Seminars and Colloquia by Series

Spectral monotonicity under Gaussian convolution

Series
Analysis Seminar
Time
Wednesday, December 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Eli Putterman – Tel Aviv University –

The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.  

On the curved trilinear Hilbert transform

Series
Analysis Seminar
Time
Wednesday, November 15, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Bingyang Hu – Auburn University –

The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator

$$

H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R

$$

is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1

 

The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:

 

1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;

 

2). a structural analysis of suitable maximal "joint Fourier coefficients";

 

3). a level set analysis with respect to the time-frequency correlation set. 

 

This is a joint work with my postdoc advisor Victor Lie from Purdue.

 

Higher dimensional fractal uncertainty

Series
Analysis Seminar
Time
Wednesday, November 1, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Alex Cohen – MIT –

The fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n. 

On the spectral synthesis for the unit circle in ${\mathcal F} L_s^q ({\mathbf R}^2)$

Series
Analysis Seminar
Time
Wednesday, October 18, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
TBA
Speaker
Masaharu Kobayashi – Hokkaido University –

Let ${\mathcal F}L^q_s ({\mathbf R}^2)$ denote the set of all tempered distributions $f \in {\mathcal S}^\prime ({\mathbf R}^2)$ such that the norm $ \| f \|_{{\mathcal F}L^q_s} = (\int_{{\mathbf R}^2}\, ( |{\mathcal F}[f](\xi)| \,( 1+ |\xi| )^s )^q\, d \xi )^{ \frac{1}{q} }$ is finite, where ${\mathcal F}[f]$ denotes the Fourier transform of $f$. We investigate the spectral synthesis for the unit circle $S^1 \subset {\mathbf R}^2$ in ${\mathcal F}L^q_s ({\mathbf R}^2)$ with $1\frac{2}{q^\prime}$, where $q^\prime$ denotes the conjugate exponent of $q$. This is joint work with Prof. Sato (Yamagata University).

Convergence of Frame Series: from Hilbert Space to Modulation Space

Series
Analysis Seminar
Time
Wednesday, October 4, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Pu-Ting Yu – Georgia Tech

It is known that if $\{x_n\}$ is a frame for a separable Hilbert space, then there exist some sequences $\{y_n\}$ such that $x= \sum x_n$, and this sum converges in the norm of H. This equation is called the reconstruction formula of x. In this talk, we will talk about the existence of frames that admit absolutely convergent and unconditionally convergent reconstruction formula. Some characterizations of such frames will also be presented. Finally, we will present an extension of this problem about the unconditional convergence of Gabor expansion in Modulation spaces.

Flag Hardy space theory—an answer to a question by E.M. Stein.

Series
Analysis Seminar
Time
Wednesday, September 20, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Ji Li – Macquarie University –


In 1999, Washington University in Saint Louis hosted a conference on Harmonic Analysis to celebrate the 70th birthday of G. Weiss. In his talk in flag singular integral operators, E. M. Stein asked “What is the Hardy space theory in the flag setting?” In our recent paper, we characterise completely a flag Hardy space on the Heisenberg group. It is a proper subspace of the classical one-parameter Hardy space of Folland and Stein that was studied by Christ and Geller. Our space is useful in several applications, including the endpoint boundedness for certain singular integrals associated with the Sub-Laplacian on Heisenberg groups, and representations of flag BMO functions.

On displacement concavity of the relative entropy

Series
Analysis Seminar
Time
Wednesday, September 6, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Liran Rotem – Technion –

It is known for many years that various inequalities in convex geometry have information-theoretic analogues. The most well known example is the Entropy power inequality which corresponds to the Brunn-Minkowski inequality, but the theory of optimal transport allows to prove even better analogues. 

At the same time, in recent years there is a lot of interest in the role of symmetry in Brunn-Minkowski type inequalities. There are many open conjectures in this direction, but also a few proven theorems such as the Gaussian Dimensional Brunn-Minkowski inequality. In this talk we will discuss the natural question — do the known information-theoretic inequalities similarly improve in the presence of symmetry?  I will present some cases where the answer is positive together with some open problems. 

Based on joint work with Gautam Aishwarya. 

A quantitative stability estimate for the Sobolev Inequality

Series
Analysis Seminar
Time
Wednesday, August 30, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Michael Loss – GaTech –

I’ll present a quantitative version of a stability estimate
for the Sobolev Inequality improving previous results of Bianchi
and Egnell. The estimate has the correct dimensional dependence
which leads to a stability estimate for the Logarithmic Sobolev inequality.
This is joint work with Dolbeault, Esteban, Figalli and Frank.

CANCELED — Multiplier weak type inequalities for maximal operators and singular integrals

Series
Analysis Seminar
Time
Wednesday, April 19, 2023 - 14:00 for 1 hour (actually 50 minutes)
Location
This seminar has been cancelled and will be rescheduled next year.
Speaker
Brandon Sweeting – University of Alabama –

This seminar has beeb cancelled and will be rescheduled next year.  We discuss a kind of weak type inequality for the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operators that was first studied by Muckenhoupt and Wheeden and later by Sawyer. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior. Such inequalities arise in the generalization of weak-type spaces to the matrix weighted setting and find applications in scalar two-weight norm inequalities via interpolation with change of measures. In this talk, I will discuss quantitative estimates obtained for $A_p$ weights, $p > 1$, that generalize those results obtained by Cruz-Uribe, Isralowitz, Moen, Pott and Rivera-Ríos for $p = 1$. I will also discuss an endpoint result for the Riesz potentials.

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