Title: Maximal operators and Hilbert transforms along variable curves
Abstract: The poster will be on work in progress with Shaoming Guo and Victor Lie. Consider the maximal operator along a variable curve in $\mathbb{R}^2$,
\[ \mathcal{M}f(x) = \sup_{\varepsilon>0} \frac1{2\varepsilon}
\int_{-\varepsilon}^\varepsilon |f(x-\gamma_{x}(t))| dt \]
and the corresponding Hilbert transform,
\[ \mathcal{H}f(x) = p.v. \int_{\mathbb{R}} f(x-\gamma_{x}(t)) \frac{dt}t. \]
In the vector field case, $\gamma_{x}(t)=(t,u(x)t)$, it is a long-standing open question to determine whether Lipschitz regularity of $u$ suffices for these operators to satisfy weak $L^2$ bounds. At this time we can prove $L^p$, $1<p<\infty$ bounds for both $\mathcal{M}$ and $\mathcal{H}$ in the case $\gamma_{x}(t)=(t,u(x_1)|t|^\alpha)$ for measurable $u$ and real $\alpha>0$ not equal to $1$. The proofs are via single-annulus estimates in the spirit of Lacey-Li and Bateman-Thiele type square function estimates. Principal ingredients include $TT^*$ estimates, a Nagel-Stein-Wainger type iterated interpolation scheme and logarithmic vector-valued estimates for the shifted maximal function. In the case $\gamma_x(t)=(t,u(x_1)t+v(x_1)|t|^\alpha)$, with $u,v$ measurable and $\alpha>0$ real and not equal to $2$, we can apply a single-annulus estimate due to Bateman to prove certain single-annulus $L^p$ estimates for $\mathcal{H}$. In $L^2$, this implies a bound for a one-dimensional polynomial Carleson operator,
\[ f\longmapsto \sup_{u,v\in\mathbb{R}}\left|p.v.\int_{\mathbb{R}}
e^{iut + iv|t|^\alpha} f(x-t) \frac{dt}{t}\right|. \]