Maximal averages along a planar vector field depending on one variable

TL;DR

This paper establishes nearly optimal $L^2$ bounds for a maximal operator linked to a planar vector field that varies solely with the horizontal coordinate, combining previous insights with Katz's directional maximal operator results.

Contribution

It provides sharp $L^2$ estimates for a specific maximal operator related to vector fields depending on one variable, extending prior understanding in harmonic analysis.

Findings

Proves sharp $L^2$ bounds for the maximal operator.

Integrates earlier work on vector fields with Katz's directional maximal operator results.

Enhances understanding of maximal functions associated with variable-dependent vector fields.

Abstract

We prove (essentially) sharp $L^2$ estimates for a restricted maximal operator associated to a planar vector field that depends only on the horizontal variable. The proof combines an understanding of such vector fields from earlier work of the author with a result of Nets Katz on directional maximal operators.