Seminar

The program of studying \emph{discrete analogues in harmonic analysis} was initiated in the late $80$s by Bourgain, who was interested in proving pointwise ergodic theorems. His result, the almost-everywhere convergence of ergodic Cesaro means along polynomial orbits for $f \in L^p, \ p > 1$, was anchored by a study of the discrete maximal function along polynomial orbits,

$$\mathcal{M}_Pf(x) := \sup_N \left| \frac{1}{N} \sum_{n \leq N} f(x-P(n)) \right|, \ P\in\mathbb{Z}[-]$$

which he showed were bounded on all $\ell^p(\mathbb{Z}), \ 1 < p \leq \infty$. Although the analoguous continuous maximal function along polynomial curves can be reduced to the Hardy-Littlewood maximal function via change of variables, understanding $\mathcal{M}_P$ required an interplay between number theory, Fourier analysis -- and even probability theory.

 

In 2015, Mirek, Stein, and Trojan essentially concluded the study of discrete radon transforms along polynomial curves. In light of their work, it has become natural to begin to study discrete analogues of \emph{Carleson} operators. While Stein observed that, due to the linear nature of the phases, the boundedness of the discrete (linear) Carleson operator

$$\sup_\lambda \left| \sum_{m \neq 0} f(x-m) \frac{e(\lambda m) }{m} \right|, \ e(t) := e^{2\pi i t}$$

can be deduced from the corresponding continuous operator, when the linear phases are replaced by higher degree polynomial phases, no such reduction is available.

 

Accordingly, Carleson-type operators with higher-degree polynomial phases have become an object of interest, namely discrete analogues of operators investigated by Stein and Wainger. We will discuss the current state of affairs for these discrete operators, which in one dimension are given by

$$\sup_{\lambda_2,\dots,\lambda_d} \left| \sum_{m \neq 0} f(x-m) \frac{e( \lambda_2 m^2 + \dots + \lambda_d m^d)}{m} \right|,$$

where $d \geq 2$ is a fixed integer.