Caltech/UCLA Joint Analysis Seminar

In the late 80s and early 90s, Bourgain, motivated by questions from pointwise ergodic theory, initiated a study of discrete radon transforms along polynomial curves. For instance, he proved that for every $p > 1$, the maximal function $$\mathcal{M} f(x) := \sup_N | \frac{1}{N} \sum_{n \leq N} f(x-n^2) |$$ is $\ell^p$ bounded. Although the analogous continuous maximal function, $$M f(x) := \sup_N | \frac{1}{N} \int_0^N f(x - t^2) \ dt | = \sup_N | \frac{1}{N} \int_0^{N^2} f(x-t) \ \frac{dt}{2\sqrt{t}} |,$$ is trivially bounded by the standard Hardy-Littlewood maximal function, subtle arithmetic issues arise in handling Bourgain's discrete maximal function, $\mathcal{M}$. In particular, whereas the zero frequency plays a distinguished role in understanding the Fourier transforms of the continuous averaging operators, subtle multi-frequency considerations arise when analyzing the analogous discrete multipliers.

In this talk, I will discuss $\ell^2$ estimates for the discrete quadratic Carleson operator, $$\mathcal{C}f(x) := \sup_\lambda | \sum_{ n \neq 0} \frac{e^{2\pi i \lambda n^2}}{n} f(x-n) |,$$ a discrete analogue of Stein's purely quadratic Carleson operator for functions on the real line $$Cf(x) := \sup_\lambda | \int \frac{e^{2\pi i \lambda t^2}}{t} f(x-t) \ dt |.$$ Stein was able to bound $Cf$ essentially by analyzing the Fourier transform of the distributions $\frac{e^{2\pi i \lambda t^2}}{t}$; this study was neatly organized by the distinguished nature the zero frequency plays in understanding the relevant Fourier multipliers. Unfortunately, as with $\mathcal{M}$, the multi-frequency nature of the problem necessitates a more involved approach -- a synthesis of Bourgain's and Stein's ideas.