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Friday, November 03, 2017
5:30 PM - 6:30 PM
Downs 103

Caltech/UCLA Joint Analysis Seminar

L^2 Estimates for the Discrete Quadratic Carleson Operator
Ben Krause, Department of Mathematics, Caltech,

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|1NnNf(xn2)| is $\ell^p$ bounded. Although the analogous continuous maximal function,Mf(x):=supN|1NN0f(xt2) dt|=supN|1NN20f(xt) dt2t|, 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|n0e2πiλn2nf(xn)|,a discrete analogue of Stein's purely quadratic Carleson operator for functions on the real line Cf(x):=supλ|e2πiλt2tf(xt) 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.

For more information, please contact Mathematics Department by phone at 626-395-4335 or by email at mathinfo@caltech.edu.