Seminars and Colloquia by Series

Fefferman--Stein type inequality in multiparameter settings and applications

Series
Analysis Seminar
Time
Wednesday, December 4, 2024 - 14:00 for
Location
Speaker
ji Li – Macquarie University –

A classical Fefferman-Stein inequality relates the distributional estimate for a square function for a harmonic function u to a non-tangential maximal function of u.   We extend this ineuality to certain multiparameter settings, including the Shilov boundaries of tensor product domains, and the Heisenberg groups  with flag structure.
Our technique bypasses the use of Fourier or the dependence of group structure. Direct applications include the  the (global) weak type endpoint estimate for multi-parameter Calderon–Zygmund operators and maximal function characterisation of multi-parameter Hardy spaces.

This talk is based on the recent progress: Ji Li, Fefferman–Stein type inequality'',  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 2024.

 

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 Borges – Brown University –

Let S2d−1 be the unit sphere in R2d, and σ2d−1 the normalized spherical measure in S2d−1. The (scale t) bilinear spherical average is given by 
At(f,g)(x):=∫S2d−1f(x−ty)g(x−tz)dσ2d−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 
M[1,2](f,g)(x)=supt∈[1,2]|At(f,g)(x)|
 are also relevant to get bounds for the bilinear spherical maximal function
M(f,g)(x):=supt>0|At(f,g)(x)|.
In a joint work with B. Foster and Y. Ou, we put that in a general framework where S2d−1 can be replaced by more general smooth surfaces in R2d, 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∈E~ where E~ is the set of all scales obtained by dyadic dilation of fixed set of scales E⊆[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 Assouline – Weizmann 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 Hsu – Purdue 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 tα term modulation admits full range Lp 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 Technau – University of Bonn –

Given x in [0,1]d, this talk is about the fine-scale distribution of the Kronecker sequence (nxmod1)n≥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 Z[i] action T, the ergodic averages weighted by the divisor function converge pointwise for all functions in Lp, for p>1.  We obtain improving and sparse bounds for these averages.

Square Functions Controlling Smoothness with Applications to Higher-Order Rectifiability

Series
Analysis Seminar
Time
Wednesday, April 17, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
John Hoffman – Florida State University

We present new results concerning characterizations of the spaces C1,α and “LIα+1” for 0<α<1.  The space LIα+1 is the space of Lipschitz functions with α-order fractional derivative having bounded mean oscillation.  These characterizations involve geometric square functions which measure how well the graph of a function is approximated by a hyperplane at every point and scale.  We will also discuss applications of these results to higher-order rectifiability.

The compactness of multilinear Calderón-Zygmund operators

Series
Analysis Seminar
Time
Wednesday, April 10, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Anastasios Fragkos – Washington University St Louis –

We prove a wavelet T(1) theorem for compactness of multilinear Calderón -Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This talk is based on joint work with Walton Green and Brett Wick.   

On the Curved Trilinear Hilbert Transform

Series
Analysis Seminar
Time
Wednesday, April 3, 2024 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
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. We show that it is bounded in the Banach range.  

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.

Pages