Lp ESTIMATES FOR MAXIMAL AVERAGES ALONG ONE-VARIABLE VECTOR FIELDS IN R2

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] -> [0, 1]. Let delta be a small parameter, and let 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 [GRAPHICS] is bounded on L-p for p > 1 with constants comparable to 1/delta.