Seminars and Colloquia by Series

Bounds for bilinear averages and its associated maximal functions

Series
Analysis Seminar
Time
Wednesday, November 6, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tainara Gobetti BorgesBrown University

Let $S^{2d-1}$ be the unit sphere in $\mathbb{R}^{2d}$, and $\sigma_{2d-1}$ the normalized spherical measure in $S^{2d-1}$. The (scale t) bilinear spherical average is given by 
$$\mathcal{A}_{t}(f,g)(x):=\int_{S^{2d-1}}f(x-ty)g(x-tz)\,d\sigma_{2d-1}(y,z).$$
There are geometric motivations to study bounds for such bilinear spherical averages, in connection to the study of some Falconer distance problem variants. Sobolev smoothing bounds for the operator 
$$\mathcal{M}_{[1,2]}(f,g)(x)=\sup_{t\in [1,2]}|\mathcal{A}_{t}(f,g)(x)|$$
 are also relevant to get bounds for the bilinear spherical maximal function
$$\mathcal{M}(f,g)(x):=\sup_{t>0} |\mathcal{A}_{t}(f,g)(x)|.$$
In a joint work with B. Foster and Y. Ou, we put that in a general framework where $S^{2d-1}$ can be replaced by more general smooth surfaces in $\mathbb{R}^{2d}$, and one can allow more general dilation sets in the maximal functions: instead of supremum over $t>0$, the supremum can be taken over $t\in \tilde{E}$ where $\tilde{E}$ is the set of all scales obtained by dyadic dilation of fixed set of scales $E\subseteq [1,2]$.

Magnetic Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on Riemannian manifolds

Series
Analysis Seminar
Time
Wednesday, October 30, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Rotem AssoulineWeizmann Institute of Science

I will present a magnetic version of the Riemannian Brunn-Minkowski and Borell-Brascamp-Lieb inequalities of Cordero-Erausquin-McCann-Schmuckenschläger and Sturm, replacing geodesics by minimizers of a magnetic action functional. Both results involve a notion of magnetic Ricci curvature.

On non-resonant planar Carleson-Radon operator along homogeneous curves

Series
Analysis Seminar
Time
Wednesday, October 9, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Martin HsuPurdue University

We go over some relevant history and related problems to motivate the study of the Carleson-Radon operator and the difficulty exhibiting in the planar case. Our main result confirms that the planar Carleson-Radon operator along homogenous curve with general monomial \(t^\alpha\) term modulation admits full range \(L^p\) bound assuming the natural non-resonant condition. In the talk, I'll provide a brief overview of the three key ingredients of the LGC based proof:

 

  1. A sparse-uniform dichotomy of the input function adapted to appropriate time-frequency foliation of the phase-space;
  2. A joint structural analysis of the linearizing stopping-time function in the phase in relation to the Gabor coefficients of the input;
  3. A level set analysis on the time-frequency correlation set.
 

Smooth Discrepancy and Littlewood's Conjecture

Series
Analysis Seminar
Time
Wednesday, October 2, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Niclas TechnauUniversity of Bonn

Given x in $[0,1]^d$, this talk is about the fine-scale distribution of the Kronecker sequence $(n x mod 1)_{n\geq 1}$.
After a general introduction, I will report on forthcoming work with Sam Chow.
Using Fourier analysis, we establish a novel deterministic analogue of Beck’s local-to-global principle (Ann. of Math. 1994),
which relates the discrepancy of a Kronecker sequence to multiplicative diophantine approximation.
This opens up a new avenue of attack for Littlewood’s conjecture.

An ergodic theorem in the Gaussian integer setting

Series
Analysis Seminar
Time
Wednesday, September 25, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker

<<>>

We discuss the Pointwise Ergodic Theorem for the Gaussian divisor function $d(n)$, that is, for a measure preserving $\mathbb Z [i]$ action $T$, the ergodic averages weighted by the divisor function converge pointwise for all functions in $L^p$, for $p>1$.  We obtain improving and sparse bounds for these averages.

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