Georgia Institute of Technology College of Sciences

 We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences $\lfloor h(p)\rfloor$, where $h$ belongs in a wide class of functions, the so-called $c$-regularly varying functions. A key ingredient of our approach are certain exponential sum estimates, which we also use for establishing a Waring-type result. Assuming that the Riemann zeta function has any zero-free strip upgrades our exponential sum estimates to polynomially saving ones and this makes a conditional result regarding the behavior of our ergodic averages on $L^1$ to not seem entirely out of reach. The talk is based on joint work with Erik Bahnson, Abbas Dohadwala and Ish Shah.
 

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.

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.