Carleson's theorem with quadratic phase

Abstract

Carleson's theorem on the pointwise convergence of Fourier series provides bounds for a maximal operator, with the maximum taken over all choices of linear functions of a phase argument. We extend this to all quadratic choices of phase functions. Specifically, we show that the maximal operator below maps $L^p$ into itself for $1<p<\infty$. $$\underset{a}{\sup }\underset{b}{\sup }|\int e^{i(a{y}^2+by)}f(x-y)dy/y|$$