Maximal operators and Hilbert transforms along variable non-flat homogeneous curves

TL;DR

This paper establishes $L^p$ boundedness for maximal operators and Hilbert transforms along variable non-flat homogeneous curves in the plane, using advanced harmonic analysis techniques.

Contribution

It proves boundedness results for these operators under minimal regularity assumptions on the variable function, extending classical results to more general curves.

Findings

Maximal operator is bounded on $L^p$ for $p>1$ with Lipschitz $u$.

Hilbert transform is bounded on $L^p$ for $p>1$ with measurable $u$ constant in second variable.

Uses stationary phase, $TT^*$, and local smoothing methods.

Abstract

We prove that the maximal operator associated with variable homogeneous planar curves $(t, u t^{\alpha})_{t\in \mathbb{R}}$, $\alpha\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\mathbb{R}^2 \to \mathbb{R}$ is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with $(t, ut^{\alpha})_{t\in \mathbb{R}}$, $\alpha\not=1$ positive, is bounded on $L^p(\mathbb{R}^2)$ for each $p>1$, under the assumption that $u:\mathbb{R}^2\to \mathbb{R}$ is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, $TT^*$ arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.