zhang
+
| Michel Alexis
| IAS (Princeton)
+
| Local
| [[#Ruixiang Zhang | The (Euclidean) Fractal Uncertainty Principle]]
+
|[[#Michel Alexis  |   The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight ]]
| Betsy, Jordan, Andreas
+
| Denisov
 
|-
 
|-
|February 27
+
|Mar 3
|Detlef Müller
+
| William Green
|University of Kiel
+
| Rose-Hulman Institute of Technology
| [[#Detlef Müller | On Fourier restriction for a non-quadratic hyperbolic surface]]
+
|[[#William Green  |   Dispersive estimates for the Dirac equation ]]
|Betsy, Andreas
+
| Betsy
 
|-
 
|-
|Wednesday, March 7, 4:00 p.m.
+
|Mar 10
| Winfried Sickel
+
| Yifei Pan
|Friedrich-Schiller-Universität Jena
+
| Indiana University-Purdue University Fort Wayne
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
|Andreas
+
| Xianghong
 
|-
 
|-
|March 13
+
|Mar 17
 +
| Spring Break!
 
|
 
|
|
 
| [[#linkofabstract | Title]]
 
 
|
 
|
|-
 
|March 20
 
|
 
 
|  
 
|  
| [[#linkofabstract | Title]]
 
|
 
 
|-
 
|-
|April 3
+
|Mar 24
|  
+
| Oscar Dominguez
|  
+
| Universidad Complutense de Madrid
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
|
+
| Andreas
 
|-
 
|-
|April 10
+
|Mar 31
|  
+
| Brian Street
|  
+
| University of Wisconsin-Madison
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
|
+
| Local
 
|-
 
|-
|Friday, April 13, 4:00 p.m. (Colloquium)
+
|Apr 7
|Jill Pipher
+
| Hong Wang
|Brown
+
| Institution
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
|WIMAW
+
| Street
 
|-
 
|-
|April 17
+
|<b>Monday, Apr 13</b>
|  
+
|Yumeng Ou
|  
+
|CUNY, Baruch College
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   TBA ]]
|
+
|Zhang
 
|-
 
|-
|April 24
+
|Apr 14
| Lenka Slavíková
+
| Tamás Titkos
| University of Missouri
+
| BBS University of Applied Sciences & Rényi Institute
| [[#linkofabstract | TBA]]
+
|[[#linktoabstract  |   Distance preserving maps on spaces of probability measures ]]
|Betsy, Andreas
+
| Street
 
|-
 
|-
|May 1
+
|Apr 21
| Xianghong Gong
+
| Diogo Oliveira e Silva
| UW
+
| University of Birmingham
| [[#linkofabstract | Title]]
+
|[[#linktoabstract  |   Title ]]
|
+
| Betsy
 
|-
 
|-
|May 15
+
|Apr 28
|Gennady Uraltsev
+
| No Seminar
|Cornell University
 
| [[#linkofabstract | TBA]]
 
|Betsy, Andreas
 
 
|-
 
|-
| May 16-18, [http://www.math.wisc.edu/~stovall/FA2018/ Workshop in Fourier Analysis]
+
|May 5
|
+
|Jonathan Hickman
|
+
|University of Edinburgh
|
+
|[[#linktoabstract  |  Title ]]
|Betsy, Andreas
+
| Andreas
 
|-
 
|-
 
|}
 
|}
  
 
=Abstracts=
 
=Abstracts=
===Brian Street===
+
===José Madrid===
  
Title: Convenient Coordinates
+
Title: On the regularity of maximal operators on Sobolev Spaces
  
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".
+
Abstract:  In this talk, we will discuss the regularity properties (boundedness and
 +
continuity) of the classical and fractional maximal
 +
operators when these act on the Sobolev space W^{1,p}(\R^n). We will
 +
focus on the endpoint case p=1. We will talk about
 +
some recent results and current open problems.
  
===Hiroyoshi Mitake===
+
===Yakun Xi===
  
Title: Derivation of multi-layered interface system and its application
+
Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities
  
Abstract:   In this talk, I will propose a multi-layered interface system which can
+
Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.
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===
 
===Joris Roos===
  
Title: A polynomial Roth theorem on the real line
+
Title: L^p improving estimates for maximal spherical averages
 +
 
 +
Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$.
 +
Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$.
 +
Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.
 +
 
 +
 
 +
 
 +
===Joao Ramos===
 +
 
 +
Title: Fourier uncertainty principles, interpolation and uniqueness sets
 +
 
 +
Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that
 +
 
 +
$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$
 +
 
 +
This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers.
  
