Maximal modulations of singular Radon transforms

Abstract: Carleson's theorem on the convergence of Fourier series is equivalent to the weak-$L^2$-boundedness of the maximally modulated Hilbert transform, and adaptions of the proof show more generally weak-$L^2$-boundedness of maximally modulated Calderón-Zygmund operators. This talk is about the open problem of whether this result can be extended to singular Radon transforms, such as the Hilbert transform along the parabola $H_P$. I will discuss the main ingredients used in the proof of Carleson's theorem, and to what extent they can be adapted for $H_P$. A corollary are improved quantitative estimates for maximal modulations of operators approximating $H_P$.

analysis of PDEsclassical analysis and ODEsfunctional analysis

Audience: researchers in the topic

OARS Online Analysis Research Seminar

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