Title: Maximal operators associated to Hilbert transforms along variable parabolas in $\mathbb{R}^2$
Abstract: For $u \in (0,\infty)$, the Hilbert transform along the parabola $\{(t,ut^2) \colon t \in \mathbb{R}\}$ is defined by $$ H^u f(x) = \text{p.v.} \int_{\mathbb{R}} f(x_1 - t, x_2 - ut^2) \frac{dt}{t}. $$ For $U \subset (0,\infty)$ we study the maximal operator $$ f(x) \mapsto \sup_{u \in U} |H^u f(x)|. $$ Via a local smoothing estimate for certain Fourier integral operators and a Chang-Wilson-Wolff inequality for martingales, we show that for $p \in (2,\infty)$, the above maximal operator is bounded on $L^p(\mathbb{R}^2)$ if and only if $U$ can be covered by finitely many dyadic intervals. We will also briefly discuss the case when $p \in (1,2]$. This is joint work with Shaoming Guo, Joris Roos and Andreas Seeger.
Slides