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.
+
We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$
  
===Michael Greenblatt===
+
In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.
  
Title:  Maximal averages and Radon transforms for two-dimensional hypersurfaces
+
===Xiaojun Huang===
  
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.
+
Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries
 +
 
 +
Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.
 +
 
 +
===Xiaocheng Li===
 +
 
 +
Title:  An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$
 +
 
 +
Abstract:  We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.
 +
 
 +
 
 +
===Xiaochun Li===
 +
 
 +
Title:  Roth’s type theorems on progressions
 +
 
 +
Abstract:  The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.
 +
 
 +
===Jeff Galkowski===
 +
 
 +
<b>Concentration and Growth of Laplace Eigenfunctions</b>
 +
 
 +
In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.
  
 
===David Beltran===
 
===David Beltran===
  
Title: Fefferman Stein Inequalities
+
Title: Regularity of the centered fractional maximal function
  
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.
+
Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.
  
===Jonathan Hickman===
+
This is joint work with José Madrid.
  
Title: Factorising X^n.
+
===Dominique Kemp===
  
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.
+
<b>Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature</b>
  
===Xiaochun Li ===
+
The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone.  Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.
  
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.
+
===Kevin O'Neill===
  
===Fedor Nazarov=== 
+
<b>A Quantitative Stability Theorem for Convolution on the Heisenberg Group </b>
  
Title: The Lerner-Ombrosi-Perez bound in the Muckenhoupt-Wheeden
+
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.
conjecture is sharp.
 
  
Abstract: We show that the upper bound $[w]_{A_1}\log (e+[w]_{A_1})$ for
+
===Francesco di Plinio===
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===
+
<b>Maximal directional integrals along algebraic and lacunary sets </b>
Title: Higher order Journé commutators
 
  
Abstract: We consider questions that stem from operator theory via Hankel and
+
I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas  of Parcet-Rogers and of  Nagel-Stein-Wainger.
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
+
===Laurent Stolovitch===
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
+
<b>Linearization of neighborhoods of embeddings of complex compact manifolds </b>
analysis, while preserving the Ansatz from classical operator theory.
 
  
===Shaoming Guo ===
+
In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.
Title: Parsell-Vinogradov systems in higher dimensions
 
  
Abstract:
+
===Bingyang Hu===
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.
 
  
===Naser Talebizadeh Sardari===
+
<b>Sparse bounds of singular Radon transforms</b>
  
Title: Quadratic forms and the semiclassical eigenfunction hypothesis
+
In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.
  
Abstract:  Let <math>Q(X)</math> be any integral primitive positive definite quadratic form in <math>k</math> variables, where <math>k\geq4</math>,  and discriminant <math>D</math>. For any integer <math>n</math>, we give an upper bound on the number of integral solutions of <math>Q(X)=n</math> in terms of <math>n</math>, <math>k</math>, and <math>D</math>. 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 <math>\mathbb{T}^d</math> for <math>d\geq 5</math>. This conjecture is motivated by the work of Berry\cite{Berry, Michael} on semiclassical eigenfunction hypothesis.
+
===Lillian Pierce===
 +
<b> On Bourgain’s counterexample for the Schrödinger maximal function </b>
  
===Xianghong Chen===
+
In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices.
 +
In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”
  
Title:  Some transfer operators on the circle with trigonometric weights
+
===Ruixiang Zhang===
  
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.
+
<b> Local smoothing for the wave equation in 2+1 dimensions </b>
  
===Bobby Wilson===
+
Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate  gains a fractional  derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.
  
Title: Projections in Banach Spaces and Harmonic Analysis
+
===Zane Li===
  
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.
+
<b> A bilinear proof of decoupling for the moment curve</b>
  
===Andreas Seeger===
+
We give a proof of decoupling for the moment curve that is inspired from nested efficient congruencing. We also discuss the relationship between Wooley's nested efficient congruencing and Bourgain-Demeter-Guth's decoupling proofs of Vinogradov's Mean Value Theorem. This talk is based on joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.
  
