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 Mf(x):=supN|1N∑n≤Nf(x−n2)| is $\ell^p$ bounded. Although the analogous continuous maximal function,Mf(x):=supN|1N∫N0f(x−t2) dt|=supN|1N∫N20f(x−t) dt2√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,Cf(x):=supλ|∑n≠0e2πiλn2nf(x−n)|,a discrete analogue of Stein's purely quadratic Carleson operator for functions on the real line Cf(x):=supλ|∫e2πiλt2tf(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.