Fall 2024 Analysis Seminar

Organizers: Shengwen Gan, Terry Harris and Andreas Seeger

Emails:

• Shengwen Gan: sgan7 at math dot wisc dot edu
• Terry Harris: tlharris4 at math dot wisc dot edu
• Andreas Seeger: seeger at math dot wisc dot edu

Time and Room: Wed 3:30--4:30 Van Vleck B119.

In some cases the seminar may be scheduled at different time to accommodate speakers.

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Links to previous seminars

Link to Spring 2025 Analysis Seminar

Date	Speaker	Institution	Title	Host(s)	Notes (e.g. unusual room/day/time)
We, 9-11	Gevorg Mnatsakanyan	UW Madison	Almost everywhere convergence of the Malmquist Takenaka series
We, 9-18	Lars Niedorf	UW Madison	Restriction type estimates and spectral multipliers on Métivier groups
Th, 9-26, 2:25-3:25, VV B215	Niclas Technau	University of Bonn	Rational points on/near homogeneous hyper-surfaces	Andreas	Note changes of time/date/room, joint with Number Theory Seminar
We, 10-2	Sergey Denisov	UW Madison	Applications of inverse spectral theory for canonical systems to NLS.
We, 10-9	Shukun Wu	Indiana University	On almost everywhere convergence of planar Bochner Riesz means	Shengwen
We, 10-16	Nathan Wagner	Brown University	Dyadic shifts, sparse domination, and commutators in the non-doubling setting	Andreas
We, 10-23	Betsy Stovall	UW Madison	Title: Incidences among flows
We, 10-30	Burak Hatinoglu	Michigan State University	Norm estimates of Chebyshev polynomials	Alexei
We, 11-6	Bingyuan Liu	University of Texas Rio Grande Valley	The Diederich--Fornaess index and the dbar-Neumann problem	Xianghong
We, 11-13	Maxim Yattselev	Indiana University (Indianapolis)	On rational approximants of multi-valued functions	Sergey
We, 11-20	Li Ji	Macquarie University	Hardy spaces associated to the multiparameter flag structure	Brian

Abstracts

Gevorg Mnatsakanyan

Title: Almost everywhere convergence of the Malmquist Takenaka series

Link to Abstract: [1]

Lars Niedorf

Title: Restriction type estimates and spectral multipliers on Métivier groups

Abstract: We present a restriction type estimate for sub-Laplacians on arbitrary two-step stratified Lie groups. Although weaker than previously known estimates for the subclass of Heisenberg type groups, these estimates turn out to be sufficient to prove an Lp-spectral multiplier theorem with sharp regularity condition s > d|1/p-1/2| for sub-Laplacians on Métivier groups.

Niclas Technau

Title: Rational points on/near homogeneous hyper-surfaces

Abstract: How many rational points are on/near a compact hyper-surface? This question is related to Serre's Dimension Growth Conjecture. We survey the state of the art, and explain a standard random model. Furthermore, we report on recent joint work with Rajula Srivastava (Uni/MPIM Bonn). Our arguments are rooted in Fourier analysis and, in particular, clarify the role of curvature in the random model.

Sergey Denisov

Title: Applications of inverse spectral theory for canonical systems to NLS

Abstract: For nice initial data, NLS can be integrated using the inverse scattering theory for the Dirac equation on the line. We will discuss the connection of the Dirac equation to canonical systems and use the recent characterization of the Szegő class of measures on the real line to obtain a new semi-conserved quantity for the NLS. The bounds for the negative Sobolev norms will be presented as an application for the L2 NLS solutions (based on joint work with Roman Bessonov).

Shukun Wu

Title: On almost everywhere convergence of planar Bochner Riesz means

Abstract: We prove that the planar Bochner Riesz mean converges almost everywhere for any L^p function in the optimal range, for 5/3<p<2. Our approach is based on a weighted L^2 estimate, which may be of independent interest. For example, up to an epsilon loss, we can reprove Cordoba's L^4 orthogonality by solely considering L^2 space and using L^2 orthogonality. This is a joint work with Xiaochun Li.