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.
+
===Sergey Denisov===
  
===Dong Dong===
+
<b> De Branges canonical systems with finite logarithmic integral </b>
  
Title: Hibert transforms in a 3 by 3 matrix and applications in number theory
+
We consider measures  m on the real line for which logarithmic
 +
integral exists and give a complete characterization of all Hamiltonians
 +
in de Branges canonical system for which m  is the spectral measure.
 +
This characterization involves the matrix A_2 Muckenhoupt condition on a
 +
fixed scale. Our result provides a generalization of the classical
 +
theorem of Szego for polynomials orthogonal on the unit circle and
 +
complements the Krein-Wiener theorem. Based on the joint work with R.
 +
Bessonov.
  
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.
 
  
===Ruixiang Zhang===
+
===Michel Alexis===
  
Title:  The (Euclidean) Fractal Uncertainty Principle
+
<b>The Steklov problem for Trigonometric Polynomials orthogonal to a Muckenhoupt weight</b>
  
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).
+
Let $\{\varphi_n\}_{n=0}^{\infty}$ be the sequence of degree $n$ polynomials on $\mathbb{T}$, orthonormal with respect to a positive weight $w$. Steklov conjectured whenever $w \geq \delta> 0$ a.e.\ then $\{\varphi_n\}$ are uniformly bounded in $L^{\infty}$. While false, this conjecture brings us to ask the following: under what regularity conditions on $w$ are $\{\varphi_n\}$ uniformly bounded in $L^p (w)$ for some $p > 2$?
 +
 +
We discuss some answers to this question using the contraction principle and operator estimates for the Hilbert transform, in particular recent joint work with Alexander Aptakarev and Sergey Denisov for when $w$ is a Muckenhoupt weight.
  
===Detlef Müller===
+
===William Green===
  
Title: On Fourier restriction for a non-quadratic hyperbolic surface
+
<b> Dispersive estimates for the Dirac equation </b>
  
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.
+
The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds. Dirac formulated a hyberbolic system of partial differential equations
 +
That can be interpreted as a sort of square root of a system of Klein-Gordon equations.
 +
 +
The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations. We will survey recent work on time-decay estimates for the solution operator. Specifically the mapping properties of the solution operator between L^p spaces. As in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution. We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay.  The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).
  
 
=Extras=
 
=Extras=
 
[[Blank Analysis Seminar Template]]
 
[[Blank Analysis Seminar Template]]

Latest revision as of 18:52, 22 February 2020

Fall 2019 and Spring 2020 Analysis Seminar Series

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

If you wish to invite a speaker please contact Brian at street(at)math

Previous Analysis seminars

Analysis Seminar Schedule

date speaker institution title host(s)
Sept 10 José Madrid UCLA On the regularity of maximal operators on Sobolev Spaces Andreas, David
Sept 13 (Friday, B139) Yakun Xi University of Rochester Distance sets on Riemannian surfaces and microlocal decoupling inequalities Shaoming
Sept 17 Joris Roos UW Madison L^p improving estimates for maximal spherical averages Brian
Sept 20 (2:25 PM Friday, Room B139 VV) Xiaojun Huang Rutgers University–New Brunswick A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries Xianghong
Oct 1 Xiaocheng Li UW Madison An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$ Simon
Oct 8 Jeff Galkowski Northeastern University Concentration and Growth of Laplace Eigenfunctions Betsy
Oct 15 David Beltran UW Madison Regularity of the centered fractional maximal function Brian
Oct 22 Laurent Stolovitch University of Côte d'Azur Linearization of neighborhoods of embeddings of complex compact manifolds Xianghong
Wednesday Oct 23 in B129 Dominique Kemp Indiana University Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature Betsy
Oct 29 Bingyang Hu UW Madison Sparse bounds of singular Radon transforms Street
Nov 5 Kevin O'Neill UC Davis A Quantitative Stability Theorem for Convolution on the Heisenberg Group Betsy
Nov 12 Francesco di Plinio Washington University in St. Louis Maximal directional integrals along algebraic and lacunary sets Shaoming
Nov 13 (Wednesday) Xiaochun Li UIUC Roth's type theorems on progressions Brian, Shaoming
Nov 19 Joao Ramos University of Bonn Fourier uncertainty principles, interpolation and uniqueness sets Joris, Shaoming
Jan 21 No Seminar
Friday, Jan 31, 4 pm, B239, Colloquium Lillian Pierce Duke University On Bourgain’s counterexample for the Schrödinger maximal function Andreas, Simon
Feb 4 Ruixiang Zhang UW Madison Local smoothing for the wave equation in 2+1 dimensions Andreas
Feb 11 Zane Li Indiana University A bilinear proof of decoupling for the moment curve Betsy
Feb 18 Sergey Denisov UW Madison De Branges canonical systems with finite logarithmic integral Street
Feb 25 Michel Alexis Local The Steklov problem for trigonometric polynomials orthogonal to a Muckenhoupt weight Denisov
Mar 3 William Green Rose-Hulman Institute of Technology Dispersive estimates for the Dirac equation Betsy
Mar 10 Yifei Pan Indiana University-Purdue University Fort Wayne Title Xianghong
Mar 17 Spring Break!
Mar 24 Oscar Dominguez Universidad Complutense de Madrid Title Andreas
Mar 31 Brian Street University of Wisconsin-Madison Title Local
Apr 7 Hong Wang Institution Title Street
Monday, Apr 13 Yumeng Ou CUNY, Baruch College TBA Zhang
Apr 14 Tamás Titkos BBS University of Applied Sciences & Rényi Institute Distance preserving maps on spaces of probability measures Street
Apr 21 Diogo Oliveira e Silva University of Birmingham Title Betsy
Apr 28 No Seminar
May 5 Jonathan Hickman University of Edinburgh Title Andreas

