L^p Estimates for Maximal Averages Along One-variable Vector Fields in R^2

TL;DR

This paper proves a conjecture regarding L^p bounds for maximal averages along vector fields depending on one variable, establishing boundedness with explicit dependence on a small parameter delta.

Contribution

It confirms a conjecture by Lacey and Li for vector fields depending on a single variable, providing explicit bounds for the associated maximal averaging operator.

Findings

Maximal operator is bounded on L^p for p>1.

Operator norm depends on delta as (delta)^(-1).

Results apply to vector fields of the form v(x,y) = (1,u(x)).

Abstract

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 from [0,1] to [0,1]. Fix a small parameter delta and let Z 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 maximal averaging operator associated to the collection Z is bounded on L^p for p>1 with constants comparable to (delta)^(-1) .