$L^p$ estimates for maximal averages along one-variable vector fields in ${\mathbf R} ^2$

We prove a conjecture of Lacey and Li in the case that the vector field depends only on one variable. Specifically: let $v$ be a vector field defined on the unit square such that $v(x,y) = (1,u(x))$ for some measurable $u: [0,1] \rightarrow [0,1]$. Let $\delta$ be a small parameter, and let $\mathcal R$ be the collection of rectangles $R$ of a fixed width such that $\delta$ much of the vector field inside $R$ is pointed in (approximately) the same direction as $R$. We show that the operator defined by

$M_{\mathcal R} f (z ) = \sup _{z\in R \in \mathcal R} {1 \over { \vert R\vert } } \int _{R} \vert f\vert$

(1)

is bounded on $L^p$ for $p>1$ with constants comparable to ${1 \over {\delta} }$.