Abstracts

José Madrid

Title: On the regularity of maximal operators on Sobolev Spaces

Abstract: In this talk, we will discuss the regularity properties (boundedness and continuity) of the classical and fractional maximal operators when these act on the Sobolev space W^{1,p}(\R^n). We will focus on the endpoint case p=1. We will talk about some recent results and current open problems.

Yakun Xi

Title: Distance sets on Riemannian surfaces and microlocal decoupling inequalities

Abstract: In this talk, we discuss the generalization of the Falconer distance problem to the Riemannian setting. In particular, we extend the recent result of Guth-Iosevich-Ou-Wang for the distance set in the plane to general Riemannian surfaces. The key new ingredient is a family of refined decoupling inequalities associated with phase functions that satisfy Carleson-Sj\”olin condition. This is joint work with Iosevich and Liu.

Joris Roos

Title: L^p improving estimates for maximal spherical averages

Abstract: For a given compact set of radii $E$ we will discuss $L^p$ improving properties of maximal spherical averages with a supremum over $E$. Our results are sharp up to endpoints for a large class of $E$. A new feature is that the optimal exponents depend on both, the upper Minkowski dimension and the Assouad dimension of the set $E$. Joint work with Tess Anderson, Kevin Hughes and Andreas Seeger.


Joao Ramos

Title: Fourier uncertainty principles, interpolation and uniqueness sets

Abstract: A classical result in the theory of entire functions of exponential type, Shannon’s interpolation formula predicates that, given a function whose Fourier transform vanishes outside the interval $[-1/2,1/2]$, it is possible to recover it from its values at the integers. More specifically, it holds, in a suitable sense of convergence, that

$$ f(x) = \sum_{n \in \mathbb{Z}} f(n) \frac{\sin(\pi(x-n))}{\pi(x -n)}. $$

This formula is unfortunately unavailable for arbitrary Schwartz functions on the real line, but a recent result of Radchenko and Viazovska provides us with an explicit construction of an interpolation basis for even Schwartz functions. It states, in a nutshell, that we can recover explicitly the function given its values at the squares of roots of integers.

We will discuss a bit these two results, and explore, in connection to classical Fourier uncertainty results, the question of determining which pairs of sets $(A,B)$ satisfy that, if a Schwartz function $f$ vanishes on A and its Fourier transform vanishes on B, then $f \equiv 0.$

In particular, we will give sufficient conditions on $(\alpha,\beta)$ pairs of positive numbers so that, if $f$ vanishes at $\pm n^{\alpha}$ and its Fourier transform vanishes at $\pm n^{\beta}$, then $f$ is identically zero.

Xiaojun Huang

Title: A generalized Kerner theorem and hyperbolic metrics on Stein spaces with compact spherical boundaries

Abstract: This is a joint work with Ming Xiao. We discuss how to construct a hyperbolic metric over a Stein space with spherical boundary. The technique we use is to employ holomorphic continuation along curves for multiple valued functions.

