Clemson Analysis Seminar Schedule

Spring 2023

Classical multiplier theorems and their sharp variants

Speaker:                Lenka Slavíková - Charles University
Time:                      Wednesday, January 18, 2023 - 11:15 am
Location:               Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
The question of finding good sufficient conditions on a bounded function $m$ guaranteeing the $L^p$-boundedness of the associated Fourier multiplier operator is a long-standing open problem in harmonic analysis. In this talk, I will recall the classical multiplier theorems of Hörmander and Marcinkiewicz and present their sharp variants in which the multiplier belongs to a certain fractional Sobolev space. The talk is based in part on a joint work with L. Grafakos and M. Mastyło.

Frames via Unilateral Iterations of Bounded Operators

Speaker:                Victor Bailey - Georgia Institute of Technology
Time:                      Friday, February 3, 2023 - 11:15 am
Location:               Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a bounded linear operator. Recent works in this area consider questions such as when can a given frame for a separable Hilbert Space, $\{f_k\}_{k \in I} \subset H$, be represented by iterations of an operator on a single vector and what are necessary and sufficient conditions for a system, $\{T^n \varphi\}_{n=0}^{\infty} \subset H$, to be a frame? In this talk, we will discuss the connection between frames given by iterations of a bounded operator and the theory of model spaces in the Hardy-Hilbert Space as well as necessary and sufficient conditions for a system generated by the orbit of a pair of commuting bounded operators to be a frame.

Muckenhoupt weights and the reverse Hölder inequality on metric measure spaces

Speaker:                Emma-Karoliina Kurki - Aalto University
Time:                      Friday, February 10, 2023 - 11:15 am
Location:               Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
It is well known that in Euclidean spaces the reverse Hölder inequality (RHI) implies the Muckenhoupt Ap condition. This statement remains true in reasonably general metric measure spaces, but the necessary condition appears to be unknown. I give an overview of what is known and not known about the RHI as a characterization of Ap weights. Furthermore, I discuss characterizations of the weak reverse Hölder inequality (WRHI), which is the nondoubling analogy of the RHI with an increasing support on the right-hand side. The weak case is based on a joint work with Carlos Mudarra.

Convergence of discrete non-linear Fourier transform via spectral problems for canonical systems

Speaker:                Ashley Zhang - University of Wisconsin, Madison
Time:                      Monday, February 20, 2023 - 11:15 am
Location:               ​TBA (and streamed on Zoom -  https://clemson.zoom.us/j/93164718636)
Abstract:
This talk will be about connections between spectral problems for canonical systems and non-linear Fourier transforms (NLFTs). Non-linear Fourier transform is closely connected to Dirac systems, which form a subclass of canonical systems of differential equations. This connection allows one to find analogs of results on inverse spectral problems for canonical systems in the area of NLFT. In particular, NLFTs of discrete sequences, discussed in the lecture notes by Tao and Thiele, are related to spectral problems for periodic measures and the theory of orthogonal polynomials.
 
I will start the talk with the basics of non-linear Fourier transforms, then connect NLFTs to canonical systems. Then I will present an explicit algorithm for inverse spectral problems developed by Makarov and Poltoratski for locally-finite periodic spectral measures, as well as an extension of their work to certain classes of non-periodic spectral measures. Finally I will return to NLFT and translate the results for inverse spectral problems to results for NLFT.

Improved spectral cluster estimates on compact manifolds

Speaker:                Xiaoqi Huang - University of Maryland
Time:                      Friday, February 24, 2023 - 3:30 pm
Location:               ​Martin Hall M-104 (and streamed on Zoom -  https://clemson.zoom.us/j/93164718636)
Abstract:
We will discuss the problem of trying to obtain improved eigenfunction / spectral projection estimates under geometric assumptions, such as the presence of negative sectional curvatures, which is based on the use of microlocal Kakeya-Nikodym estimates along with a combination of local and global harmonic analysis that further exploit geometric assumptions. Using a similar approach, we will also discuss some improved Strichartz estimates on compact manifolds with negative curvature. This is based on joint work with Chris Sogge and Matthew Blair.

Operator-free sparse domination

Speaker:                Emiel Lorist - Delft University of Technology
Time:                      Friday, March 3, 2023 - 11:15 am
Location:               Virtual on Zoom -  https://clemson.zoom.us/j/93164718636
Abstract:
Sparse domination is a recent technique, allowing to estimate (in norm, pointwise or dually) many operators in harmonic analysis by simple, positive expressions. This technique has led to many new results in harmonic analysis over the past decade. In this talk I will discuss a sparse domination principle for an arbitrary family of functions f(x,Q), where x ∈ Rn and Q is a cube in Rn. When applied to operators, this result recovers various recent sparse domination results. In contrast to preceding results, our sparse domination principle can also be applied to non-operator objects, which allows one to use sparse techniques in new areas. I will discuss new applications of sparse domination to weighted Poincaré inequalities and to potential theory.
This talk is based on joint work with Andrei Lerner and Sheldy Ombrosi.

Exponential frames and point configurations

Speaker:                Alex Iosevich - University of Rochester
Time:                      Friday, March 17, 2023 - 11:15 am
Location:               Martin M-102 (and streamed on Zoom -  https://clemson.zoom.us/j/93164718636)
Abstract:
We are going to discuss the existence and non-existence of  exponential bases and frames corresponding to domains in Euclidean space. A particular emphasis is placed on connections with techniques involving finite point configurations in geometric measure theory.