# Square functions for bi-Lipschitz maps and directional operators

## TL;DR

This paper establishes new square function bounds for bi-Lipschitz perturbations and directional operators, providing insights into Hilbert transforms along curves and a novel proof related to direction fields.

## Contribution

It introduces a Littlewood-Paley diagonalization for bi-Lipschitz maps and derives new bounds for directional operators, advancing understanding of related harmonic analysis problems.

## Key findings

- Diagonalization result for bi-Lipschitz maps
- Square function bounds for directional operators
- Alternative proof of Katz's theorem on direction fields

## Abstract

First we prove a Littlewood-Paley diagonalization result for bi-Lipschitz perturbations of the identity map on the real line. This result entails a number of corollaries for the Hilbert transform along lines and monomial curves in the plane. Second, we prove a square function bound for a single scale directional operator. As a corollary we give a new proof of part of a theorem of Katz on direction fields with finitely many directions.