Xiaocheng Li

Title: An Estimate for Spherical Functions on $\mathrm{SL}(3,\mathbb{R})$

Abstract: We prove an estimate for spherical functions $\phi_\lambda(a)$ on $\mathrm{SL}(3,\mathbb{R})$, establishing uniform decay in the spectral parameter $\lambda$ when the group parameter $a$ is restricted to a compact subset of the abelian subgroup $\mathrm{A}$. In the case of $\mathrm{SL}(3,\mathbb{R})$, it improves a result by J.J. Duistermaat, J.A.C. Kolk and V.S. Varadarajan by removing the limitation that $a$ should remain regular. As in their work, we estimate the oscillatory integral that appears in the integral formula for spherical functions by the method of stationary phase. However, the major difference is that we investigate the stability of the singularities arising from the linearized phase function by classifying their local normal forms when the parameters $\lambda$ and $a$ vary.


Xiaochun Li

Title: Roth’s type theorems on progressions

Abstract: The arithmetic progression problems were posed by Erd\”os-Turan, answered affirmatively by Semer\’edi. However, there are still many questions remained on precise quantitative description on how large a subset shall be in oredr to guarantee a progression in it. Involving with Fourier analysis, considerable work had been accomplished recently. We will give a survey on those progress, and report our recent progress on quantitative version of Roth’s type theorem on (polynomial) progressions of short length.

Jeff Galkowski

Concentration and Growth of Laplace Eigenfunctions

In this talk we will discuss a new approach to understanding eigenfunction concentration. We characterize the features that cause an eigenfunction to saturate the standard supremum bounds in terms of the distribution of L^2 mass along geodesic tubes emanating from a point. We also show that the phenomena behind extreme supremum norm growth is identical to that underlying extreme growth of eigenfunctions when averaged along submanifolds. Finally, we use these ideas to understand a variety of measures of concentration; in each case obtaining quantitative improvements over the known bounds.

David Beltran

Title: Regularity of the centered fractional maximal function

Abstract: I will report some recent progress regarding the boundedness of the map $f \mapsto |\nabla M_\beta f|$ from the endpoint space $W^{1,1}(\mathbb{R}^d)$ to $L^{d/(d-\beta)}(\mathbb{R}^d)$, where $M_\beta$ denotes the fractional version of the centered Hardy--Littlewood maximal function. A key step in our analysis is a relation between the centered and non-centered fractional maximal functions at the derivative level, which allows to exploit the known techniques in the non-centered case.

This is joint work with José Madrid.

Dominique Kemp

Decoupling for Real Analytic Surfaces Exhibiting Zero Curvature

The celebrated l^2 decoupling theorem of Jean Bourgain and Ciprian Demeter presented a new perspective on a range of problems related to hypersurfaces with nonzero Gaussian curvature, such as exponential sum estimates, additive energy estimates, local smoothing, and counting solutions to Diophantine inequalities. The same authors also extended their theory to the n-dimensional cone. Following their steps, we prove optimal l^2 decoupling results for the remaining class of zero-curvature two-dimensional surfaces without umbilical points (the so-called tangent surfaces). We are also able to prove a decoupling theorem for the real analytic surfaces of revolution. These results should be viewed as partial progress toward the goal of proving a decoupling theorem for arbitrary real analytic hypersurfaces.


Kevin O'Neill

A Quantitative Stability Theorem for Convolution on the Heisenberg Group

Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to characterize near-maximizers. Specifically, any near-maximizing triple of the trilinear form for convolution on the Heisenberg group must be close to a particular type of triple of ordered Gaussians after adjusting by symmetry. In this talk, we will use the expansion method to prove a quantitative version of this characterization.

Francesco di Plinio

Maximal directional integrals along algebraic and lacunary sets

I will discuss two recent results obtained in collaboration with (partly) Natalia Accomazzo and Ioannis Parissis (U Basque Country). The first is a sharp $L^2$ estimate for the maximal averaging operator associated to sets of directions from algebraic sets in R^n of arbitrary codimension. The proof uses a new scheme of polynomial partitioning on manifolds which extends ideas by Larry Guth. The second result is a sharp estimate in all dimensions for the maximal directional singular integrals along lacunary directions. This settles a question of Parcet and Rogers. The proof uses a combination of two-dimensional and $n$-dimensional coverings combining seemingly contrasting ideas of Parcet-Rogers and of Nagel-Stein-Wainger.