Nathan Wagner

Title: Dyadic shifts, sparse domination, and commutators in the non-doubling setting

Abstract: In this talk, we will discuss a dyadic variant of the Hilbert transform, which is a useful model of its continuous counterpart and the prototypical example of a so-called "Haar shift". After discussing some background and motivation in the Lebesgue measure case, we will turn to the situation where the L2 Haar functions are defined with respect to a locally finite Borel measure μ, which may not satisfy the dyadic doubling condition. In this more general setting, Lopez-Sanchez, Martell, and Parcet identified a weak regularity condition on the measure μ which characterizes weak-type and Lp estimates for this dyadic Hilbert transform. I then will discuss joint work with Jose Conde Alonso and Jill Pipher, where we obtain a domination of the dyadic Hilbert transform (and more generally, Haar shifts) by a modified sparse form. As an application, we characterize the class of weights where the dyadic Hilbert transform and related operators are bounded. A surprising novelty is that the usual (dyadic) Muckenhoupt A2 condition is necessary, but no longer sufficient in the non-doubling setting, and our modified weight condition reflects the "complexity" of the underlying Haar shift. Finally, we will examine a different dyadic Haar shift model of the Hilbert transform and its relationship to BMO (bounded mean oscillation) functions via commutators in the non-doubling setting (joint with Tainara Borges, Jose Conde Alonso, and Jill Pipher).

Betsy Stovall

Title: Incidences among flows

Abstract: In this talk, we will discuss a family of combinatorial problems that may be viewed as discretized and (possibly) perturbed versions of various continuum incidence problems in harmonic analysis. Namely, given a finite collection L of integral curves of some “small” family of “nice” vector fields, how many incidences can be formed under various definitions of “small,” “nice,” and “incidence”? Well-studied special cases include the Joints Problem in R^n, the Szeméredi—Trotter Theorem for point-line intersections in the plane, and work of Magyar—Stein—Wainger, Pierce, and others, in which the discretization is to the integers. In this talk, we will discuss results and examples both old and new; the new results are joint with Huang and Tammen.

Burak Hatinoglu

Title: Norm estimates of Chebyshev polynomials

Abstract: The Chebyshev polynomial of a compact set in the complex plane is the unique monic polynomial of a given degree minimizing the sup-norm among the monic polynomials of the same degree. As a classical object in the approximation theory, its history and connections with potential theory go back to the first half of the 20th century, Fekete, Szego and Faber. However, in the last 20 years many remarkable improvements were made in the study of Chebyshev polynomials. In this talk, after reviewing basics and classical theorems, I will consider recent results on norm estimates of Chebyshev polynomials, discuss a Cantor-type set construction approach, and an application to the spectral theory of periodic operators.

Bingyuan Liu

Title: The Diederich--Fornaess idex and the dbar-Neumann problem

Abstract: Introduced in 1977, the Diederich–Fornaess index was developed to help construct bounded plurisubharmonic functions on bounded pseudoconvex domains. For the past three decades, it is believed that the Diederich–Fornaess index is linked with the global regularity of the dbar-Neumann operator. A longstanding open question has been whether a Diederich–Fornaess index of 1 implies this global regularity. In this talk, we will overview the background and present and sketch a proof of a recent theorem, jointly proved by Emil Straube and me. This theorem answers this open question affirmatively for (0, n-1) forms.

Maxim Yattselev

Title: On Rational Approximants of Multi-Valued Functions

Abstract: Let D be a bounded Jordan domain and A be its complement on the Riemann sphere. The asymptotic behavior in D of the best rational approximants in the uniform norm on A of functions holomorphic on A that admit a multi-valued continuation to quasi every point of D with finitely many possible branches will be discussed.

Ji Li

Title: Hardy spaces associated to multiparameter flag structures

The theory of multi-parameter flag singular integral originates from the study of the ∂ ̄-problem on the Heisenberg group by D. Phong and E.M. Stein. In our recent work, we established a complete flag Hardy space theory on the Heisenberg group, including characterisations via Littlewood–Paley area function, square function, non-tangential and radial maximal functions, atoms, and the flag Riesz transforms. It provided a unified approach for proving the $L^p$ boundedness of different types of singular integrals, and led to the endpoint $L\log L\to L^{1,\infty}$ estimates. The representations of flag BMO functions are also provided.