# Lipschitz linearization of the maximal hyperbolic cross multiplier

## TL;DR

This paper investigates the boundedness of a linearized maximal operator related to the hyperbolic cross multiplier in two dimensions, establishing $L^p$ bounds under Lipschitz and lower bound conditions.

## Contribution

It introduces new $L^p$ bounds for the linearized maximal operator of the hyperbolic cross multiplier under Lipschitz assumptions, extending previous results.

## Key findings

- Established $L^p$ bounds for the operator for all $1<p<
finite$
- Provided conditions under which the bounds hold, including Lipschitz and lower bounds
- Discussed related results and implications in harmonic analysis

## Abstract

We study the linearized maximal operator associated with dilates of the hyperbolic cross multiplier in dimension two. Assuming a Lipschitz condition and a lower bound on the linearizing function, we obtain $L^{p}(\mathbb{R}^{2}) \to L^{p}(\mathbb{R}^{2})$ bounds for all $1<p <\infty$. We discuss various related results.