Laurent Stolovitch

Linearization of neighborhoods of embeddings of complex compact manifolds

In this work, we address the following question due to Grauert: if a neighborhood M of a holomorphically embedded complex compact manifold C is formally equivalent to another one, are two neighborhoods biholomorphically equivalent? We shall present the case where the other neighborhood is the neighborhood of the zero section of the normal bundle of C in M. The solution to this problem involves "small divisors problems". This is joint work with X. Gong.

Bingyang Hu

Sparse bounds of singular Radon transforms

In this talk, we will first briefly talk about the general theory of sparse domination, and then talk about the sparse bounds of singular Radon transforms, which strengths the $L^p$ boundedness of such operators due to Christ, Nagel, Stein and Wainger in 1999.

Lillian Pierce

On Bourgain’s counterexample for the Schrödinger maximal function

In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space H^s must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices. In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”

Ruixiang Zhang

Local smoothing for the wave equation in 2+1 dimensions

Sogge's local smoothing conjecture for the wave equation predicts that the local L^p space-time estimate gains a fractional derivative of order almost 1/p compared to the fixed time L^p estimates, when p>2n/(n-1). Jointly with Larry Guth and Hong Wang, we recently proved the conjecture in $\mathbb{R}^{2+1}$. I will talk about a sharp square function estimate we proved which implies the local smoothing conjecture in dimensions 2+1. A key ingredient in the proof is an incidence type theorem.

Zane Li

A bilinear proof of decoupling for the moment curve

We give a proof of decoupling for the moment curve that is inspired from nested efficient congruencing. We also discuss the relationship between Wooley's nested efficient congruencing and Bourgain-Demeter-Guth's decoupling proofs of Vinogradov's Mean Value Theorem. This talk is based on joint work with Shaoming Guo, Po-Lam Yung, and Pavel Zorin-Kranich.


Sergey Denisov

De Branges canonical systems with finite logarithmic integral

We consider measures m on the real line for which logarithmic integral exists and give a complete characterization of all Hamiltonians in de Branges canonical system for which m is the spectral measure. This characterization involves the matrix A_2 Muckenhoupt condition on a fixed scale. Our result provides a generalization of the classical theorem of Szego for polynomials orthogonal on the unit circle and complements the Krein-Wiener theorem. Based on the joint work with R. Bessonov.


Michel Alexis

The Steklov problem for Trigonometric Polynomials orthogonal to a Muckenhoupt weight

Let $\{\varphi_n\}_{n=0}^{\infty}$ be the sequence of degree $n$ polynomials on $\mathbb{T}$, orthonormal with respect to a positive weight $w$. Steklov conjectured whenever $w \geq \delta> 0$ a.e.\ then $\{\varphi_n\}$ are uniformly bounded in $L^{\infty}$. While false, this conjecture brings us to ask the following: under what regularity conditions on $w$ are $\{\varphi_n\}$ uniformly bounded in $L^p (w)$ for some $p > 2$?

We discuss some answers to this question using the contraction principle and operator estimates for the Hilbert transform, in particular recent joint work with Alexander Aptakarev and Sergey Denisov for when $w$ is a Muckenhoupt weight.

William Green

Dispersive estimates for the Dirac equation

The Dirac equation was derived by Dirac in 1928 to model the behavior of subatomic particles moving at relativistic speeds. Dirac formulated a hyberbolic system of partial differential equations That can be interpreted as a sort of square root of a system of Klein-Gordon equations.

The Dirac equation is considerably less well studied than other dispersive equations such as the Schrodinger, wave or Klein-Gordon equations. We will survey recent work on time-decay estimates for the solution operator. Specifically the mapping properties of the solution operator between L^p spaces. As in other dispersive equations, the existence of eigenvalues and/or resonances at the edge of the continuous spectrum affects the dynamics of the solution. We classify the threshold eigenvalue and resonance structure in two and three spatial dimensions and study their effect on the time decay. The talk with survey joint works with B. Erdogan (Illinois), M. Goldberg (Cincinnati) and E. Toprak (Rutgers).

Extras

Blank Analysis Seminar Template